The Bitcoin Backbone Protocol

blockchain

2019/08/04 发布于 研究 分类

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1. The Bitcoin Backbone Protocol: Analysis and Applications∗ Aggelos Kiayias† University of Athens aggelos@di.uoa.gr Juan A. Garay Yahoo Labs garay@yahoo-inc.com Nikos Leonardos† LIAFA, Université Paris Diderot–Paris 7 nikos.leonardos@gmail.com July 7, 2015 Abstract Bitcoin is the first and most popular decentralized cryptocurrency to date. In this work, we extract and analyze the core of the Bitcoin protocol, which we term the Bitcoin backbone, and prove two of its fundamental properties which we call common prefix and chain quality in the static setting where the number of players remains fixed. Our proofs hinge on appropriate and novel assumptions on the “hashing power” of the adversary relative to network synchronicity; we show our results to be tight under high synchronization. Next, we propose and analyze applications that can be built “on top” of the backbone protocol, specifically focusing on Byzantine agreement (BA) and on the notion of a public transaction ledger. Regarding BA, we observe that Nakamoto’s suggestion falls short of solving it, and present a simple alternative which works assuming that the adversary’s hashing power is bounded by 1/3. The public transaction ledger captures the essence of Bitcoin’s operation as a cryptocurrency, in the sense that it guarantees the liveness and persistence of committed transactions. Based on this notion we describe and analyze the Bitcoin system as well as a more elaborate BA protocol, proving them secure assuming high network synchronicity and that the adversary’s hashing power is strictly less than 1/2, while the adversarial bound needed for security decreases as the network desynchronizes. 1 Introduction Bitcoin, introduced in [29], is a decentralized payment system that is based on maintaining a public transaction ledger in a distributed manner. The ledger is maintained by anonymous participants (“players”) called miners, executing a protocol that maintains and extends a distributed data structure called the blockchain. The protocol requires from miners to solve a “proof of work” (POW, aka “cryptographic puzzle” — see, e.g., [16, 38, 4, 24]), which essentially amounts to brute-forcing a hash inequality based on SHA-256, in order to generate new blocks for the blockchain. The blocks that comprise the blockchain contain sets of transactions that are generated at will by owners of bitcoins, who issue transactions that credit any entity of their choice who accepts payments in bitcoin. Payers broadcast transactions and miners include the transactions they receive into the blocks ∗ † An abridged version of this paper appears in Proc. Eurocrypt 2015. Research partly supported by ERC project CODAMODA. 1
2. they generate. Miners are rewarded for maintaining the blockchain by receiving bitcoins; it is in this manner bitcoins are created and distributed among the miners who are the first recipients of newly minted bitcoins. An important concern in Bitcoin (or any e-payment system for that matter) is the prevention of double-spending attacks. Specifically, in the context of Bitcoin, a double-spending attack can occur when the attacker initially credits an account, receives service or goods by the account holder, but then manages to reorganize the transaction ledger so that the transaction that credits the account holder is reverted. In this way, the attacker keeps her bitcoin while receiving services and thus she is able to spend it again somewhere else. In [29], Nakamoto provides an initial set of arguments of why the Bitcoin system will prevent double-spending attacks. Specifically, he argues that if a payee waits for the transaction that gives her credit to advance into the blockchain a number of k blocks, then the probability that an attacker can build an alternative blockchain that “reorganizes” the public blockchain (which contains the credit transaction) drops exponentially with k. Nakamoto argues this by modeling the attacker and the set of honest players as two competing actors performing a random walk moving toward a single direction with probabilistic steps. He demonstrates that the k blocks the payee waits are enough to ensure a negligible (in k) probability of the attacker catching up with the honest players. Nevertheless, the above analysis can be easily seen to be oversimplified: in particular, it does not account for the fact that in Bitcoin’s decentralized setting the attacker may attempt to introduce disagreement between the honest miners, thus splitting their hashing power on different POW instances. Nakamoto himself appeared to recognize the relevance of agreement in the context of Bitcoin, arguing in a forum post [30] that actually “Bitcoin’s basic concept” of building and exchanging a blockchain is capable of solving Byzantine agreement (BA) [36, 27] in the presence of an actively malicious adversary.1 However a thorough analysis establishing the exact security properties of the Bitcoin system has yet to appear. Our results. In this paper we extract, formally describe, and analyze the core of the Bitcoin protocol. We call this protocol the Bitcoin backbone, as we describe it in a way that is versatile and extensible and can be used to solve other problems as well — not just the problem of maintaining a public transaction ledger. The Bitcoin backbone protocol is executed by players that build a blockchain following the Bitcoin source code [31] and allows a set of players to maintain a blockchain in a distributed fashion. The protocol is parameterized by three external functions V (·), I(·), R(·) which we call the input validation predicate, the input contribution function, and the chain reading function, respectively. At a high level, V (·) determines the proper structure of the information that is stored into the blockchain, I(·) specifies how the contents of the blocks are formed by the players, and R(·) determines how a blockchain is supposed to be interpreted in the context of the application. Note that the structure, contents, and interpretation of the blockchain are not important for the description of the backbone protocol and are left to be specified by the three external functions above, which are application-specific (we provide examples of these functions in Section 5). We analyze the Bitcoin backbone protocol in a static setting when the participants operate in a synchronous communication network (more details below and in Section 2) in the presence of an adversary that controls a subset of the players. We assume that the protocol is executed by a fixed number n of players; note, however, that this number is not necessarily known to the protocol participants. The players themselves cannot authenticate each other and therefore there is no way 1 In [30] Nakamoto refers to the problem as “Byzantine Generals,” which is often used to refer to the single-source version of the problem. Note that since more than one general may propose a time to attack this in fact is the case where every party has an input value, i.e., Byzantine agreement. In fact, in an anonymous setting such as Bitcoin’s, the single-source version is nonsensical. Note that in the cryptographic setting, the two problems are not equivalent in terms of the number of tolerated misbehaving parties t (t < n vs. t < n/2, respectively). 2
3. to know the source of a message; we capture this by allowing the adversary to “spoof” the source address of any message that is delivered. We assume that messages are eventually delivered and all parties in the network are able to synchronize in the course of a “round.” The notion of round is not important for the description of the backbone protocol (which can also be executed in a loose and asynchronous fashion in the same way that Bitcoin works), however, it is important in terms of Bitcoin’s inherent computational assumption regarding the players’ ability to produce POWs. Specifically, we assume that in a single round, all parties involved are allowed the same number of queries to a cryptographic hash function, as well as to communicate with the other participants. The hash function is modeled as a random oracle [6]. For simplicity we assume a “flat model,” where all parties have the same quota of hashing queries per round, say q; the non-flat model where parties have differing hashing power capabilities can be easily captured by clustering the flat-model parties into larger virtual entities that are comprised by more than one flat-model player. In fact “mining pools” in Bitcoin can be thought of such aggregations of flat-model players. The adversary itself represents such pool as it controls t < n players; for this reason, the adversary’s quota per round is t · q hashing queries. Note that in this setting, the fact t < n/2 directly corresponds to the adversary controlling strictly less than half of the system’s total “hashing power” that all players collectively harness, thus, we will use terms such as “honest majority” and “(1/2)-bounded adversary” interchangeably. In our analysis of the Bitcoin backbone protocol we formalize and prove two fundamental properties it possesses. The properties are quantified by three parameters γ, β and f ; γ and β roughly correspond to the collective hashing power per round of the honest players and the adversary, respectively, while f represents the expected number of POWs that may be found per round by the Bitcoin network participants as a whole. The common prefix property. We prove that if γ > λβ for some λ ∈ [1, ∞) that satisfies λ2 − f λ + 1 ≥ 0, then the blockchains maintained by the honest players will possess a large common prefix. More specifically, if two honest parties “prune” (i.e., cut off) k blocks from the end of their local chains, the probability that the resulting pruned chains will not be mutual prefixes of each other drops exponentially in k (see Definition 2 for the precise formulation). Provided that f is very close to 0 this enables us to choose λ very close to 1 and thus establish the common prefix property as long as an honest majority of participants in the flat-model setting is guaranteed (equivalently, when the adversary controls strictly less than 50% of the hashing power). On the other hand, when the network “desynchronizes” and f gets closer to 1, achieving a common prefix requires λ → φ, where φ is the golden ratio, which in turn suggests much stricter bounds on the adversarial behavior (in fact, the upper bound on the adversary for our analysis approaches 0). The chain-quality property. We prove that if γ > λβ, for some λ ∈ [1, ∞), then the ratio of blocks in the chain of any honest player that are contributed by honest players is at least (1− λ1 ). Again observe that if λ is close to 1, we obtain that the blockchain maintained by honest players is guaranteed to have few, but still some, blocks contributed by honest players; a higher λ would be necessary to guarantee bigger percentages of blocks contributed by honest players in the blockchain. We also observe that this result is basically tight, i.e., that the adversary is capable of following a strategy (that deviates from the strategy of honest players) that enables the introduction of that many blocks in the blockchain, under a favorable (for the adversary) assumption on the propagation of adversarial blocks in the network. While the above two security properties may seem rather abstract since they refer to properties of the data structure that is maintained distributively by the parties, we demonstrate that they are in fact quite powerful and show that the Bitcoin backbone protocol armed with the above 3
4. Figure 1: An overview of the backbone protocol’s applications: Nakamoto’s BA protocol Πnak BA , our 1/3 1/2 BA protocols ΠBA and ΠBA , and the public ledger protocol ΠPL . All properties must be satisfied with overwhelming probability. In each box we state the name of the property as well as the maximum ratio of the adversarial hashing power that we can prove the protocol withstands (based on the corresponding backbone property). The value  stands for a negligible quantity. properties can be used as a basis for solving other problems, including the problem of distributively maintaining a “robust” public transaction ledger. In Figure 1 we show how the two properties imply the properties of the applications that are explained below. Byzantine agreement for (1/3)-bounded adversaries. As a first application, we show how a randomized BA protocol can be built on top of the Bitcoin backbone protocol more or less directly, and based solely on the POW assumption. We instantiate the V (·), I(·), R(·) functions so that parties form blockchains and act according to the following rules:'>rules: each party i attempts to insert its own input vi ∈ {0, 1} into the blockchain; a blockchain is valid only if blocks contain elements in {0, 1}; the protocol terminates when the blockchain has reached a sufficient length; and, the blockchain is read by the honest parties by pruning k elements from its end and returning the majority bit appearing in the resulting blockchain’s prefix. We show how the common prefix property and the chain-quality property of the backbone protocol ensure Agreement and Validity (BA’s basic properties; see Section 2) with high probability, thus turning the Bitcoin backbone protocol into a probabilistic BA protocol. Observe that for the above protocol to work the chain-quality property should ensure that a majority of blocks in the blockchain originate from the honest players (otherwise Validity is lost). Our chain quality property enables this with overwhelming probability assuming the adversarial power is bounded by 1/3. This approach is different from Nakamoto’s proposal [30] for BA, which, as we also show, only guarantees Validity with overwhelming probability if the adversary has a negligible amount of hashing power. On the positive side, we stress that Nakamoto’s protocol fails gracefully when the adversarial power gets close to 50% as Validity can be shown with constant probability (but not overwhelming). Public transaction ledgers and BA for honest majority. Next, we focus on how a “robust public transaction ledger” can be built on top of the Bitcoin backbone. We instantiate the V (·), I(·), R(·) functions so that parties form blockchains and act according to the following rules:'>rules: each party (which in this context is called a “miner”) receives a set S of transactions on its input tape and attempts to insert those in its blockchain, omitting any transactions in S that are already included in it. (A Bitcoin transaction is, for example, a statement of the type “account A credits account 4
