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(201) c(I, i) ≡ u(j) ≡ ETHEREUM: A SECURE DECENTRALISED GENERALISED TRANSACTION LEDGER BERLIN VERSION 21 suffix, two optimising nodes are provided. Thus while traversing, one may potentially acquire multiple nibbles from each of the other two node types, extension and leaf. There are three kinds of nodes in the trie: Leaf: A two-item structure whose first item corresponds to the nibbles in the key not already accounted for by the accumulation of keys and branches traversed from the root. The hex-prefix encoding method is used and the second parameter to the function is required to be 1. Extension: A two-item structure whose first item corresponds to a series of nibbles of size greater than one that are shared by at least two distinct keys past the accumulation of the keys of nibbles and the keys of branches as traversed from the root. The hex-prefix encoding method is used and the second parameter to the function is required to be 0. Branch: A 17-item structure whose first sixteen items correspond to each of the sixteen possible nibble values for the keys at this point in their traversal. The 17th item is used in the case of this being a terminator node and thus a key being ended at this point in its traversal. A branch is then only used when necessary; no branch nodes may exist that contain only a single non-zero entry. We may formally define this structure with the structural composition function c: HP(I0[i..(∥I0∥ − 1)],1),I1 HP(I0[i..(j −1)],0),n(I,j) if ∥I∥ = 1 where ∃I : I ∈ I if i ̸= j otherwise where j = max{x : ∃l : ∥l∥ = x∧∀I ∈ I : I0[0..(x−1)] = l} n({I : I ∈ I ∧ I0[i] = j}, i + 1) I1 if ∃I:I∈I∧∥I0∥=i () otherwise (u(0), u(1), ..., u(15), v) v= where D.1. Trie Database. Thus no explicit assumptions are made concerning what data is stored and what is not, since that is an implementation-specific consideration; we simply define the identity function mapping the key-value set I to a 32-byte hash and assert that only a single such hash exists for any I, which though not strictly true is accurate within acceptable precision given the Keccak hash’s collision resistance. In reality, a sensible implementation will not fully recompute the trie root hash for each set. A reasonable implementation will maintain a database of nodes determined from the computation of various tries or, more formally, it will memoise the function c. This strategy uses the nature of the trie to both easily recall the contents of any previous key-value set and to store multiple such sets in a very efficient manner. Due to the dependency relationship, Merkle-proofs may be constructed with an O(logN) space requirement that can demonstrate a particular leaf must exist within a trie of a given root hash. Appendix E. Precompiled Contracts For each precompiled contract, we make use of a template function, ΞPRE, which implements the out-of-gas checking. (∅,0,A,()) if g < gr (202) ΞPRE(σ, g, A, I) ≡ (σ, g − gr, A, o) otherwise The precompiled contracts each use these definitions and provide specifications for the o (the output data) and gr, the gas requirements. We define ΞECREC as a precompiled contract for the elliptic curve digital signature algorithm (ECDSA) public key recovery function (ecrecover). See Appendix F for the definition of the function ECDSARECOVER. We also define d to be the input data, well-defined for an infinite length by appending zeroes as required. In the case of an invalid signature (ECDSARECOVER(h, v, r, s) = ∅), we return no output. (203) (204) (205) (206) (207) (208) (209) (210) (211) (212) (213) (214) ΞECREC ≡ gr = ∥o∥ = if ∥o∥=32: o[0..11] = o[12..31] = d[0..(∥Id∥ − 1)] = d[∥Id ∥..] = h = v = r = s = ΞPRE where: 3000 0 if ECDSARECOVER(h, v, r, s) = ∅ 32 otherwise 0 KECECDSARECOVER(h, v, r, s)[12..31] Id (0, 0, ...) d[0..31] d[32..63] d[64..95] d[96..127] where: We define ΞSHA256 and ΞRIP160 as precompiled contracts implementing the SHA2-256 and RIPEMD-160 hash functions respectively. Their gas usage is dependent on the input data size, a factor rounded up to the nearest number of words.PDF Image | ETHEREUM: A SECURE DECENTRALISED GENERALISED TRANSACTION
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