Approximate Bitcoin Mining

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Approximate Bitcoin Mining ( approximate-bitcoin-mining )

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Figure 1: Mining Process Block Diagram • Σ0(x)≡x≫2⊕x≫13⊕x≫22 • Σ1(x)≡x≫6⊕x≫11⊕x≫25 • σ0(x)≡x≫7⊕x≫18⊕x≫3 • σ1(x)≡x≫17⊕x≫19⊕x≫10 Algorithm 2 SHA-256 block header H(0) nonce SHA-256 SHA-256 SHA-256 digest threshold solution? 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 2.2 function SHA-256(M) for i from 0 to N − 1 do forjfrom0to15do Wj =M(i) j end for forjfrom16to63do Wj = σ1(Wj−2)+Wj−7 +σ0(Wj−15)+Wj−16 end for forjfrom0to63do t0 ← h+Σ1(e)+Ch(e,f,g)+Kj +Wj t1 ←Σ0(a)+Maj(a,b,c) h←g;g←f;f←e;e←d+t1 d←c;c←b;b←a;a←t1 +t2 H(i)←H(i−1)+a;H(i)←H(i−1)+b 0011 H(i)←H(i−1)+c;H(i)←H(i−1)+d 2233 H(i)←H(i−1)+e;H(i)←H(i−1)+f 4455 H(i)←H(i−1)+g;H(i)←H(i−1)+h 6677 end for end for return H(N−1) end function Related Work comparator As a result, miners naturally compete against each other to gain a higher fraction of the network’s hash rate in or- der to maximize reward. In a race to capture the network’s rewards, miners have developed increasingly sophisticated solutions, culminating in the development of Bitcoin ASIC accelerators [13]. A miner’s revenue is determined by the accelerator’s hash rate (GHash/s); operating costs are de- termined by its energy efficiency (GHash/J). The mining algorithm is shown in Algorithm 1. In short, mining is a search for the nonce value that results in a double SHA-256 hash digest (Algorithm 2) value less than a given threshold. The nonce is a 32-bit field within a 1024-bit block header. In order to verify transactions at a steady rate, this threshold varies over time as a function of difficulty D(t). Difficulty is adjusted by the network regularly such that a solution is expected to be found approximately every 10 minutes, regardless of the network’s collective hash rate. By their very nature, hash functions are designed to be non-invertible, so mining is performed by brute force, guess- ing nonce values and comparing the hash output. This task is perfectly parallel as multiple hashes may be computed at once. It follows that one’s probability of finding a solution is proportional to one’s hash rate. The first miner to find a valid nonce broadcasts the value on the network for verifica- tion and is rewarded with newly minted digital (bit)coins. Algorithm 1 Mining Process 1: nonce ← 0 2: while nonce < 232 do 3: threshold ← ((216 − 1) ≪ 208)/D(t) 4: digest ← SHA-256(SHA-256(header)) 5: if digest < threshold then 6: return nonce 7: else 8: nonce ← nonce + 1 9: end if 10: end while Algorithm 2 presents a basic description of SHA-256. For details on message padding, initial hash values H(0), and constants Kj, see [12]. • The message M is divided into N 512-bit blocks M(0),M(1),...,M(N−1). Each of these blocks is fur- thersubdividedinto1632-bitwordsM(i),M(i),...,M(i). 0 1 15 • The intermediate hash value H(i) is composed of 8 32- bit words H(i), H(i), . . . , H(i). • Ch(x,y,z) ≡ (x∧y)⊕(¬x∧z) • Maj(x, y, z) ≡ (x ∧ y) ⊕ (x ∧ z) ⊕ (y ∧ z) Although some work has been done to improve the per- formance of SHA-256 ASICs in context of other applica- tions [6] [10], no published research, to the best of our knowl- edge, attempts to optimize ASICs for Bitcoin mining. The most closely related work is by Courtois et al. [5] who ex- plore mining optimizations from an algorithmic perspective. Their central observation is that the first half of the block header (shown in gray in Figure 1) does not change across nonce iterations, so its hash may be precomputed. Since this precomputation cost is amortized across 232 nonce iter- ations, it halves the cost of the first SHA-256 round (shown within the dashed line in Figure 1). Our paper is the first work to explore hardware optimizations, specifically approx- imation-based optimizations theorized by [8], unique to Bit- coin mining2. 2.3 Baseline Hardware 2.3.1 Implementation For our studies, we selected as baseline the SHA-256 ASIC design outlined by Dadda et al. [6]. A summary of SHA-256 is provided in Algorithm 2. The hashing core in this design is implemented as two parallel pipelines, the Compressor (Line 9 of Algorithm 2) and the Expander (Line 3 of Algorithm 2) shown in Figure 2. The logic functions Ch, Maj, Σ0, Σ1, σ0, and σ1 in the figure are defined with Algorithm 2. 2A form of functional approximation was discussed in a re- cent blog post by Sergio Lerner [8]. 017

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