5. B a z number of bitcoins,” which is signed using the secret key that corresponds to account A’s Bitcoin address; each account has a unique Bitcoin address.) Reading a blockchain, on the other hand, amounts to returning the total sequence of transactions that is contained in the blockchain of the miner (and note that miners may disagree about the chain they report). We show how the common prefix property and the chain-quality property ensure two properties needed by the ledger, which we call Persistence and Liveness, assuming an honest majority and arbitrary adversarial behavior. Persistence states that once a transaction goes more than k blocks “deep” into the blockchain of one honest player, then it will be included in every honest player’s blockchain with overwhelming probability, and it will be assigned a permanent position in the ledger. On the other hand, Liveness says that all transactions originating from honest account holders will eventually end up at a depth more than k blocks in an honest player’s blockchain, and hence the adversary cannot perform a selective denial of service attack against honest account holders. For both properties to hold we require an honest majority (i.e., that the adversary’s hashing power is strictly less than 50%) assuming high network synchronicity (i.e., that the expected number of POW solutions per round satisfies2 f → 0). If this is violated, Persistence requires stricter bounds on adversarial hashing power in order to be preserved following the bounds of the common prefix property. In the context of Bitcoin, our analysis implies that the Bitcoin backbone provides an operational transaction ledger under the assumptions: (i) the adversary controls less than half of the total hashing power, and (ii) the network synchronizes much faster relative to the POW solution rate, (iii) digital signatures cannot be forged. On the other hand, when the network desynchronizes our results cannot support that the ledger is maintained by assuming an honest majority. This negative result is consistent with the experimental analysis provided by Decker and Wattenhoffer [15], who predicted a drop below 50% in the required adversarial bound for any setting when information propagation is problematic. Our result also provides some justification for the “slow” rate of 10minute increments used in Bitcoin block generation. Specifically, information propagation in the Bitcoin network is on the order of seconds3 so the ratio (essentially f ) of this time window over the average 10-minute period is reasonably close to “small” and thus transaction persistence can be shown for roughly an honest majority. On the other hand, cryptocurrencies including Litecoin, Primecoin and others, reacting to the demand to offer faster transaction processing, opted for a faster response rate (some as small as 1 minute), which results in more precarious situations, e.g., f > 0.1, which is far from being “negligible” and thus cannot support our analysis that a common prefix would be guaranteed by merely assuming an honest majority. We finally note that the Persistence and Liveness properties we put forth and prove should not be interpreted as proofs that all Bitcoin’s objectives are met. In particular, they do not guarantee that miners are properly incentivized to carry out the backbone protocol, and they can only offer guarantees in a setting of an honest majority amongst a fixed number of players as opposed to a setting where there is an ever changing population of parties acting rationally; see related work below as well as Section 7 for further discussion. Finally, we present a BA protocol assuming an honest majority, by suitably exploiting the properties of the robust transaction ledger above. The protocol substitutes Bitcoin’s transactions with a type of transactions that are themselves based on POWs, and hence uses POWs in two distinct ways: for the maintenance of the ledger and for the generation of the transactions. We show that the ledger’s Persistence implies Agreement, and that Liveness implies Validity, because assuming the ledger is maintained for long enough, a majority of transactions originating from 2 3 Note that we use the notation f → 0 to mean that “f is close to 0” since f will be a constant in our analysis. See, for example, http://bitcoinstats.com/network/propagation/. 5
6. the honest parties will be included (despite the fact that honest parties may control a minority of blocks in the blockchain). The protocol requires special care in the way it employs POWs since the adversary should be incapable of “shifting” work between the two POW tasks that it faces in each round. To solve this problem, we introduce a special strategy for POW-based protocol composition which we call “2-for-1 POWs.” Related work. Realizing a digital currency with a centralized entity but while achieving strong privacy was proposed early on by Chaum in [13]. A number of other works improved various aspects of this concept, however the approach remained centralized. Nakamoto [29] proposed the first decentralized currency system based on POWs while relaxing the anonymity property of the payment system to mere pseudonymity. This work was followed by a multitude of other related proposals including Litecoin4 , Primecoin [26], and Zerocash [8], to mention a few. Our analysis of the Bitcoin backbone covers all these works as well, since they are based on exactly the same protocol. It is interesting to juxtapose our positive results to the results of Eyal and Sirer [17], who introduce an attack strategy called “selfish mining” that shows how the number of blocks contributed to the blockchain by an adversary can exceed the percentage of the hashing power the adversary possesses. Their results are consistent and complementary to ours. The crux of the issue is (in our terminology) in terms of the chain-quality property, as its formulation is quite permissive: in particular we show that if the adversary controls a suitably bounded amount of hashing power, then it is also suitably bounded in terms of the number of blocks it has managed to insert in the blockchain that honest players maintain. Specifically, recall that we prove that if the hashing power of the adversary satisfies β < λ1 γ (where γ roughly corresponds to the hashing power of the honest players), then the adversary may control at most a λ1 percentage of the blocks in the chain. For instance, if the adversary controls up to 1/3 of the hashing power (i.e., λ = 2), then it will provably control less than 50% of the blocks in the honest players’ blockchain. As it can be easily seen, this does not guarantee that the rate of a party’s hashing power translates to an equal rate of rewards (recall that in Bitcoin the rewards are linearly proportional to the number of blocks that a party contributes in the chain). We define as ideal chain quality the property that for any coalition of parties (following any mining strategy) the percentage of blocks in the blockchain is exactly proportional to their collective hashing power. The chain quality property that we prove is not ideal and the results of [17] show that in fact there is a strategy that magnifies the percentage of a malicious coalition. Still, their mining attack does much worse than our bound. To close the gap, we sketch (cf. Remark 3) a simple selfish mining strategy that matches our upper bound and hence our chain quality result is tight in our model5 assuming the number of honest parties is large. Byzantine agreement (BA, aka distributed consensus) [36, 27] considers a set of n parties connected by reliable and authenticated pair-wise communication links and with possible conflicting initial inputs that wish to agree on a common output in the presence of the disruptive (even malicious) behavior of some of them. The problem has received a considerable amount of attention under various models. In this paper we are interested in randomized solutions to the problem (e.g., [7, 37, 11, 18, 20, 25])6 as in the particular setting we are in, deterministic BA algorithms are not possible. In more detail, we consider BA in the anonymous synchronous setting, i.e., when pro4 http://www.litecoin.com. Our model allows the unfavorable event of adversarial messages winning all head-to-head races in terms of delivery with honestly generated messages in any given round. 6 We remark that, in contrast to the approach used in typical randomized solutions to the problem, where achieving BA is reduced to (the construction of) a shared random coin, the probabilistic aspect here stems from the parties’ likelihood of being able to provide proofs of work. In addition, as our analysis relies on the random oracle model [6], we are interested in computational/cryptographic solutions to the problem. 5 6
7. cessors do not have identifiers and cannot correlate messages to their sources, even across rounds. This model for BA was considered by Okun, who classified it as “anonymous model without port awareness,” and proved the aforementioned impossibility result, that deterministic algorithms are impossible for even a single failure [34, 33]. In addition, Okun showed that probabilistic BA is feasible by suitably adapting Ben-Or’s protocol [7] for the standard, non-anonymous setting (cf. [34])7 ; the protocol, however, takes exponentially many rounds. It turns out that by additionally assuming that the parties are “port-aware” (i.e., they can correlate messages to sources across rounds), deterministic protocols are possible and some more efficient solutions were proposed in [35]. The anonymous synchronous setting was also considered by Aspnes et al. [2] who pointed to the potential usefulness of proofs of work (e.g., [16, 38, 4, 24]) as an identity assignment tool, in such a way that the number of identities assigned to the honest and adversarial parties can be made proportional to their aggregate computational power, respectively. For example, by assuming that the adversary’s computational power is less than 50%, one of the algorithms in [2] results in a number of adversarial identities less than half of that obtained by the honest parties. By running this procedure in a pre-processing stage, it is then suggested that a standard authenticated BA protocol could be run. Such protocols, however, would require the establishment of a consistent PKI (as well as of digital signatures), details of which are not laid out in [2]. In contrast, and as mentioned above, building on our analysis of the Bitcoin backbone protocol, we propose two BA protocols solely based on POWs that operate in O(k) rounds with error probability e−Ω(k) . The protocols solve BA with overwhelming probability under the assumption that the adversary controls less than 1/3 and 1/2 of the computational power, respectively. The connection between Bitcoin and probabilistic BA was also considered by Miller and LaViola in [28] where they take a different approach compared to ours, by not formalizing how Bitcoin works, but rather only focusing on Nakamoto’s suggestion for BA [30] as a standalone protocol. As we observe here, and also recognized in [28], Nakamoto’s protocol does not quite solve BA since it does not satisfy Validity with overwhelming probability. The exact repercussions of this fact are left open in [28], while with our analysis, we provide explicit answers regarding the transaction ledger’s actual properties and the level of security that the backbone realization can offer. Finally, related to the anonymous setting, the feasibility of secure computation without authenticated links was considered by Barak et al. in [5] in a more extreme model where all messages sent by the parties are controlled by the adversary and can be tampered with and modified (i.e., not only source addresses can be “spoofed,” but also messages’ contents can be altered and messages may not be delivered). It is shown in [5] that it is possible to limit the adversary so that all he can do is to partition the network into disjoint sets, where in each set the computation is secure, and also independent of the computation in the other sets. Evidently, in such system, one cannot hope to build a global ledger. Organization of the paper. The rest of the paper is organized as follows. In Section 2 we present our model within which we formally express the Bitcoin backbone protocol and prove its basic properties. The backbone protocol builds “blockchains” based on a cryptographic hash function; we introduce notation for this data structure as well as the backbone protocol itself in Section 3, followed by its analysis in Section 4. Sections 5 and 6 are dedicated to the applications built on top of the backbone protocol — (simple) BA protocols and robust transaction ledger, respectively. Specifically, Section 5 covers Nakamoto’s (insufficient) suggestion for BA as well as our solution for 1/3 adversarial power, while in Section 6 we present our treatment of a robust public ledger formalizing the properties of Persistence and Liveness and how they apply to Bitcoin. Finally, we also include in this section our BA protocol for 1/2 adversarial power. Some directions for future 7 Hence, BA in this setting shares a similar profile with BA in the asynchronous setting [19]. 7
