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(285) (286) (287) (288) (289) (290) (291) (292) (293) (294) (295) (296) (297) (298) (299) ΞBLAKE2 ΞBLAKE2 F F(σ, g, A, I) gr o (h′0,...,h′7) BE4(r) LE8 (h0 ) LE8 (h7 ) LE8 (m0 ) LE8 (m15 ) LE8 (tlow ) LE8 (thigh ) f ≡ ΞPRE except: = (∅,0,A,()) if∥Id∥≠ 213∨f∈/{0,1} = r ≡ LE8(h′0) · ... · LE8(h′7) ≡ F(h, m, tlow, thigh, f) ≡ Id [0..4] ≡ Id [4..12] ... ≡ Id [60..68] ≡ Id [68..76] ... ≡ Id [188..196] ≡ Id [196..204] ≡ Id [204..212] ≡ Id [212] ETHEREUM: A SECURE DECENTRALISED GENERALISED TRANSACTION LEDGER BERLIN VERSION 25 function is specified in RFC 7693 by Saarinen and Aumasson [2015]. wherer∈B32,∀i∈0..7:hi ∈B64,∀i∈0..15:mi ∈B64,tlow ∈B64,thigh ∈B64,f∈B8,BEk isthek-bytebig-endian representation—compare with(188): k−1 (300) BEk(x) ≡ (b0, b1, ..., bk−1) : x = bn · 256k−1−n n=0 and LEk is the k-byte little-endian representation: (301) LEk(x) ≡ (b0, b1, ..., bk−1) : x = bn · 256n Appendix F. Signing Transactions Transactions are signed using recoverable ECDSA signatures. This method utilises the SECP-256k1 curve as described by Courtois et al. [2014], and is implemented similarly to as described by Gura et al. [2004] on p. 9 of 15, para. 3. It is assumed that the sender has a valid private key pr, which is a randomly selected positive integer (represented as a byte array of length 32 in big-endian form) in the range [1, secp256k1n − 1]. We assume the existence of functions ECDSAPUBKEY, ECDSASIGN and ECDSARECOVER. These are formally defined in the literature, e.g. by Johnson et al. [2001]. (302) (303) (304) ECDSAPUBKEY(pr ∈ B32) ≡ ECDSASIGN(e ∈ B32,pr ∈ B32) ≡ ECDSARECOVER(e ∈ B32,v ∈ B1,r ∈ B32,s ∈ B32) ≡ pu ∈ B64 (v ∈ B1,r ∈ B32,s ∈ B32) pu ∈ B64 Where pu is the public key, assumed to be a byte array of size 64 (formed from the concatenation of two positive integers each < 2256), pr is the private key, a byte array of size 32 (or a single positive integer in the aforementioned range) and e is the hash of the transaction, h(T ). It is assumed that v is the ‘recovery identifier’. The recovery identifier is a 1 byte value specifying the parity and finiteness of the coordinates of the curve point for which r is the x-value; this value is in the range of [27, 30], however we declare the upper two possibilities, representing infinite values, invalid. The value 27 represents an even y value and 28 represents an odd y value. We declare that an ECDSA signature is invalid unless all the following conditions are true6: (305) (306) (307) where: (308) 0 < r < secp256k1n 0 < s < secp256k1n÷2+1 v ∈ {27, 28} secp256k1n = 115792089237316195423570985008687907852837564279074904382605163141518161494337 For a given private key, pr, the Ethereum address A(pr) (a 160-bit value) to which it corresponds is defined as the rightmost 160-bits of the Keccak hash of the corresponding ECDSA public key: (309) A(pr) = B96..255KECECDSAPUBKEY(pr) 6A signature of a transaction can be valid not only with a recovery identifier but with some other numbers. See how the component Tw of a transaction is interpreted. with r rounds and w = 64 k−1 n=0PDF Image | ETHEREUM: A SECURE DECENTRALISED GENERALISED TRANSACTION
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