def pow_e

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/secp256k1/fieldpow.bend:62

import Base
import bend-collections-laws-crypto@1.0.0.0/spec/crypto/secp256k1/field.bend as FS
import bend-collections-laws-crypto@1.0.0.0/src/crypto/secp256k1/field.bend as F
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/logic.bend as Lg
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/limbs.bend as LV
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/consts.bend as K
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/secp256k1/bitsv.bend as BV

62# x^e for the exponent spelled by the bits, from acc = x^e0
63def pow_e(+one: Nat, +h1: {one == 1n : Nat}, +x: List<&2, Nat>, +hx: {FS.reduced(one, FS.prime(one), x) == True{} : Bool}, bs: List<&2, Nat>, +acc: List<&2, Nat>, +ha: {FS.reduced(one, FS.prime(one), acc) == True{} : Bool}, +e: Nat, +he: {LV.ev(acc) == Nat.mod(Nat.pow(LV.ev(x), e), 1n+K.pp(one)) : Nat}, +hb: {BV.all01(bs) == True{} : Bool}) -> {LV.ev(F.pow_go(x, bs, acc)) == Nat.mod(Nat.pow(LV.ev(x), BV.hv(bs, one, e)), 1n+K.pp(one)) : Nat}:
64  match bs:
65    case Nil{}:
66      he
67    case +b <> +t:
68      +h0 = Lg.and_left(Nat.is_lt(b, 2n), BV.all01(t), hb)
69      pow_e(one, h1, x, hx, t, F.pow_step(b, x, F.sq(acc)), step_r(one, h1, x, hx, acc, ha, b, h0), Nat.add(Nat.mul(b, one), Nat.double(e)), step_e(one, h1, x, hx, acc, ha, e, he, b, h0), Lg.and_right(Nat.is_lt(b, 2n), BV.all01(t), hb))