8. research are offered in Section 7. 2 Model and Definitions We describe our protocols in a standard multiparty synchronous communication setting (e.g., Canetti’s formulation of “real world” execution [12]) with the relaxation that the underlying communication graph is not fully connected and messages are delivered through a “diffusion” mechanism that reflects Bitcoin’s peer-to-peer structure. Our adversarial model in the network is “adaptive,” meaning that the adversary is allowed to take control of parties on the fly, and “rushing,” meaning that in any given round the adversary gets to see all honest players’ messages before deciding his strategy, and, furthermore, also allows the adversary to change the source information on every message. Note that the adversary cannot change the contents of the messages nor prevent them from being delivered. Effectively, this parallels communication over TCP/IP in the Internet where messages between parties are delivered reliably, but nevertheless malicious parties may “spoof” the source of a message they transmit and make it appear as originating from an arbitrary party (including another honest party) in the view of the receiver. This aspect of the communication model, where processors cannot correlate messages to their sources, even across arounds, was considered by Okun [33], who classified it as “anonymous model without port awareness.” In this setting we use Broadcast as the message transmission command that captures the “send-to-all” functionality allowed by our communication model. Note that an adversarial sender may abuse Broadcast and attempt to confuse honest parties by sending and delivering inconsistent messages to them. The parties’ inputs are provided by the environment Z which also receives the parties’ outputs. Parties that receive no input from the environment remain inactive, in the sense that they will not act when their turn comes in each round. The environment may provide input to a party at any round and may also modify that input from round to round. We denote by Input() the input tape of each party. In each round, parties are able to read their input tape Input() and communication tape Receive(), perform some computation that will be suitably restricted (see below) and issue8 a Broadcast message that is guaranteed to be delivered to all parties in the beginning of the next round. As stated above the adversary can do multiple broadcasts per round and in fact deliver to each honest party a different message or even multiple messages. The term {viewPΠ,A,Z (κ, z)}κ∈N,z∈{0,1}∗ denotes the random variable ensemble describing the view of party P after the completion of an execution with environment Z, running protocol Π, and adversary A, on auxiliary input z ∈ {0, 1}∗ . We often drop the parameters κ and z and simply refer to the ensemble by viewPΠ,A,Z if the meaning is clear from the context. If n parties i P1 , . . . , Pn execute Π, the concatenation of the view of all parties hviewPΠ,A,Z ii=1,...,n is denoted by viewΠ,A,Z . With foresight, we note that, in contrast to the standard setting where parties are aware of the number of parties executing the protocol, we are interested in protocols Π that do not make explicit use of the number of parties n or their identities. Further, note that because of the unauthenticated nature of the communication model the parties may never be certain about the number of participants in a protocol execution. Nonetheless note that the number of parties is fixed during the course of the protocol execution. In order to capture the parties’ limited ability to produce POWs, we assume that all parties may have access to an oracle H(·) and allowed to perform a number of queries q per round, where q is a function of the security parameter κ; we refer to such parties as q-bounded. Note that this 8 For simplicity, we assume that the broadcast operation is atomic and hence the corruption of a party may not happen while the operation is taking place (cf. [23, 21]). 8
9. is a “flat-model” interpretation of the parties’ computation power, where all parties are assumed equal. In the real world, different honest parties may have different “hashing power;” nevertheless, our flat-model does not sacrifice generality since one can imagine that real honest parties are simply clusters of some arbitrary number of honest flat-model parties. The adversary A is allowed to perform t · q queries per round, where t ≤ n is the number of corrupted parties. The environment Z, on the other hand, is not permitted any queries to H(·). The rationale for this is that we would like to bound the “CPU power” [29] of the adversary to be proportionate to the number of parties it controls while making it infeasible for them to be aided by external sources or by transferring the hashing power potentially invested in concurrent or previous protocol executions. It follows that in our analysis we will focus on the “standalone” setting, where a single protocol instance is executed in isolation. We refer to the above restrictions on the environment, the parties and the adversary as the q-bounded synchronous setting. The view of the parties participating in the protocol will be denoted P,H(·) H(·) by viewΠ,A,Z (κ, q, z) and the concatenation of all parties’ views by viewΠ,A,Z (κ, q, z). In our theorems we will be concerned with properties of protocols Π in the q-bounded synchronous H(·) setting. Such properties will be defined as predicates over the random variable viewΠ,A,Z (κ, q, z) by quantifying over all possible adversaries A and environments Z. Note that all our protocols will only satisfy properties with a small probability of error in κ as well as in a parameter k that can be freely selected in {1, . . . , κ}. The probability space is determined by the oracle H(·) as well as any random choices made by the protocol itself (if any). Further details about the model are given in Appendix A. Byzantine agreement. As a simple illustration of the formulation above we define the properties of a Byzantine agreement (BA) protocol. Definition 1. A protocol Π solves BA in the q-bounded synchronous setting provided it satisfies the following two properties: Agreement. There is a round after which all honest parties return the same output if queried by the environment. Validity. The output returned by an honest party P equals the input of some party P 0 that is honest at the round P ’s output is produced. We note that in our protocols, the participants are capable of detecting agreement and furthermore they can also detect whether other parties detect agreement, thus termination can be easily achieved by all honest parties. The formulation of Validity above is intended to capture security/correctness against adaptive adversaries. The notion (specifically, the requirement that the output value be one of the honest parties’ inputs) has also been called “Strong Validity” [32], but the distinction is only important in the case of non-binary inputs. In either case, it is known that in the synchronous cryptographic setting the problem has a solution if and only if n > V t, where V is the input/decision domain [20]. Our POW-based protocols work for both versions of the problem. 3 The Bitcoin Backbone Protocol We start by introducing blockchain notation. Let G(·), H(·) be cryptographic hash functions with output in {0, 1}κ . A block is any triple of the form B = hs, x, ctri where s ∈ {0, 1}κ , x ∈ {0, 1}∗ , ctr ∈ N are such that satisfy predicate validblockD q (B) defined as (H(ctr, G(s, x)) < D) ∧ (ctr ≤ q). 9
10. The parameter D ∈ N is also called the block’s difficulty level. The parameter q ∈ N is a bound that in the Bitcoin implementation determines the size of the register ctr; in our treatment we allow this to be arbitrary, and use it to denote the maximum allowed number of hash queries in a round. We do this for convenience and our analysis applies in a straightforward manner to the case that ctr is restricted to the range 0 ≤ ctr < 232 and q is independent of ctr. A blockchain, or simply a chain is a sequence of blocks. The rightmost block is the head of the chain, denoted head(C). Note that the empty string ε is also a chain; by convention we set head(ε) = ε. A chain C with head(C) = hs0 , x0 , ctr0 i can be extended to a longer chain by appending a valid block B = hs, x, ctri that satisfies s = H(ctr0 , G(s0 , x0 )). In case C = ε, by convention any valid block of the form hs, x, ctri may extend it. In either case we have an extended chain Cnew = CB that satisfies head(Cnew ) = B. The length of a chain len(C) is its number of blocks. Given a chain C that has length len(C) = n > 0 we can define a vector xC = hx1 , . . . , xn i that contains all the x-values that are stored in the chain such that xi is the value of the i-th block. Consider a chain C of length m and any nonnegative integer k. We denote by C dk the chain resulting from the “pruning” the k rightmost blocks. Note that for k ≥ len(C), C dk = ε. If C1 is a prefix of C2 we write C1  C2 . We note that Bitcoin uses chains of variable difficulty, i.e., the value D may change across different blocks within the same chain according to some rule that is determined by the x values stored in the chain9 . This is done to account for the fact that the number of parties (and hence the total hashing power of the system) is variable from round to round (as opposed to the unknown but fixed number of parties n we assume). See Section 7 for further discussion. We are now ready to describe the protocol. 3.1 The backbone protocol The Bitcoin backbone protocol is executed by an arbitrary number of parties over an unauthenticated network. For concreteness, we assume that the number of parties running the protocol is n; however, parties need not be aware of this number when they execute the protocol. As mentioned in Section 2, communication over the network is achieved by utilizing a send-to-all Broadcast functionality that is available to all parties (and maybe abused by the adversary in the sense of delivering different messages to different parties). Each party maintains a blockchain, as defined above. Each party’s chain may be different, but, as we will prove, under certain well-defined conditions, the chains of honest parties will share a large common prefix. (Figure 2 depicts the local view of each party as well as the shared portion of their chains.) In the protocol description we intentionally avoid specifying the type of values that parties try to insert in the chain, the type of chain validation they perform (beyond checking for its structural properties with respect to the hash functions G(·), H(·)), and the way they interpret the chain. These functions are handled by the external functions V (·), I(·), R(·) which are specified by the application that runs “on top” of the backbone protocol. The Bitcoin backbone protocol is specified as Algorithm 4. Before describing it in detail we first introduce the protocol’s three supporting algorithms. Chain validation. The first algorithm, called validate performs a validation of the structural properties of a given chain C. It is given as input the values q and D, as well as a hash function H(·). It is parameterized by a predicate V (·), called the input validation predicate. For each block of 9 In Bitcoin every 2016 blocks the difficulty is recalibrated according to the time-stamps stored in the blocks so that the block generation rate remains at approximately 10 minutes per block. 10
11. Figure 2: Overview of the basic operation of the Bitcoin backbone protocol. Miner M1 receives from the environment a Read instruction that results in the application of the R(·) function on the contents of its chain which are equal to the vector hx1 , x2 , x3 , x4 , x5 i. Miner M2 receives from the environment an Insert instruction and uses the function I(·) to determine the value y5 that it subsequently successfully inserts in its local block chain by solving a proof of work; this results in a broadcast of the newly extended chain. Finally miner M3 receives the newly extended chain and validates it both structurally as well as using the input validation predicate V (·). M3 will adopt this chain if M3 deems it better than its local chain as specified by the backbone protocol. Note that the joint view of M1 , M2 , M3 is inconsistent but there is agreement on the prefix hx1 , x2 , x3 i. the chain, the algorithm checks that the proof of work is properly solved, that the counter ctr does not exceed q and that the hash of the previous block is properly included in the block. It further collects all the inputs from the chain’s blocks and assembles them into a vector xC . If all blocks verify and V (xC ) is true then the chain is valid; otherwise it is rejected. Note that we purposely leave the predicate V (·) undetermined. 11
12. Algorithm 1 The chain validation predicate, parameterized by q, D, the hash functions G(·), H(·), and the input validation predicate V (·). The input is C. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: function validate(C) b ← V (xC ) ∧ (C 6= ε) if b = True then . The chain is non-empty and meaningful w.r.t. V (·) hs, x, ctri ← head(C) s0 ← H(ctr, G(s, x)) repeat hs, x, ctri ← head(C) 0 if validblockD q (hs, x, ctri) ∧ (H(ctr, G(s, x)) = s ) then 0 s ←s . Retain hash value C ← C d1 . Remove the head from C else b ← False end if until (C = ε) ∨ (b = False) end if return (b) end function Chain comparison. The objective of the second algorithm, called maxvalid, is to find the “best possible” chain when given a set of chains. The algorithm is straightforward and is parameterized by a max(·) function that applies some ordering in the space of chains. The most important aspect is the chains’ length, in which case max(C1 , C2 ) will return the longest of the two. In case len(C1 ) = len(C2 ), some other characteristic can be used to break the tie. In our case, max(·, ·) will always return the first operand10 ; alternatively, other options exist, such as lexicographic order or picking a chain at random. The analysis we will perform will essentially be independent of the tie-breaking rule11 . Algorithm 2 The function that finds the “best” chain, parameterized by function max(·). The input is {C1 , . . . , Ck }. 1: 2: 3: 4: 5: 6: 7: 8: 9: function maxvalid(C1 , . . . , Ck ) temp ← ε for i = 1 to k do if validate(Ci ) then temp ← max(C, temp) end if end for return temp end function 10 Note that the way we deploy maxvalid, amounts to parties always giving preference to their local chain as opposed to any incoming chain. This is consistent with current Bitcoin operation; however, some debate about alternate tiebreaking rules has ensued in Bitcoin forums, e.g., see [14]. 11 It is worth to point out that the behavior of maxvalid(·) is associated with some stability aspects of the backbone protocol and currently there are proposals to modify it (e.g., by randomizing it — cf. [17]). It is an interesting question whether any improvement in our results can be achieved by randomizing the maxvalid operation. 12
13. Algorithm 3 The proof of work function, parameterized by q, D and hash functions H(·), G(·). The input is (x, C). 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: function pow(x, C) if C = ε then s←0 else hs0 , x0 , ctr0 i ← head(C) s ← H(ctr0 , G(s0 , x0 )) end if ctr ← 1 B←ε h ← G(s, x) while (ctr ≤ q) do if (H(ctr, h) < D) then B ← hs, x, ctri break end if ctr ← ctr + 1 end while C ← CB return C end function . Determine proof of work instance . Proof of work succeeded . Extend chain Proof of work. The third algorithm, called pow, is the main “workhorse” of the backbone protocol. It takes as input a chain and attempts to extend it via solving a proof of work. This algorithm is parameterized by two hash functions H(·), G(·) (which in our analysis will be modeled as random oracles)12 , as well as two positive integers q, D; q represents the number of times the algorithm is going to attempt to brute-force the hash function inequality that determines the POW instance, and D determines the “difficulty” of the POW. The algorithm works as follows. Given a chain C and a value x to be inserted in the chain, it hashes these values to obtain h and initializes a counter ctr. Subsequently, it increments ctr and checks to see whether H(ctr, h) ≤ D; if a suitable ctr is found then the algorithm succeeds in solving the POW and extends chain C by one block inserting x as well as ctr (which serves as the POW). If no suitable ctr is found, the algorithm simply returns the chain unaltered. (See Algorithm 3.) The backbone protocol. Given the three algorithms above, we are now ready to describe the Bitcoin backbone protocol. This is the protocol that is executed by the miners and which is assumed to run “indefinitely” (our security analysis will apply when the total running time is polynomial in κ). It is parameterized by two functions, the input contribution function I(·) and the chain reading function R(·), which is applied to the values stored in the chain. Each miner starts a round with a local chain C (we say that the miner has chain C at this round) and checks its communication tape Receive() to see whether a “better” chain has been received and in such case it adopts it resulting in chain C˜ (we say that the miner adopts chain C˜ at this ˜ round). Choosing the chain C˜ is done using the maxvalid function; note that it could be that C = C. ˜ Then, the miner attempts to extend C by running the POW algorithm pow described above. 12 In reality the same hash function (SHA-256) instantiates both G and H; however, it is notationally more convenient to consider them as distinct. 13
