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(31) Hr Ho ≡ ETHEREUM: A SECURE DECENTRALISED GENERALISED TRANSACTION LEDGER BERLIN VERSION 6 We define the Bloom filter function, M, to reduce a log entry into a single 256-byte hash: The component types are defined thus: (26) M(O) ≡ x∈{Oa }∪Ot M3:2048(x) (37) Hp ∈ B32 Hr ∈ B32 Hb∈B256 Hl ∈ N Hx ∈ B where (38) Bn = ∧Ho∈B32 ∧Ht∈B32 ∧Hd∈N ∧ Hg∈N ∧ Hm∈B32 ∧Hc∈B20∧ ∧He∈B32∧ ∧Hi∈N ∧ ∧Hs∈N256∧ ∧ Hn∈B8 where M3:2048 is a specialised Bloom filter that sets three bits out of 2048, given an arbitrary byte sequence. It does this through taking the low-order 11 bits of each of the first three pairs of bytes in a Keccak-256 hash of the byte sequence.3 Formally: {B : B ∈ B ∧ ∥B∥ = n} We now have a rigorous specification for the construc- (27)M3:2048(x : x ∈ B) ≡ (28) y= (29) ∀i ∈ {0, 2, 4} : (30) m(x,i) ≡ y : y ∈ B256 (0, 0, ..., 0) where: except: tion of a formal block structure. The RLP function RLP (see Appendix B) provides the canonical method for trans- forming this structure into a sequence of bytes ready for transmission over the wire or storage locally. 4.3.4. Block Header Validity. We define P(BH) to be the parent block of B, formally: (39) P (H) ≡ B′ : KEC(RLP(BH′ )) = Hp The block number is the parent’s block number incre- mented by one: (40) Hi≡P(H)Hi+1 The canonical difficulty of a block of header H is defined as D(H): Ht ≡ He ≡ TRIE({∀i<∥BT∥,i∈N: p(i, LT (BT [i]))}) TRIE({∀i<∥BR∥,i∈N: p(i, BR [i])}) ≡ TRIE(LS (Π(σ, B))) KEC(RLP(L∗H (BU ))) Bm(x,i) (y) = 1 KEC(x)[i, i + 1] mod 2048 where B is the bit reference function such that Bj(x) equals the bit of index j (indexed from 0) in the byte array x. 4.3.2. Holistic Validity. We can assert a block’s validity if and only if it satisfies several conditions: it must be in- ternally consistent with the ommer and transaction block hashes and the given transactions BT (as specified in sec 11), when executed in order on the base state σ (derived from the final state of the parent block), result in a new state of the identity Hr: (41) D0 D(H)≡ maxD0,P(H)Hd +x×ς2 +ε where: (42) D0 ≡ 131072 (43) x≡ P(H)Hd 2048 ∧ ∧ ∧ ∧ Hb ≡ where p(k, v) is simply the pairwise RLP transformation, in this case, the first being the index of the transaction in the block and the second being the transaction receipt: (32) p(k, v) ≡ RLP(k), RLP(v) Furthermore: (33) TRIE(LS(σ)) = P (BH )H r (45) Thus TRIE(LS(σ)) is the root node hash of the Merkle Patricia tree structure containing the key-value pairs of the state σ with values encoded using RLP, and P(BH) is the parent block of B, defined directly. (46) if Hi=0 otherwise ,−99 rb r∈BR The values stemming from the computation of transac- tions, specifically the transaction receipts, BR, and that defined through the transaction’s state-accumulation func- tion, Π, are formalised later in section 11.4. Hi′ ≡ (47) max(Hi − κ, 0) 4.3.3. Serialisation. The function LB and LH are the prepa- κ≡ ration functions for a block and block header respectively. We assert the types and order of the structure for when the RLP transformation is required: 3000000 5000000 9000000 9700000 if FByzantium Hi < FConstantinople if FConstantinople Hi < FMuirGlacier if FMuirGlacier Hi < FLondon if Hi FLondon (34) (35) LH(H) ≡ LB(B) ≡ ( Hp, Ho, Hc, Hr, Ht, He, Hb, Hd, Hi,Hl,Hg,Hs,Hx,Hm,Hn ) LH(BH), L∗T(BT), L∗H(BU) Note that D0 is the difficulty of the genesis block. The Homestead difficulty parameter, ς2, is used to affect a dy- namic homeostasis of time between blocks, as the time between blocks varies, as discussed below, as implemented in EIP-2 by Buterin [2015]. In the Homestead release, the exponential difficulty symbol, ε causes the difficulty to slowly increase (every 100,000 blocks) at an exponential rate, and thus increasing the block time difference, and putting time pressure on transitioning to proof-of-stake. With L∗T and L∗H being element-wise sequence transfor- mations, thus: (36) f∗(x0, x1, ...) ≡ f(x0), f(x1), ... for any function f (44) ς2≡max y− Hs −P(H)Hs 9 ε≡ 2⌊Hi′ ÷100000⌋−2 y≡ 1 if∥P(H)U∥=0 2 otherwise 311 bits = 22048, and the low-order 11 bits is the modulo 2048 of the operand, which is in this case is “each of the first three pairs of bytes in a Keccak-256 hash of the byte sequence.”PDF Image | ETHEREUM: A SECURE DECENTRALISED GENERALISED TRANSACTION
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