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ETHEREUM: A SECURE DECENTRALISED GENERALISED TRANSACTION LEDGER BERLIN VERSION 40 Where a single round modifies each subset of the cache as follows: (338) ERMH (x) = Ermh (x, 0), Ermh (x, 1), ..., Ermh (x, n − 1) (339) Ermh(x, i) = KEC512(x′[(i − 1 + n) mod n] ⊕ x′[x′[i][0] mod n]) with x′ = x except x′[j] = Ermh(x,j) ∀ j < i J.3.3. Full dataset calculation. Essentially, we combine data from Jparents pseudorandomly selected cache nodes, and hash that to compute the dataset. The entire dataset is then generated by a number of items, each Jhashbytes bytes in size: (340) d[i] = Edatasetitem(c,i) ∀ i < In order to calculate the single item we use an algorithm inspired by the FNV hash (Glenn Fowler [1991]) in some cases dsize Jhashbytes as a non-associative substitute for XOR. (341) EFNV (x, y) = (x · (0x01000193 ⊕ y)) mod 232 The single item of the dataset can now be calculated as: (342) Edatasetitem (c, i) = Eparents (c, i, −1, ∅) Eparents(c,i,p + 1,Emix(m,c,i,p + 1)) if p < Jparents − 2 otherwise J.4. Proof-of-work function. Essentially, we maintain a “mix” Jmixbytes bytes wide, and repeatedly sequentially fetch Jmixbytes bytes from the full dataset and use the EFNV function to combine it with the mix. Jmixbytes bytes of sequential access are used so that each round of the algorithm always fetches a full page from RAM, minimizing translation lookaside buffer misses which ASICs would theoretically be able to avoid. If the output of this algorithm is below the desired target, then the nonce is valid. Note that the extra application of KEC at the end ensures that there exists an intermediate nonce which can be provided to prove that at least a small amount of work was done; this quick outer PoW verification can be used for anti-DDoS purposes. It also serves to provide statistical assurance that the result is an unbiased, 256 bit number. The PoW-function returns an array with the compressed mix as its first item and the Keccak-256 hash of the concatenation of the compressed mix with the seed hash as the second item: (345) PoW(Hn, Hn, d) = {mc(KEC(RLP(LH(Hn))), Hn, d), KEC(sh(KEC(RLP(LH(Hn))), Hn) + mc(KEC(RLP(LH(Hn))), Hn, d))} With Hn being the hash of the header without the nonce. The compressed mix mc is obtained as follows: (343) (344) Eparents (c, i, p, m) = KEC512(c[i mod csize] ⊕ i) (350) Eaccesses (d, m, s, i) = if i = Jaccesses − 2 Eaccesses (Emixdataset (d, m, s, i), s, i + 1) otherwise Enewdata returns an array with nmix elements: (352) Enewdata(d, m, s, i)[j] = d[EFNV(i ⊕ s[0], m[i mod ]) mod dsize/Jhashbytes nmix Emix (m, c, i, p + 1) if p = 0 Emix(m, c, i, p) = EFNVm, c[EFNV(i ⊕ p, m[p mod ⌊Jhashbytes/Jwordbytes⌋]) mod csize] otherwise nmix (346) mc (h, n, d) = Ecompress (Eaccesses (d, sh (h, n), sh (h, n), −1), −4) i=0 The seed hash being: (347) sh (h, n) = KEC512(h + Erevert (n)) Erevert(n) returns the reverted bytes sequence of the nonce n: (348) Erevert(n)[i] = n[∥n∥ − i] We note that the “+”-operator between two byte sequences results in the concatenation of both sequences. The dataset d is obtained as described in section J.3.3. The number of replicated sequences in the mix is: Jmixbytes (349) nmix = Jhashbytes In order to add random dataset nodes to the mix, the Eaccesses function is used: Emixdataset (d, m, s, i) (351) Emixdataset (d, m, s, i) = EFNV (m, Enewdata (d, m, s, i)) Jmixbytes Jwordbytes · nmix + j] ∀ j < nmixPDF Image | ETHEREUM: A SECURE DECENTRALISED GENERALISED TRANSACTION
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