14. The value that the miner attempts to insert in the chain is determined by function I(·). The input to I(·) is the state st, the current chain C, the contents of the miner’s input tape Input() (recall that they can be written by the environment Z at the beginning of any round) and communication tape Receive(), as well as the current round number round. The protocol expects two types of entries in the input tape, Read and (Insert, value); other inputs are ignored. We purposely leave the functions I(·), R(·) undetermined in the description of the backbone protocol, as their specifics will vary according to the application. One may choose, for example, I(·) to be as simple as copying the contents of the Insert input symbols from Input() into x and keeping st = , or performing a complex operation parsing C and maintaining old inputs in st. We provide explicit examples of I(·) and R(·) in Section 5. When the input x is determined, the protocol attempts to insert it into the chain C by invoking pow. In case the local chain C is modified during the above steps, the protocol transmits (“broadcasts”) the new chain to the other parties. Finally, in case a Read symbol is present in the communication tape, the protocol applies function R(·) to its current chain and writes the result onto the output tape Output(). This way, the round ends and a new round begins, continuing indefinitely. Algorithm 4 The Bitcoin backbone protocol, parameterized by the input contribution function I(·) and the chain reading function R(·). 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 3.2 C←ε st ← ε round ← 0 while True do C˜ ← maxvalid(C, any chain C 0 found in Receive()) ˜ round, Input(), Receive()) hst, xi ← I(st, C, ˜ Cnew ← pow(x, C) if C 6= Cnew then C ← Cnew Broadcast(C) end if round ← round + 1 if Input() contains Read then write R(xC ) to Output() end if end while . Determine the x-value. (Desired) Properties of the backbone protocol We next define the two main properties of the backbone protocol that we will prove. The first property is called the common prefix property and is parameterized by a value k ∈ N. It considers an arbitrary environment and adversary in the q-bounded setting, and it holds as long as any two honest parties’ chains are different only in its most recent k blocks. Definition 2 (Common Prefix Property). The common prefix property Qcp with parameter k ∈ N H(·) states that for any pair of honest players P1 , P2 maintaining the chains C1 , C2 in viewΠ,A,Z (κ, q, z), it holds that dk dk C1  C2 and C2  C1 . 14
15. The second property, which we call the chain quality property, aims at expressing the number of honest-player contributions that are contained in a sufficiently long and continuous part of an honest player’s chain. Specifically, for parameters k ∈ N and µ ∈ (0, 1), the rate of adversarial input contributions in a continuous part of an honest party’s chain is bounded by µ. This is intended to capture that at any moment that an honest player looks at a sufficiently long part of its blockchain, that part will be of sufficient “quality,” i.e., the number of adversarial blocks present in that portion of the chain will be suitably bounded. Definition 3 (Chain Quality Property). The chain quality property Qcq with parameters µ ∈ R H(·) and ` ∈ N states that for any honest party P with chain C in viewΠ,A,Z (κ, q, z), it holds that for any ` consecutive blocks of C the ratio of adversarial blocks is at most µ. It is easy to see that any set of, say, h honest parties, obtain as many blocks as their proportion of the total hashing power, i.e., h/n. We say that a protocol Π satisfies ideal chain quality if this is the case for adversarial parties as well, i.e., µ = t/n with respect to those parties. The ideal chain quality is not achieved by the Bitcoin backbone protocol, cf. Remark 3. 4 Analysis of the Bitcoin Backbone We now proceed to the analysis of the protocol presented in the previous section. Let {0, 1}κ be the range of H(·). Each party tries to provide a POW by issuing queries to H(·), which succeed with probability p = D/2κ , where D is the difficulty level. By the properties of the random oracle H(·), any collection of queries will be treated as a collection of independent Bernoulli trials with success probability p. In order to support this we will assume that the function I(·) (which determines the input of the players that is to be inserted in the blockchain) ensures (at least with overwhelming probability) that the inputs are unique. There are two simple ways to enforce this: either have I(·) add a sufficiently long random nonce to x, or, in case parties have unique identities, it may be parameterized by it and introduce it as part of x. In either case, this value will be ignored by the other functions V (·), R(·) as it need not be useful in the application. It is easy to see that if a κ-long nonce is used the output will be unique except for probability at most q 2 · 2−κ where q is the total number of queries submitted to the random oracle; we will ignore this small term in our analysis. 4.1 Definitions and preliminary lemmas Recall that n is the number of parties, t of which can be corrupted by the adversary. We introduce the following parameters for notational convenience: α = pq(n − t), γ = α − α2 , β = pqt, f = α + β. The first parameter, α, reflects the hashing power of the honest parties. It is an upper bound on the expected number of solutions that the honest parties compute in one round. Similarly, β, is the expected number of solutions that the corrupted parties compute in one round. Notice the asymmetry that while the honest parties will not compute more than one solution per round, a corrupted party may use all its q queries and potentially compute more than one solution. The parameter γ will serve as a lower bound on the following two probabilities. The first one is that at least one honest party computes a solution in a round: 1 − (1 − p)q(n−t) ≥ 1 − e−α ≥ γ; 15
16. we will call such round a successful round. The second one is the probability that exactly one honest party does so; we will call such round a uniquely successful round. We lower bound the probability of such a round by the probability that out of q(n − t) coin tosses exactly one comes up heads. Thus, the probability is at least: (n − t)qp(1 − p)q(n−t)−1 ≥ α(1 − α + p) ≥ γ. The ratio α/β = (n − t)/t will be of interest for the analysis. When α is small (as it will be when f is small), then γ ≈ α and we will be justified to concentrate on the ratio γ/β. To understand how well γ estimates the probability of a uniquely successful round, call it γ 0 , we observe the following upper bound: γ 0 = (n − t)(1 − (1 − p)q )(1 − p)q(n−t−1) ≤ (n − t)pqe−α+pq ≤ α(1 − α + pq + (α − pq)2 /2) = α − α2 (1 − 1 n−t ) + α3 2 (1 − 1 2 n−t ) , where we use Facts 1 and 2 (see Appendix B). From this it follows that γ 0 ≤ α − α2 + α3 /2 + O(1/(n − t)). The following definition will be crucial in the analysis of the common-prefix property. Definition 4 (Uniform rounds). We call a round uniform if, at that round, every honest party ˜ at line 7 of Algorithm 4 invokes the pow(·) algorithm with a chain of the same length (i.e., len(C) is the same for all honest parties). We will call a query of a party successful if it submits a pair (ctr, h) such that H(ctr, h) ≤ D. Without loss of generality, let P1 , . . . , Pt be the set of corrupted parties (knowledge of this set will not be used in any argument). For each round i, j ∈ [q], and k ∈ [t], we define Boolean random variables Xi and Zijk ∈ {0, 1} as follows. If at round i an honest party obtains a POW, then Xi = 1, otherwise Xi = 0. Regarding the adversary, if at round i, the j-th query of the k-th corrupted party is successful, then Zijk = 1, otherwise Zijk = 0. Further, if Xi = 1, we call i a successful round. If a round is uniform (Def. 4) and uniquely successful, we say it is a uniquely successful uniform round. Next, we will prove two preliminary lemmas that will be helpful in our analysis. The first one states that, at any round, the length of any honest party’s chain will be at least as large as the number of successful rounds. As a consequence, the chain of honest parties will grow at least at the rate of successful rounds. The second lemma is a simple application of Chernoff bounds and states that, with high probability, the honest parties will have, at any round, at least λ as many successful rounds as the adversary has. The usefulness of this lemma will be in showing that honest parties will be building a blockchain at a rate the adversary will find it hard to overcome. Lemma 5. Suppose that at round r the chain of an honest partyP is of length `. Then, after round s ≥ r, the chain of any honest party will have length at least ` + si=r Xi . Proof. By induction on s − r ≥ 0. For the basis, observe that at the round at which the honest party’s chain reaches length ` (round r or earlier), he broadcasts it. Therefore, after round r every honest party would have a chain of length at least `. For the inductive step, if it is the case Xs = 0, then the inductive hypothesis suffices. Otherwise (Xs = 1), observe that by hypothesis every honest P party has a chain of length at least `0 = `+ s−1 X i . Therefore, i=1 Psat least one honest party broadcasts 0 0 at round s a chain of length at least ` + 1. Since ` + 1 = ` + i=1 Xi , this completes the proof. Lemma 6. Assume γ ≥ (1 + δ)λβ for some δ ∈ (0, 1) and λ ≥ 1. The probability that during s rounds the number of successful rounds exceeds by a factor (1+ 2δ )λ the number of solutions computed 2 by the adversary is at least 1 − e−Ω(δ s) . 16
17. Ps Proof. Without loss of generality we assume the s rounds start at round 1. Let X = i=1 Xi and Ps P P Z = i=1 j∈[q] k∈[t] Zijk . By an application of Chernoff bounds (Appendix B) we obtain Pr[X ≤ (1 − 4δ )γs] ≤ e−Ω(δ 2 s) and Pr[Z ≥ (1 + 5δ )βs] ≤ e−Ω(δ 2 s) . It follows that the union of these events has a measure exponentially small in s. However, if none of them hold, then X > (1 − 4δ )γs ≥ (1 − 4δ )(1 + δ)λβs > (1 + 2δ )(1 + 5δ )λβs > (1 + 2δ )λZ. We are now ready for the treatment of the protocol’s properties outlined in Section 3.2. 4.2 The common-prefix property This property is established in Theorem 9, whose main argument is in turn given in Lemma 8. We start with a lemma leading to that argument. The lemma will be used to argue that uniform rounds favor the honest parties. Informally, the idea is that a uniquely successful uniform round forces an adversary trying to make honest parties’ chains “diverge” to produce POWs. In the second lemma we take advantage of this, to show that if the adversary has appropriately bounded computational power, then there will be enough uniquely successful uniform rounds to prevent him from mounting a successful attack on the common-prefix property. Lemma 7. Consider a uniquely successful uniform round where the honest parties have chains of length ` − 1. Then, in any subsequent round, there can be at most one chain C where the `-th block was contributed by an honest party. Proof. Let r be a uniquely-successful uniform round and C, with len(C) = `, be the chain computed by the party that solves the proof of work and extends its local chain of length ` − 1 to `. At round r + 1 every honest party will receive C and will either adopt it or adopt another chain sent by the adversary. In any case, every honest party will have a chain of length at least `, and will never query the pow(·) function with a chain of length ` − 1 again. The statement of the lemma thus follows. Note that in order for the common-prefix property to be violated at round r, at least two honest dk dk parties should have chains C1 and C2 such that C1  C2 or C2  C1 . Therefore, the existence of many blocks computed at uniform rounds forces the adversary to provide as many blocks of its own. We need to show that, with high probability the adversary will fail to collect as many solutions by round r. We say that two chains diverge at a given round, if the last block of their common prefix was computed before that round. Our main lemma below asserts the following. Suppose the protocol is halted at round r and two honest parties have distinct chains C1 and C2 . Then, for s large enough, the probability that C1 and C2 diverge at round r − s is negligible. The idea of the proof is to upper bound the number of (valid) broadcasts that the adversary can perform during these last s rounds. Note that they are in the order of βs in expectation. The crucial observation here is that if at a given round the adversary is silent, then a uniform round follows. Therefore we expect about (1 − β)s uniform rounds, and consequently γ(1 − β)s uniquely-successful uniform rounds. Recalling Lemma 7, the adversary needs to collect γ(1 − β)s POWs. Thus, in the lemma’s condition we choose the relation between β and γ suitably so that the adversary is incapable of accomplishing this task, except with probability exponentially decreasing in s. 17
