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ETHEREUM: A SECURE DECENTRALISED GENERALISED TRANSACTION LEDGER BERLIN VERSION 23 We define a binary operation + on C1 for distinct elements (X1, Y1), (X2, Y2) with (243) (X1, Y1) + (X2, Y2) ≡ λ≡ (X,Y) ifX1̸=X2 (0, 0) otherwise Y2 − Y1 X2 − X1 X ≡ λ2−X1−X2 Y ≡ λ(X1−X)−Y1 In the case where (X1, Y1) = (X2, Y2), we define + on C1 with (244) (X1, Y1) + (X2, Y2) ≡ λ≡ (X,Y) ifY1̸=0 X ≡ λ2−2X1 Y ≡ λ(X1−X)−Y1 (C1,+) is known to form a group. We define scalar multiplication · with (245) n·P ≡(0,0)+P +···+P n for a natural number n and a point P in C1. We define P1 to be a point (1,2) on C1. Let G1 be the subgroup of (C1,+) generated by P1. G1 is known to be a cyclic group of order q. For a point P in G1, we define logP1(P) to be the smallest natural number n satisfying n·P1 = P. logP1 (P ) is at most q − 1. Let Fp2 be a field Fp[i]/(i2 + 1). We define a set C2 with (246) C2 ≡ {(X,Y) ∈ Fp2 ×Fp2 | Y2 = X3 +3(i+9)−1}∪{(0,0)} We define a binary operation + and scalar multiplication · with the same equations (243), (244) and (245). (C2,+) is also known to be a group. We define P2 in C2 with (247) P2 ≡ (11559732032986387107991004021392285783925812861821192530917403151452391805634 × i +10857046999023057135944570762232829481370756359578518086990519993285655852781, 4082367875863433681332203403145435568316851327593401208105741076214120093531 × i +8495653923123431417604973247489272438418190587263600148770280649306958101930) We define G2 to be the subgroup of (C2,+) generated by P2. G2 is known to be the only cyclic group of order q on C2. ForapointP inG2,wedefinelogP2(P)bethesmallestnaturalnumbernsatisfyingn·P2 =P. Withthisdefinition, logP2 (P ) is at most q − 1. Let GT be the multiplicative abelian group underlying Fq12. It is known that a non-degenerate bilinear map e : G1 × G2 → GT exists. This bilinear map is a type three pairing. There are several such bilinear maps, it does not matterwhichischosentobee. LetPT =e(P1,P2),abeasetofkpointsinG1,andbbeasetofkpointsinG2. It follows from the definition of a pairing that the following are equivalent (248) logP1(a1)×logP2(b1)+···+logP1(ak)×logP2(bk) ≡ 1 k (249) e(ai,bi) = PT i=0 Thus the pairing operation provides a method to verify (248). A 32 byte number x ∈ P256 might and might not represent an element of Fp. g1 if g1 ∈ G1 ∅ otherwise (x,y) ifx̸=∅∧y̸=∅ ∅ otherwise δp (x[0..31]) δp (x[32..63]) mod q (250) δp(x) ≡ A 64 byte data x ∈ B512 might and might not represent an element of G1. (251) (252) (253) (254) δ1(x) ≡ g1 ≡ x ≡ y ≡ (0, 0) 3X12 2Y1 otherwise x ifxPDF Image | ETHEREUM: A SECURE DECENTRALISED GENERALISED TRANSACTION
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