def mul_small_mod

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/curve25519/mulw.bend:179

import Base
import bend-collections-laws-crypto@1.0.0.0/spec/crypto/curve25519/field.bend as FS
import bend-collections-laws-crypto@1.0.0.0/src/crypto/curve25519/field.bend as F
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/word.bend as WD
import bend-collections-laws-crypto@1.0.0.0/proofs/lib/lemmas/proofs/nat_algebra.bend as NA
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/num.bend as M
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/limbs.bend as LM
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/poly.bend as PL
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/reduce.bend as RD

179def mul_small_mod(+one: Nat, +h1: {one == 1n : Nat}, +pp: Nat, +hP: {Nat.add(1n+pp, 19n) == WD.sc(255n, one) : Nat}, +a: List<&2, U32>, +k: U32, +ha: {LM.okb(32n, a, 255n) == True{} : Bool}, +hk: {Nat.is_le(v(k), WD.sc(17n, one)) == True{} : Bool}) -> {Nat.mod(FS.value(F.mul_small(a, k)), 1n+pp) == Nat.mod(Nat.mul(FS.value(a), v(k)), 1n+pp) : Nat}:
180  +e1 = RD.reduce_mod(one, h1, pp, hP, F.scal(k, a), scal_in(one, h1, a, k, ha, hk))
181  +e2 = PL.scal_val(one, h1, k, a, WD.sc(17n, one), 255n, hk, LM.okb_lea(32n, a, 255n, ha), ms_lt(one, h1))
182  +e3 = Equal.trans(Nat, FS.value(F.scal(k, a)), Nat.mul(v(k), FS.value(a)), Nat.mul(FS.value(a), v(k)), e2, NA.mul_comm(v(k), FS.value(a)))
183  Equal.trans(Nat, Nat.mod(FS.value(F.mul_small(a, k)), 1n+pp), Nat.mod(FS.value(F.scal(k, a)), 1n+pp), Nat.mod(Nat.mul(FS.value(a), v(k)), 1n+pp), e1, M.cong_mod(1n+pp, FS.value(F.scal(k, a)), Nat.mul(FS.value(a), v(k)), e3))