18. Lemma 8. Assume f < 1 and γ ≥ (1 + δ)λβ, for some real δ ∈ (0, 1) and λ ≥ 1 such that λ2 − f λ − 1 ≥ 0. Suppose two honest parties at round r have or adopt chains C1 and C2 . Then, for 3 any s ≤ r, the probability that C1 and C2 diverge at round r − s is at most e−Ω(δ s) . Proof. We define three bad events, A, B and C, which we show to hold with probability exponentially small in s. We conclude the proof by showing that if none of these bad events happens, then there cannot exist C1 and C2 diverging at round r − s. The bad event A occurs if, at some round r0 ≥ r − s, the adversary broadcasts a chain C with the following properties. (1) C is returned by the function maxvalid of an honest party; (2) the block head(C) was computed by the adversary before round r − (1 + 8δ )s. We now give an upper bound on the probability that event A occurs. Let r∗ ≤ r −(1+ 8δ )s be the latest round at which a block of C was computed by an honest party (if none exists, then r∗ = 0), and let ` denote the length of the chain up to that block. If any other block computed by an honest party exists among the blocks from length ` up to len(C), then such block was computed in rounds r − (1 + 8δ )s up to r0 , and it follows that the probability that the adversary’s block can extend it at round r0 is negligible in (κ − log D). Therefore, we infer that with overwhelming probability the adversary has computed all the blocks from length ` to len(C), and done so during the rounds r∗ to r0 . Let Z denote the total number of solutions the adversary obtained in r0 − r∗ rounds. Let also X denote the total number of successful rounds for the honest parties in r0 − r∗ rounds. We have Z ≥ len(C) − ` ≥ X. The first inequality was argued above and the second one follows from Lemma 5. Finally, note that, by Lemma 6, the event Z ≥ X has measure exponentially small in the number of rounds r0 − r∗ . 3 Since that number satisfies r0 − r∗ ≥ δs/8, we conclude that Pr[A] ≤ e−Ω(δ s) . The second bad event occurs if the adversary has obtained a large number of solutions during (1 + 8δ )s rounds. Specifically, let Z denote the number of successful calls to the oracle by the adversary, for a total of (1 + 8δ )s rounds. Define B to be the event Z ≥ (1 + 9δ )(1 + 8δ )βs. An application of Chernoff bounds gives Pr[Z ≥ (1 + 9δ )(1 + 8δ )βs] ≤ e−Ω(βδ 2 s) . The third bad event occurs when the honest parties do not obtain enough solutions from the oracle during uniform rounds. Consider any number, say, s0 of rounds (not necessarily consecutive), and denote by X the number of them that were uniquely successful. We have Pr[X ≤ (1 − 4δ )γs0 ] ≤ e−Ω(γδ 2 s0 ) . From now on we assume that none of the events A, B and C occurs. It is easy to see that if at any round the adversary does not broadcast a (new) POW, then the next round will be uniform. Using this observation for a given s consecutive rounds, we will calculate a lower bound on the number of rounds that will be uniform. The adversary may prevent a round among the s consecutive rounds from being uniform by broadcasting a solution that was found during the s consecutive rounds as well as in the past for an extended period of (1 + 8δ )s rounds. Note that, since A does not occur, he 3 may not use even older solutions with probability at least 1 − e−Ω(δ s) . The negation of the second bad event bounds the number of solutions the adversary can obtain. This implies that at least s0 = s − (1 + 9δ )(1 + 8δ )βs ≥ s − (1 + 4δ )βs = (1 − β)s − 4δ βs 18
19. rounds among the s rounds will be uniform. Given the negation of the third bad event, there were X > (1− 4δ )γs0 uniquely successful uniform rounds during the s rounds of the protocol. By Lemma 7, it is necessary for the adversary, in order to maintain the concurrent existence of C1 and C2 , to obtain at least X solutions. Thus, for the adversary to succeed, it should hold that Z ≥ X. Substituting in this inequality the bounds on Z ≤ (1 + 4δ )βs and X > (1 − 4δ )γs0 given by ¬B and ¬C, respectively, and rearranging we obtain (1 + 2δ )β ≥ (1 − 4δ )γ(1 − β). (1) Recall that β + γ < f . Since γ ≥ (1 + δ)λβ, this implies 1 − β > (1 + λ − f )/(1 + λ). Using the last two inequalities to simplify (1), this implies λ2 − f λ − 1 < 0, contradicting the choice of λ in the statement of the lemma. We conclude that if A ∪ B ∪ C does not occur, then C1 and C2 cannot diverge at round r − s. Finally, an application of the union bound on A ∪ B ∪ C implies that the adversary can successfully maintain such C1 and C2 with probability at most exponentially small in s and the statement of the lemma follows. The above lemma is almost what we need, except that it refers to number of rounds instead of number of blocks. In order to obtain the common-prefix property we should use the properties of the blockchains of the parties themselves as the sole measure of divergence. The next theorem establishes the connection. Theorem 9. Assume f < 1 and γ ≥ (1 + δ)λβ, for some real δ ∈ (0, 1) and λ ≥ 1 such that λ2 − f λ − 1 ≥ 0. Let S be the set of the chains of the honest parties at a given round of the backbone protocol. Then the probability that S does not satisfy the common-prefix property with parameter k 3 is at most e−Ω(δ k) . Remark 1. Observe that as f → 0, λ → 1. On the other hand, if f → 1 then λ → φ, where φ is the √ golden ratio ( 1+2 5 ). Proof. If there is only one chain in S then the property is satisfied trivially. Consider two chains C1 and C2 in S and the least integer k ∗ such that dk∗ C1 dk∗  C2 and C2  C1 . (2) We need to show that the event k ∗ ≥ k happens with probability exponentially small in k. Let r be the current round and let r − s be the round at which the last common block of C1 and C2 was computed. The length of the chains cannot be greater than the number of solutions Y obtained from the oracle in s rounds. By the Chernoff bound, Pr[Y ≥ (1 + δ)f s] ≤ e−δ 2 f s/3 . 2 It follows that, with probability 1 − e−δ f s/3 , s > k ∗ /((1 + δ)f ). Thus, if k ∗ ≥ k, we have a sequence of s = Ω(k) consecutive rounds with chains C1 and C2 diverging, and the theorem follows from Lemma 8. Remark 2. Recall that in our analysis we are interested in the relationship between α and β. In particular, the ratio α/β reflects the power the honest parties have against the power of the adversary. α ≤ β implies that the adversary can, with constant probability, preclude the honest 19
20. Figure 3: The degradation of the adversarial bound of Theorem 9 as f ranges in (0, 1) in the x-axis (lower curve). When ties are broken following lexicographic order the analysis can be improved (upper curve). parties that follow the protocol from doing anything useful. This is simply because such an adversary has enough power to build a chain that will often be longer than the chain the honest parties are building. Therefore, it is to be expected that the statements are meaningful only when α/β is bounded away from 1 by a constant δ. In case the network is synchronized (f → 0), the value of α gets very close to the value of γ = α − α2 , and hence our result is tight. In case of a larger f , our analysis shows that the upper bound on the adversarial hashing power devolves and in fact approaches 0 as f → 1 — in other words, in a network were a POW becomes relatively easy compared to network synchronization time, Theorem 9 provides no security guarantee whatsoever. In practice, this underscores the importance of calibrating the difficulty of the proof of work to maintain a small value of f (such calibration takes place in the Bitcoin system every 2016 blocks). It is an interesting question to further explore the behavior of the backbone protocol in desynchronized networks. We remark that with our analysis we can prove a much better behavior for f → 1 for a modified backbone protocol that has a deterministic tie-breaking rule (e.g., chooses a chain that is the lexicographically smallest from those received13 ). In this case we can prove, for example, that our analysis enables the common prefix property to hold when f = 1 assuming the adversary controls less than about 29% of the hashing power. In Figure 3 we show how the bound of Theorem 9 degenerates when the parameter f ranges in the (0, 1) range as well as the improvement in the analysis that can be achieved by lexicographic tie-breaking (we omit the details of this analysis). 4.3 The chain-quality property We now turn to the chain-quality property (Definition 3), which the theorem below establishes for a suitable bound on the number of blocks introduced by the adversary. Theorem 10. Assume f < 1 and γ ≥ (1+δ)λβ for some δ ∈ (0, 1). Suppose C belongs to an honest party and consider any ` consecutive blocks of C. The probability that the adversary has contributed 2 more than (1 − 3δ ) λ1 ` of these blocks is less than e−Ω(δ `) . From the above theorem, it follows immediately that the chain quality is satisfied with parameter µ = λ1 for any segment length ` and probability that drops exponentially in `. 13 This has in fact been debated in an number of occasions; see, e.g., [14]. 20
21. Proof. Let us denote the i-th block of C by Bi so that C = B1 . . . Blen(C) and consider the ` consecutive blocks Bu , . . . , Bv that correspond to some suffix of C dk that has length at least k. We let L denote the least number of consecutive blocks Bu0 , . . . , Bv0 that include the ` given ones (i.e., u0 ≤ u and v ≤ v 0 ) and have the properties (1) that the block Bu0 was computed by an honest party or is B1 , in case such block does not exist, and (2) that there exists a round at which an honest party was trying to extend the chain ending at block Bv0 . Observe that number L is well defined since Blen(C) is at the head of a chain that an honest party is trying to extend. Now let x denote the number of inputs from honest parties that are included in the ` blocks and, towards a contradiction, asume that     x ≤ 1 − (1 − 3δ ) λ1 ` ≤ 1 − (1 − 3δ ) λ1 L. Let s be the number of rounds during which the L blocks were incorporated to the chain. Given that f is bounded away from 1 by a constant, the Chernoff bound implies that in s rounds the total number of solutions is greater than s with probability at most e−Ω(s) . It follows that s ≥ L with high probability. Observe that the adversary computed L − x of these blocks. Let Z be the random variable that corresponds to the POWs obtained by the adversary during these s rounds and X the successful rounds of the honest players in the same sequence of rounds. The probability that the adversary guesses the hash of block Bu0 and is able to extend it using solutions from previous rounds is negligible in (κ − log D) and we ignore it. We now have: Z ≥ L − x ≥ (1 − 3δ ) λ1 L ≥ (1 − 3δ ) λ1 X ≥ 1 X δ (1+ 2 )λ where the penultimate inequality follows from Lemma 5. By Lemma 6 this has measure e−Ω(δ 2 `) . Remark 3. We are able to argue that Theorem 10 is tight under the simplification that ties between blockchains of equal length always favor the adversary. In particular, we assume that the function maxvalid at line 5 of Algorithm 4, in case of chains of equal length, will always return the suggestion of the adversary if there is one. This simplification is made without loss of generality in our model since the adversary is rushing and hence in case two chains are transmitted in a single round the adversary can always arrange it so that its own solution arrives first14 . Furthermore, if the number of honest parties is large, when an honest party discovers a solution in a round, all other honest parties will prefer the one transmitted by the adversary and thus the effect of a single honest party opting for its own block will be negligible. The attack below is a type of “selfish mining” attack (it is a variation of the one in [17] and appears to be folklore in bitcoin circles) that accomplishes the stated bound. The attack is as follows. Initially, the adversary works on the same chain as every honest party. However, whenever it finds a solution it keeps it private and keeps on extending a private chain. Whenever an honest party finds a solution, the (rushing) adversary releases one block from the private chain; if the private chain is depleted the adversary returns to the public chain. We now argue that this strategy exploits the conditions stated above and maximizes the adversarial blocks in the blockchain up to the upper bound of Theorem 10. Consider s rounds of the protocol. With high probability, the adversary will obtain more than (1 − )βs solutions for some small  > 0. With each one of them it will try to block the blocks that are broadcast by honest parties. At the end of the s rounds, there may be a few “unused” blocks but these will be, with high probability, at most βs. This is because during the rounds that 14 In fact, this rushing capability was argued to be realistic in [17] through the dispersion of sybil nodes in the Bitcoin peer-to-peer network that echo the adversary’s messages. 21
22. the adversary acquired the blocks that it did not broadcast, none of the honest players obtained a solution; this is a low probability event. Now, the honest parties will have—with high probability— at most (1 + )γs successful rounds. It follows that, for a small constant δ, the quality of the chain is 1 − 1−δ λ . Note that the Chernoff bound can be used to make the argument more formal and replace the expression “with high probability” with 1 − eΩ(s) . From this it follows that in order to obtain better chain quality one should consider mechanisms that result in more favorable (for the honest parties) behavior in the function maxvalid. 5 Simple POW-based Byzantine Agreement Protocols We now turn to applications of the Bitcoin backbone protocol, showing how it can be used as a basis to solve other problems. We start in this section by analyzing Nakamoto’s suggestion for solving BA, observing that it falls short of satisfying Definition 1; we then present our simple instantiation which solves BA. This protocol, however, only tolerates an adversarial hashing power less than 1/3, which takes us to the next section, where we present Bitcoin’s essential task, namely, distributively maintaining a public transaction ledger, as well as a more elaborate BA protocol tolerating an adversarial power strictly less than 1/2. An overview of our applications and the way their properties depend on those of the backbone protocol was already presented in Figure 1. 5.1 Nakamoto’s suggestion for Byzantine agreement As our first illustration of how the Bitcoin backbone can be used we present Nakamoto’s suggestion for solving BA, as presented in a forum post [30]15 . We describe his solution (call it Πnak BA ) via the backbone protocol by specifying the functions V (·), I(·), R(·) in a suitable way (see Figure 4). The input validation predicate V (·) will be defined to require that all valid chains contain the same input value together with a nonce. The chain reading function R(·) simply returns this value (ignoring the nonce) in case the chain has length at least k (which is the security parameter); otherwise it is undefined. The input contribution function I(·) examines the contents of the current chain C and the contents of the input tape Input(). In case C = ε the input contribution for the next block is taken verbatim from the input tape; otherwise, the input contribution is determined as the (unique) value that is already present in the C (and in this case the local input is ignored). Note that we will only consider environments Z that provide an input symbol to all parties. Note that the nonce is added to ensure “work independence”: the parties need to introduce a fresh random κ-bit nonce at each block (cf. the beginning of Sec. 4). It follows that initially the protocol builds various chains all containing the same value. The intuition is that Agreement will follow from the fact that the honest players will eventually agree on a single chain, as long as the majority of the hashing power lies with the honest parties. While this is true, as we will demonstrate, the second necessary property does not hold: this protocol cannot provide Validity (with high probability). As we now show, Agreement follows easily from the common-prefix property. Indeed, as long as there is a common prefix (irrespective of its length), it is ensured that when R(·) becomes defined and all honest parties will produce the same output. Lemma 11 (Agreement). Suppose f < 1 and γ ≥ (1 + δ)λβ, for some real δ ∈ (0, 1) and λ ≥ 1 such that λ2 − f λ − 1 ≥ 0. Protocol Πnak BA from Fig. 4 satisfies Agreement (cf. Definition 1) with 3 k) −Ω(δ probability at least 1 − e . 15 Note that Nakamoto’s description is quite informal. We make the most plausible interpretation of it in our formal framework. 22
23. Input validation predicate V (·) Chain reading function R(·) (parameterized by k) Input contribution function I(·) V (hx1 , . . . , xn i) is true if and only if it holds that v1 = . . . = vn ∈ {0, 1}, ρ1 , . . . , ρn ∈ {0, 1}κ where xi = hvi , ρi i. If V (xC ) = True and len(C) ≥ k, the value of R(xC ) is the (unique) value v that is present in each block of C, while it is undefined if V (xC ) = False or len(C) < k. If C = ∅ and (Insert, v) is in the input tape then I(st, C, round, Input()) is equal to hv, ρi where ρ ∈ {0, 1}κ is a random value; otherwise (i.e., the case C 6= ∅), it is equal to hv, ρi where v is the unique v ∈ {0, 1} value that is present in C and ρ ∈ {0, 1}κ is a random value. The state st always remains . Figure 4: Expressing Nakamoto’s BA protocol Πnak BA over the Bitcoin backbone protocol via the specification of V (·), R(·), I(·). Proof. Observe that chains contain unique values (ignoring the nonces), therefore a disagreement between honest parties implies that two parties have disjoint chains (essentially, this is equivalent to a fork that happens at the onset). It follows from the common prefix property (Theorem 9) that the event of any two chains of length at least k that are completely disjoint happens with probability 3 at most e−Ω(δ k) . On the other hand, it is easy to see that Validity cannot be guaranteed with overwhelming probability unless the hashing power of the adversary is negligible compared to the honest players, i.e., t/n is negligible. This is because in case the adversary finds a solution first, then every honest player will extend the adversary’s solution and switch to the adversarial input hence abandoning the original input. While one can still show that Validity can be ensured with non-zero probability (and thus the protocol fails gracefully assuming honest majority), Πnak BA falls short from providing a solution to BA. Interestingly, by appropriately modifying the way the backbone protocol is used, we show in the next section how a solution can be derived. 5.2 A Byzantine agreement protocol for (1/3)-bounded adversaries We now show that the Bitcoin backbone can be directly used to satisfy BA’s properties with an error that decreases exponentially in the length of the chain, assuming however that the adversary’s hashing power is less than 1/3. There are two important differences with respect to the approach in the previous section: (i) parties never abandon their original input but instead they do insist in inserting it into the blockchain, and (ii) when the chain becomes of length 2k, they output the majority of their local length-k prefix (note that here we consider binary BA). The protocol (i.e., the specification of the functions V (·), I(·), R(·)) is presented in Figure 5. 1/3 Lemma 12 (Agreement). Suppose f < 1 and γ ≥ 2(1 + δ)β, for some real δ ∈ (0, 1). Protocol ΠBA 3 of Fig. 5 satisfies Agreement in O(k) rounds with probability at least 1 − e−Ω(δ k) . Proof. In order for agreement to be violated, at least two honest players should have upon terminadk dk tion chains C1 and C2 such that C1 6= C2 . In particular, the set {C1 , C2 } should be a set of chains that belong to honest parties and does not satisfy the common-prefix property. Thus, the statement of the lemma follows directly from Theorem 8. We now turn to the Validity property. In order to prove it we need to show that, upon termination of the protocol, the chain of any honest party will contain among the first k inputs more 23
24. Input validation predicate V (·) Chain reading function R(·) (parameterized by k) Input contribution function I(·) V (hx1 , . . . , xn i) is true if and only if v1 , . . . , vn ∈ {0, 1}, ρ1 , . . . , ρn ∈ {0, 1}κ where vi , ρi are the values from the pair xi = hvi , ρi i. If V (hx1 , . . . , xn i) = True and n ≥ 2k, the value R(xC ) is the majority bit of v1 , . . . , vk where xi = hvi , ρi i; otherwise (i.e., the case V (hx1 , . . . , xn i) = False or n < 2k) the output value is undefined. I(st, C, round, Input()) is equal to hv, ρi if the input tape contains (Insert, v); ρ is a random κ-bit string. The state st remains always . 1/3 Figure 5: Protocol ΠBA over the Bitcoin backbone via the specification of V (·), R(·), I(·). inputs from honest players than provided by the adversary. As we will see, this is a consequence of the chain-quality property. 1/3 Lemma 13 (Validity). Suppose f < 1 and γ ≥ 2(1 + δ)β, for some real δ ∈ (0, 1). Protocol ΠBA 2 satisfies Validity in O(k) rounds with probability at least 1 − e−Ω(δ k) . Proof. For the property to be satisfied we only need to ensure that in Cdk the majority of the inputs 1/3 was computed by the honest parties. As in protocol ΠBA we have len(Cdk ) = k, Theorem 10 with λ = 2 provides exactly what we want. 1/3 Note that ΠBA solves BA only in case the adversary’s hashing power is bounded by 1/3. In case adversarial blocks win all head-to-head races within a round, the result is tight, as argued in Remark 3. In the next section we show a more elaborate construction based on a transaction ledger which tolerates less than 1/2. Remark 4. As mentioned in Section 2, “Strong Validity” refers to the requirement that the output value be one of the honest parties’ inputs, and the distinction is relevant in the case of non-binary inputs, i.e., coming from an arbitrary set V , V > 2. It is easy to modify the above algorithm to also satisfy this property by making the chain reading function the element with highest plurality in the chain (ties broken favoring the lexicographically smallest element in V ), as opposed to majority, and by imposing a more stringent bound on the adversary, namely, that γ ≥ V (1 + δ)β. This ensures that the expected number of blocks in the blockchain that are controlled by the adversary is less than V1 , and maintains validity even in the worst case that the honest parties’ inputs are equally split among all possible values but one (i.e., there are V − 1 inputs equally proportioned among the honest parties). Agreement is ensured in the same way as before via the common-prefix property. The bound is in-line with the known bounds for the standard (computational) setting, n > V t, cf. [20]. 6 Public Transaction Ledgers We now come to the application which the Bitcoin backbone was designed to solve: maintaining a public transaction ledger. We first formally introduce this object — a “book” where transactions are recorded — and its properties, and then we show how it can be used to implement the Bitcoin ledger and BA in the honest majority setting by properly instantiating the notion of a transaction. 24
25. 6.1 Robust public transaction ledgers A public transaction ledger is defined with respect to a set of valid ledgers L and a set of valid transactions T , each one possessing an efficient membership test. A ledger x ∈ L is a vector of sequences of transactions tx ∈ T . Each transaction tx may be associated with one or more accounts, denoted a1 , a2 , . . . The backbone protocol parties, called miners in the context of this section, process sequences of transactions of the form x = tx1 . . . txe that are supposed to be incorporated into their local chain C. The input inserted at each block of the chain C is the sequence x of transactions. Thus, a ledger is a vector of transaction sequences hx1 , . . . , xm i, and a chain C of length m contains the ledger xC = hx1 , . . . , xm i if the input of the j-th block in C is xj . The description and properties of the ledger protocol will be expressed relative to an oracle Txgen which will control a set of accounts by creating them and issuing transactions on their behalf. In an execution of the backbone protocol, the environment Z as well as the miners will have access to Txgen. Specifically, Txgen is a stateful oracle that responds to two types of queries (which we purposely only describe at a high level): GenAccount(1κ ): It generates an account a. ˜ It returns a transaction tx provided that tx ˜ is some suitably formed string, IssueTrans(1κ , tx): or ⊥. We also consider a symmetric relation on T , denoted by C(·, ·), which indicates when two transactions tx1 , tx2 are conflicting. Valid ledgers x ∈ L can never contain two conflicting transactions. We call oracle Txgen unambiguous if it holds that for all PPT A, the probability that ATxgen produces a transaction tx0 such that C(tx0 , tx) = 1, for tx issued by Txgen, is negligible in κ. Finally, a transaction tx is called neutral if C(tx, tx0 ) = 0 for any other transaction tx0 . The presence of neutral transactions in the ledger can be helpful for a variety of purposes, as we will see next and in the BA protocol that we build on top of the ledger. For convenience we will assume that a single random nonce ρ ∈ {0, 1}κ is also a valid transaction. Nonces will be neutral transactions and may be included in the ledger for the sole purpose of ensuring independence between the POW instances solved by the honest parties. Next, we determine the three functions V (·), I(·), R(·) that will turn the backbone protocol into ΠPL , a protocol realizing a public transaction ledger. Input validation predicate V (·) Chain reading function R(·) Input contribution function I(·) V (hx1 , . . . , xm i) is true if and only if the vector hx1 , . . . , xm i is a valid ledger, i.e., hx1 , . . . , xm i ∈ L. If V (hx1 , . . . , xm i) = True, the value R(xC ) is equal to hx1 , . . . , xm i; undefined otherwise. I(st, C, round, Input()) operates as follows: if the input tape contains (Insert, v), it parses v as a sequence of transactions and retains the largest subsequence x0  v that is valid with respect to xC (and whose transactions are not already included in xC ). Finally, x = tx0 x0 where tx0 is a neutral random nonce transaction. Figure 6: The public transaction ledger protocol ΠPL , built on the Bitcoin backbone. We now introduce two essential properties for a protocol maintaning a public transaction ledger: (i) Persistence and (ii) Liveness. In a nutshell, Persistence states that once an honest player reports a transaction “deep enough” in the ledger, then all other honest players will report it indefinitely whenever they are asked, and at exactly the same position in the ledger (essentially, this means 25
26. that all honest players agree on all the transactions that took place and in what order). In a more concrete Bitcoin-like setting, Persistence is essential to ensure that credits are final and that they happened at a certain “time” in the system’s timeline (which is implicitly defined by the ledger itself). Note that Persistence is useful but not enough to ensure that the ledger makes progress, i.e., that transactions are eventually inserted in a chain. This is captured by the Liveness property, which states that as long as a transaction comes from an honest account holder and is provided by the environment to all honest players, then it will be inserted into the honest players’ ledgers, assuming the environment keeps providing it as an input for a sufficient number of rounds16 . Definition 14. A protocol Π implements a robust public transaction ledger in the q-bounded synchronous setting if it satisfies the following two properties: Persistence: Parameterized by k ∈ N (the “depth” parameter), if in a certain round an honest player reports a ledger that contains a transaction tx in a block more than k blocks away from the end of the ledger, then tx will always be reported in the same position in the ledger by any honest player from the next round on. Liveness: Parameterized by u, k ∈ N (the “wait time” and “depth” parameters, resp.), provided that a transaction either (i) issued by Txgen, or (ii) is neutral, is given as input to all honest players continuously for u consecutive rounds, then there exists an honest party who will report this transaction at a block more than k blocks from the end of the ledger. We prove the two properties separately, starting with Persistence. The proof is based on the common prefix property of the backbone protocol (recall Definition 2 and Theorem 9). Lemma 15 (Persistence). Suppose f < 1 and γ ≥ (1 + δ)λβ, for some real δ ∈ (0, 1) and λ ≥ 1 3 such that λ2 − f λ − 1 ≥ 0. Protocol ΠPL satisfies Persistence with probability 1 − e−Ω(δ k) , where k is the depth parameter. Proof. Let C1 be the chain of some honest player P1 . We show that if a transaction tx is included d in C1k at round r∗ then this transaction will be always included in every honest player’s chain with 3 probability at least 1 − e−Ω(δ k) . Let r1 be an arbitrary round where P1 reports a transaction tx more than k blocks deep in the dk ledger, and let r2 > r1 the smallest round that an honest party P2 has a chain C2 , such that C2 does not include tx. Call this event B(r2 ). Suppose tx was inserted in C1 at round r∗ . It follows that C1 and C2 diverge at round r∗ . If it holds that at round r2 player P1 possesses a continuation of chain C1 , then by Lemma 8 3 ∗ this occurs with probability e−Ω(δ (r2 −r )) . If not, then this means that P1 abandoned C1 at some round following r1 and prior to r2 for a chain not containing tx. It follows that by Lemma 8 this 3 ∗ occurs with probability e−Ω(δ (r1 −r )) . So in either case we conclude that the probability of event 3 ∗ B(r2 ) is e−Ω(δ (r1 −r )) . The claim then follows by a union bound over all rounds r > r∗ . Letting s = r − r∗ and  be an appropriate constant, the probability that there r > r∗ such that B(r) occurs (i.e., that P is an 3 s0 3 −δ Persistence is violated at round r) is at most s0 ≥s e = e−Ω(δ s) . Finally, as in the proof Theorem 9 we can argue that s = Ω(k). 16 Observe that here we take the view that new transactions are available to all honest players and the way they are propagated is handled by the environment that feeds the backbone protocol. While this makes sense in the honest/malicious cryptographic model, it has been challenged in a model where all players are rational [3]. Analysis of the backbone protocol in a setting where transaction propagation is governed by rational players is beyond the scope of our paper. 26
27. We next prove Liveness, which is based on the chain-quality property (recall Definition 3 and Theorem 10). Lemma 16 (Liveness). Assume f < 1 and γ ≥ (1 + δ)λβ, for some δ ∈ (0, 1), λ ∈ [1, ∞) and let k ∈ N. Further, assume oracle Txgen is unambiguous. Then protocol ΠPL satisfies Liveness with 2 wait time u = 2k/(1 − δ)γ and depth parameter k with probability at least 1 − e−Ω(δ k) . Proof. We prove that assuming all honest players receive as input the transaction tx for at least u = 2k/[(1 − δ)γ] rounds, there exists an honest party with chain C such that tx is included in C dk . Indeed, after u rounds the Chernoff bound implies that the honest parties had at least 2k 2 successful rounds with probability at least 1 − e−Ω(δ k) . Invoking Lemma 5, we infer that the chain’s length of any honest party has increased by at least 2k blocks. Finally, the chain-quality property (Theorem 10) implies that at least one of the blocks in the length-k suffix of C dk was computed by an honest party. Such a block would include tx since it is infeasible for adversarial Z, A to produce a conflicting transaction tx0 (which would be the only event making an honest player drop tx from the sequence of transactions x that it attempts to insert in the blockchain). Thus, the lemma follows. 6.2 Bitcoin-like transactions and ledger Next, we show how to instantiate the public transaction ledger for Bitcoin, by defining the sets of transactions and valid ledgers. Transactions and accounts are defined with respect to a digital signature scheme that is comprised of three algorithms hKeyGen, Sign, Verifyi. An account will be a pair a = (vk, G(vk)) where G(·) is a hash function and G(vk) is the “address” corresponding to the account. A transaction tx is of the form “{a1 , a2 , . . . , ai } → (σ, {(a01 , b01 ), . . . , (a0o , b0o )}),” where a1 , . . . , ai are the accounts to be debited, a01 , . . . , a0o are the addresses of the accounts17 to be credited with funds b01 , . . . , b0o , respectively, and σ is a vector h(vk1 , σ1 ), . . . , (vki , σi )i of verification keys and digital signatures issued under them, on the same message {(a01 , b01 ), . . . , (a0o , b0o )}. (We note that Bitcoin transactions can be more expressive but the above description is sufficient for the purpose of our analysis). Next, we specify the Txgen oracle: GenAccount(1κ ): It generates an account a by running KeyGen and computing the hash G(·) on the verification key. The account is the pair (vk, G(vk)), where G(vk) is the account’s address. The corresponding secret key, sk, is kept in the state of Txgen. ˜ It returns a transaction tx provided that tx ˜ is a transaction that is only missIssueTrans(1κ , tx): ing the signatures by accounts that are maintained by Txgen. (Recall the format of transactions above.) Each account is only allowed a single transaction. Note that the above restriction on IssueTrans is without loss of generality, as in Bitcoin, entities typically maintain a number of accounts and are allowed (although not forced) to move their balances forward to a new account as they make new transactions. The conflict relation C(·, ·) over T satisfies that C(tx1 , tx2 ) = 1 if and only if tx1 6= tx2 and tx1 , tx2 have an input account in common18 . Thus, we can easily prove the unambiguity of the Txgen oracle based on the unforgeability of the underlying digital signature. 17 In bitcoin terminology every account has an address that is used to uniquely identify it. Payments directed to an account require only this “bitcoin address.” The actual verification key corresponding to the account will be revealed only when the account makes a payment. 18 The conflict relation is more permissive in the actual Bitcoin ledger. We adopt the more simplified version given above as it does not change the gist of the analysis. 27
28. Lemma 17. Assume that hKeyGen, Sign, Verifyi is an existentially unforgeable signature scheme. Then oracle Txgen is unambiguous. In order to define the set of valid Bitcoin ledgers we first need to determine in what sense a transaction may be valid with respect to a ledger. Then we will define the set of valid ledgers recursively as the maximal set of vectors of sequences of transactions that satisfy this condition. So here it goes. A transaction tx is valid with to a Bitcoin ledger x = hx1 , . . . , xm i provided that all Pi respect P o 0 b ≥ digital signatures verify and j=1 bj , where bj is the balance that was credited to j=1 j P P account aj in the latest transaction involving aj in x. In case e = ij=1 bj − oj=1 b0j > 0, then e is a transaction fee that may be claimed separately in a special transaction of the form “∅ → . . .,” called a coinbase transaction. In more detail, a coinbase transaction has no inputs and its purpose is to enable miners to be rewarded Pofor maintaining the legder. The transaction is of the form “∅ → {(a1 , b1 ), . . . , (ao , bo )},” and j=1 bj is determined based on the other transactions that are “bundled” in the block as well as a flat reward fee, as explain below. A sequence of transactions x = h∅ → {(a1 , b1 ), . . . , (ao , bo )}, tx1 , . . . , txl i is said to be valid with respect to a ledger x = hx1 , . . . , xm i, if each transaction txj is valid with respect to the ledger x extended by the transactions tx1 , . . . , txj−1 . I.e., for all j = 1, . . . , l the transaction txj should be valid with respect to ledger hx1 , . . . , xm , tx1 . . . txj−1 i, Po and furthermore, the total P fee e = j=1 bj collected in the transaction ∅ → {(a1 , b1 ), . . . , (ao , bo )} does not exceed rm + m j=1 ej , which includes all the individual fees corresponding to transactions tx1 , . . . , txe , plus a value rm that is the flat reward given for extending a ledger of length m to a ledger of length m + 1.19 The set of valid ledgers L with respect to a reward progression {rj }j∈N contains ε (the empty ledger), and any ledger x which extends a ledger in L by a valid sequence of transactions. Note that the first transaction sequence of any Poledger x ∈ L contains a single transaction of the form ∅ → {(a1 , b1 ), . . . , (ao , bo )} that satisfies j=1 bj = r0 , where r0 is the initial flat reward. This first transaction “distributes an initial amount of money” to the ledger’s initiator(s).20 It is easy to see that L has an efficient membership test. Given the existence of coinbase transactions in this application we can do away with random nonces as standalone transactions and the description of the input contribution function I in Fig. 6, is modified to include their generation each time an input sequence of transactions is determined to be inserted P in the ledger. Specifically, I(·) will form a coinbase transaction ∅ → {(a, b)}, where b = rlen(C) + m j=1 ej and ej is the fee corresponding to x’s j-th transaction. Account a is a freshly created account that is obtained via running KeyGen. I(·) will append account a and the corresponding (vk, sk) to its private state st. We will refer to the modified ΠPL protocol by the moniker ΠBTC . ΠBTC inherits from ΠPL the properties of Persistence and Liveness which will ensure the following with overwhelming probability in k. Apart from its latest k blocks, the transaction ledger is fixed and immutable for all honest miners. 19 Currently, the flat reward for extending the Bitcoin chain is 25BTC. The sequence r0 , r1 , . . . for Bitcoin follows a geometric progression with large constant intervals. 20 In the case of Bitcoin, it was supposedly Nakamoto himself who collected this first reward of 50BTC. 28
29. If a majority of miners21 receive an honest transaction and attempt to insert it following the protocol for a sufficient number of rounds (equal to parameter u, the “wait time”), it will become a permanent entry in the ledger (no matter the adversarial strategy of the remaining miners). 6.3 Byzantine agreement for honest majority We now use the public transaction ledger formulation to achieve POW-based BA for an honest majority by properly instantiating the notion of a transaction, thus improving on the simple BA protocol tolerating a (1/3)-bounded adversary presented in Section 5. Here we consider a set of valid ledgers L that contain sequences of transactions of the form hnonce, v, ctri, and satisfy the predicate: (H1 (ctr, G(nonce, v)) < D) ∧ (ctr ≤ q), (3) where H1 (·), G(·) are two hash functions as in the definition of the backbone protocol, and v ∈ {0, 1} is a party’s input. (Recall that D is the difficulty level and q determines how many calls to H1 (·) a party is allowed to make per round.) To distinguish the oracles, in this section we will use H0 (·) to refer to the oracle used in the backbone protocol. For the ledger we consider in this section, there will be no accounts and all transactions will be neutral — i.e., the conflict predicate C(·, ·) will be false for all pairs of transactions. We first provide a high level description of the BA protocol assuming parties have q queries per round to each oracle H0 (·), H1 (·). We then show how to use a single oracle H(·) to achieve the combined functionality of both of them while only using q queries per round. 1/2 At a high level, the protocol, ΠBA , works as follows: Operation: In each round, parties run two protocols in parallel. The first protocol is protocol ΠPL (Fig. 6), which maintains the transaction ledger and requires q queries to the oracle H0 (·). The second process is a “transaction production” protocol Πtx (Fig. 7), which continuously generates transactions satisfying predicate (3). The protocol makes q queries to the H1 (·) oracle. Termination: When the ledger reaches 2k blocks, a party prunes the last k blocks, collects all the unique POW transactions that are present in the ledger and returns the majority bit from the bits occuring in these transactions (note that uniqueness takes also the nonce of each transaction into account). 1/2 As described, protocol ΠBA does not conform to the q-bounded setting since parties require q queries to oracle H0 (·) and q queries to oracle H1 (·) to perform the computation of a single round (the setting imposes a bound of q queries to a single oracle for all parties). Note that a naïve simulation of H0 (·), H1 (·) by a single oracle H(·) in the (2q)-bounded setting (e.g., by setting Hb (x) = H(b, x)) would violate the restriction imposed on each oracle individually, since nothing would prevent the adversary, for example, from querying H0 (·) 2q times. Next, we show how we can combine the two protocols into a single protocol that utilizes at most q queries to a single random oracle in a way that the adversary will remain q-bounded for each oracle. This transformation, 1/2 explained below, completes the description of ΠBA . 2-for-1 POWs. We now tackle the problem of how to turn a protocol operation that uses two separate POW subprocedures involving two distinct and independent oracles H0 (·), H1 (·) into a protocol that utilizes a single oracle H(·) for a total number of q queries per round. Our transformation is general and works for any pair of protocols that utilize H0 (·), H1 (·), provided that certain 21 Recall that we assume a flat model w.r.t. hashing power; a majority of miners corresponds to a set of parties controlling the majority of the hashing power. 29
30. Algorithm 5 The POW-based transaction production protocol Πtx , parameterized by q, D and hash functions H1 (·), G(·). 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: v ← Input() ctr ← 1 tx ← ε h ← G(nonce, v) while (ctr ≤ q) do if (H1 (ctr, h) < D) then tx ← hnonce, v, ctri break end if ctr ← ctr + 1 end while Broadcast(tx) . nonce is a random κ-bit string . Proof of work succeeded Figure 7: The transaction production protocol Πtx . 1/2 conditions are met (which are satisfied by protocol ΠBA above). In more detail, we consider two protocols Π0 , Π1 that utilize a POW step as shown in Algorithm 6 in Figure 8. In order to achieve composition of the two protocols Π0 , Π1 in the q-bounded setting with access to a single oracle H(·), we will substitute steps 2-11 in both protocols with a call to a new function, double-pow, defined below. First, observe that in Πb , b ∈ {0, 1}, the POW steps 2-11 operate with input wb and produce output in Bb if the POW succeeds. The probability of obtaining a solution is D · 2−κ . The modification consists in changing the structure of the POWs from pairs of the form (w, ctr) to triples of the form (w, ctr, label), where label is a κ-bit string that is neutral from the point of view of the proof. This will further require the modification of the verification step for POWs in both protocols Π0 , Π1 in the following manner. Any verification step in Π0 of a POW hw0 , ctri which is of the form H(ctr, G(w0 )) < D, will now operate with a POW of the form hw0 , ctr, labeli and will verify the relation H(ctr, hG(w0 ), labeli) < D. Any verification step in Π1 of a POW hw1 , ctri which is of the form H(ctr, G(w1 )) < D, will now operate with a POW of the form hw1 , ctr, labeli and will verify the relation [H(ctr, hlabel, G(w1 )i)]R < D, where [a]R denotes the reverse of the bitstring a. This parallel composition strategy in the form of function double-pow is shown in Algorithm 7. Either or both the solutions it returns, B0 , B1 , may be empty if no solution is found. 1/2 Protocol ΠBA will employ double-pow, which will substitute the individual POW operation of the two underlying protocols Π0 , Π1 as defined in lines 2-11 of Algorithm 6. The correctness of the above composition strategy follows from the following simple observation. 30
31. Algorithm 6 POW-based protocol fragment of Πb , b ∈ {0, 1} parameterized by q, D and hash functions Hb (·), G(·), b ∈ {0, 1}. The value wb is determined from the protocol’s context. 1: . . . . Value wb is determined 2: ctr ← 1 3: B ← ε 4: hb ← G(wb ) 5: while (ctr ≤ q) do 6: if (H(ctr, hb ) < D) then 7: Bb ← hwb , ctri 8: break 9: end if 10: ctr ← ctr + 1 11: end while 12: . . . . The POW B is exploited here Algorithm 7 The double proof of work function, parameterized by q, D and hash functions H(·), G(·) that substitutes steps 2-11 of two POW-based protocols. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: function double-pow(w0 , w1 ) B0 , B1 ← ε ctr ← 1 h ← hG(w0 ), G(w1 )i while (ctr ≤ q) do u ← H(ctr, h) if (u < D) ∧ (B0 = ε) then B0 ← hw0 , ctr, G(w1 )i end if if ([u]R < D) ∧ (B1 = ε) then B1 ← hw1 , ctr, G(w0 )i end if ctr ← ctr + 1 end while return hB0 , B1 i end function Figure 8: The 2-for-1 POW transformation. Lemma 18. Consider a uniform random variable U over the integers in [0, 2κ ) and an integer D such that D = 2t for some positive integer t < κ/2. Then, the events (U < D) and ([U ]R < D) are independent and they both occur with probability D · 2−κ . Proof. It is easy to see that each event happens with probability D · 2−κ . The conjunction of the two events involves the choice of an integer U which satisfies U < D and [U ]R < D. Observe that because D = 2t , it follows that the conditioning on U < D leaves the t least significant bits of U uniformly random while fixing the remaining κ − t bits. It follows that the t most significant bits of [U ]R are uniformly random in the conditional space U < D. The event [U ]R < D has probability (D/2κ−t )/D = D2−κ and thus the two events are independent. Theorem 19. Assume f < 1 and γ ≥ (1 + δ)λβ, for some real δ ∈ (0, 1) and λ ≥ 1 such that 1/2 3 λ2 − f λ − 1 ≥ 0. Protocol ΠBA solves BA in O(k) rounds with probability at least 1 − e−Ω(δ k) . Proof. First observe that due to Lemma 18, the success probability for all parties to solve a proof of work of either kind in each round is q · D2−κ and the events are independent with each other. Regarding Agreement, observe that it follows directly from Theorem 9 (common prefix) that all 3 parties will return the majority of the same set with probability at least 1 − e−Ω(δ k) . To show Validity, let C be the chain of an honest party upon termination of the protocol. Let r denote the greatest round on which a block of C dk was computed by an honest party. We argue that the rest of the blocks in C dk , that must have been inserted by the adversary, were computed by round (1 + 2δ )r. Assume the contrary and let r0 > (1 + 2δ )r denote the least round on which an honest player adopted the chain C (after round (1 + 2δ )r). Let X denote the successful rounds 31
32. from round r to round r0 and Z the number of POWs the adversary obtained in these rounds. Lemma 5 implies that the chain of every honest player advanced in length by X blocks at least. By the definition of r0 , the adversary inserted all the blocks of C computed in these s = r0 − r rounds. 2 It follows that Z ≥ X. By Lemma 6 this occurs with probability at most e−Ω(δ s) . To finish the proof, recall that each block contains the aggregation of all broadcast transactions up to the round it was computed. Thus, C dk contains POWs computed by honest parties during r rounds and, with high probability, POWs computed by the adversary during at most (1 + 2δ )r rounds. By Lemma 6, 2 the honest parties have computed the majority of the blocks with probability at least 1 − e−Ω(δ s) and Validity is satisfied. Since s > δr/2, we need to argue that r = Ω(k). To see this, note that in (1 + 2δ )r rounds the parties created a chain of length len(C dk ) = k. An application of the Chernoff bound shows that r = Ω(k) with probability at least 1 − e−Ω(δk) . Finally note that it is easy to infer from Lemma 5 that the length of chain of all honest parties will reach 2k blocks in O(k) rounds with probability 1 − e−Ω(δk) . Remark 5. Regarding strong validity in the multivalued BA setting, i.e., where the input domain is V and has a constant cardinality strictly larger than 2 we can adapt the above protocol to return the plurality from the values stored in the transactions that are found in the ledger. In order to ensure strong validity by this modification we restrict the hashing power of the adversary to (1 − δ)/ V since this will ensure that the adversary’s number of transactions cannot overturn the plurality value as defined by the honest parties’ inputs (even if those are evenly distributed amongst them). 7 Summary and Directions for Future Work In this paper we presented a formal treatment of the Bitcoin backbone, the protocol used at the core of Bitcoin’s transaction ledger. We expressed and proved two properties of the backbone protocol — “common prefix” and “chain quality” — and showed how they can be used as foundations for designing Byzantine agreement and robust public transaction ledger protocols. Our results show that an honest majority among the (equally equipped) participants suffices, assuming the network synchronizes much faster than the proof of work rate (f → 0 in our notation) and the proper inputs (e.g., transactions) are available to the honest majority22 , while the bound on the adversary for honest parties to reach agreement degenerates as f gets larger. While these are encouraging results, we have demonstrated deviations that are of concern for the proper operation of Bitcoin. Importantly, we show that as the network ceases to synchronize fast enough compared to the proof-of-work rate (i.e., the worst-case time that takes honest players to “hear” each other becomes substantial compared to the time it takes to solve a proof of work), the honest majority property ceases to hold and the bound offered by our analysis that is required to obtain a robust transaction ledger approaches 0 as f approaches 1. Note that the effects of bad synchronization is in the maintenance of the common prefix property, which is the critical property for showing agreement. A second important concern is regarding the chain quality property, where our results show that if an adversary controls a hashing power corresponding to β then the ratio of the blocks it can contribute to the blockchain is bounded but can be strictly bigger than β. When β gets close to 1/2, our bounds show that the honest players’ contributions approach 0 in our security model. The above caveats in the two basic properties of the backbone have repercussions on the Persistence and Liveness properties of the Bitcoin ledger. Firstly, they illustrate that fast information 22 Our formalization is a way to formally express what perhaps was Nakamoto’s intuition when he wrote about Bitcoin that “it takes advantage of the nature of information being easy to spread but hard to stifle” [31]. 32
33. propagation amongst honest players is essential for transaction persistence. Secondly, they show that transaction liveness becomes more fragile as the adversarial power gets close to 1/2. Note that we achieve Liveness for any adversarial bound less than 1/2 but we do not assume any upper bound on the number of transactions that may be inserted in a block23 ; it is obvious that the fewer blocks the honest miners get into the blockchain the harder may be for a transaction to get through. Furthermore, the fact that chain quality demonstrably fails to preserve a one-to-one correspondence between a party’s hashing power and the ratio of its contributions to the ledger point to the fact that Bitcoin’s rewarding mechanism is not incentive compatible (cf. [17]). Assuming the hashing power of the honest parties γ exceeds the adversary’s hashing power β by a factor λ, we show that the adversary’s contributions to the ledger are bounded by 1/λ — a result we show to be tight in our rushing adversary model. In this way our results flesh out the incentive compatibility problems of the Bitcoin backbone, but (on a more positive note) they also point to the fact that honest hashing-power majority is sufficient to maintain the public ledger (under favorable network conditions), and hence suggest that the Bitcoin protocol can work as long as the majority of the miners want it to work (without taking into account the rationality of their decision). The above observations apply to the setting where the number of participants is fixed. In the dynamic setting (where the number of parties running the protocol may change from round to round), given the flat model that we consider, the difficulty D of the blockchain may be calibrated according to the number of players n that are active in the system. If D is set by an omniscient trusted party then the analysis carries in a straightforward way but otherwise, if D is somehow calculated by the parties themselves, the adversary can try to exploit its calculation. Note that in this case the maxvalid function would need to take the difficulty’s variability into account and thus choose the “most difficult” chain (as opposed to the longest). Comparing chains based on difficulty is simply done by computing the length of a chain by counting blocks proportionally to how difficult they are (for example, a block whose difficulty is two times larger than a given difficulty value would contribute twice as much in “length”). Interesting open questions include the security analysis of the Bitcoin backbone protocol in a rational setting as opposed to honest/malicious, in the dynamic setting where the parties themselves attempt to recalibrate the difficulty based on some metric (e.g., the time that has passed during the generation of a certain number of blocks), and in a concurrent/universal composition setting as opposed to standalone. Furthermore, the substitution of the random oracle assumption with a suitable computational assumption, as well as the development of backbone modifications that improve its characteristics in terms of common prefix and chain quality. In terms of the ledger application, transaction processing times (i.e., reducing the wait time parameter u in the Liveness property) is also an interesting question with implications to practice (since real world payment systems benefit greatly from fast transaction confirmation and verification). In all these cases, our work offers a formal foundation that allows analyzing the security properties of “tweaks” on the backbone protocol (such as the randomization rule of [17] or the “GHOST” rule in [39] used in ethereum24 ) towards meeting the above goals. Another set of interesting directions include the development of other applications that may be built on top of the backbone protocol such as secure multiparty computation with properties such as fairness and guaranteed output delivery (current works in this direction, e.g., [1, 9, 10], assume an idealized version of the Bitcoin system). 23 In the current Bitcoin implementation there is an upper bound of 1MB for blocks, hence the number transactions per block is limited. 24 https://www.ethereum.org/ 33
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36. A Model (cont’d) All entities involved in a protocol execution are described by interactive Turing machines (ITMs); an instance of an execution is described by a collection of instances of these ITM instances or ITI s. The set of all polynomial-time ITMs is denoted by ITM. We specify our protocols Π in a way similar to Canetti’s synchronous model for “real-world” execution [12]. The entities involved in an execution are the set of parties P = {P1 , . . . , Pn } running the protocol Π, the adversary denoted by A and the environment denoted by Z that provides inputs to the parties and receives outputs from them. The communication of ITIs is defined in [12] as having the sending ITI write directly to a communication input tape of the receiving ITI; the set of potential receivers is not fixed when an ITI is instantiated. This means that we have to include the messages generated by an ITI in its view explicitly. We denote by Receive() the communication input tape of each party. The communication between parties running the protocol Π will be non-authenticated but reliable25 . We capture this by allowing the adversary A to modify the sender information in any message that is written to the communication input tape of an ITI. The adversary A is allowed to corrupt parties P1 , . . . , Pn by writing a special message in their input communication tape. When that happens the adversary has complete control of the ITI of the corrupted party. We bound the number of corruptions by t ≤ n. Following [12] we assume a synchronous mode of interaction between parties. In particular, the protocol execution will be divided in rounds, and in each round all parties will be given the opportunity to act. The order of party activation is controlled by the adversary A which is adaptive and rushing. The term {viewPΠ,A,Z (κ, z)}κ∈N,z∈{0,1}∗ denotes the random variable ensemble describing the view of party P after the completion of an execution with environment Z, running protocol Π, and adversary A, on auxiliary input z ∈ {0, 1}∗ . We often drop the parameters κ and z and simply refer to the ensemble by viewPΠ,A,Z if the meaning is clear from the context. The concatenation of the i (κ, z)ii=1,...,n . is denoted by viewΠ,A,Z . view of all parties hviewPΠ,A,Z The parties’ inputs are provided by the environment Z which also receives the parties’ outputs. Parties that receive no input from the environment remain inactive, in the sense that they will not act when their turn comes in each round. The environment may provide input to a party at any round and may also modify that input from round to round. We denote by Input() the input tape of each party. We will consider ITM’s that may have access to an oracle H(·) and perform a number of queries q per round where q is a function of the security parameter κ; we refer to such parties as q-bounded. The adversary A is allowed to perform t·q queries per round where t ≤ n is the number of corrupted parties. The environment Z is not permitted any queries to H(·). The rationale for this is that we would like to bound the “CPU power” [29] of the adversary to be proportionate to the number of parties it controls while making it infeasible for the adversary be aided by external sources in the environment. We express this restriction on the environment, the parties and the adversary as the q-bounded setting. The view of the parties participating in the protocol will be denoted by P,H(·) H(·) viewΠ,A,Z (κ, q, z) and the concatenation of all parties’ views by viewΠ,A,Z (κ, q, z). We now define formally a property of an execution which is expressed as a predicate over the view of all parties in P. 25 This parallels communication over TCP/IP in the Internet where, assuming all intermediate nodes in a communication path are honest and functioning, messages between honest parties are delivered reliably, nevertheless malicious parties may “spoof” the source of a message they transmit and make it appear as originating from an arbitrary party (including an honest party) in the view of the receiver. 36
37. Definition 20. Given a predicate Q and a bound q, we say that the protocol Π satisfies property Q in the q-bounded setting provided that for all polynomial-time Z, A and strings z the probability H(·) that Q(viewΠ,A,Z (κ, q, z)) is false is negligible in κ. Note that we will only consider properties that are polynomial-time computable predicates. B Useful Inequalities We will require the following inequalities. Fact 1 (Bernoulli’s inequality). For q ≥ 1 and 0 ≤ p ≤ 1, (1 − p)q ≥ 1 − pq. Fact 2. For any real α > 0, 1 − α < e−α < 1 − α + α2 2 . Theorem 21 (Chernoff bounds). Suppose {Xi : i ∈ [n]} are P mutually independent Boolean random variables, with Pr[Xi = 1] = p, for all i ∈ [n]. Let X = ni=1 Xi and µ = pn. Then, for any δ ∈ (0, 1], 2 2 Pr[X ≤ (1 − δ)µ] ≤ e−δ µ/2 and Pr[X ≥ (1 + δ)µ] ≤ e−δ µ/3 . 37