def subg_mod

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/curve25519/fieldops.bend:155

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/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/reduce.bend as RD
import bend-collections-laws-crypto@1.0.0.0/proofs/crypto/curve25519/consts.bend as K

155def subg_mod(+one: Nat, +h1: {one == 1n : Nat}, +pp: Nat, +hP: {Nat.add(1n+pp, 19n) == WD.sc(255n, one) : Nat}, +a: List<&2, U32>, +b: List<&2, U32>, +ks: List<&2, U32>, +ha: {LM.okb(32n, a, 255n) == True{} : Bool}, +hb: {LM.okb(32n, b, 255n) == True{} : Bool}, +hk: {LM.okb(32n, ks, 2040n) == True{} : Bool}, +hg: {gea(ks, 255n) == True{} : Bool}, +hv: {K.valo(one, ks) == Nat.mul(8n, 1n+pp) : Nat}) -> {Nat.mod(FS.value(F.reduce(F.subl(a, b, ks))), 1n+pp) == Nat.mod(Nat.sub(Nat.add(FS.value(a), 1n+pp), Nat.mod(FS.value(b), 1n+pp)), 1n+pp) : Nat}:
156  +e1 = RD.reduce_mod(one, h1, pp, hP, F.subl(a, b, ks), subg_okb(one, h1, a, b, ks, ha, hb, hk, hg))
157  +es = subl_val(one, h1, 32n, a, b, ks, 255n, 255n, 2040n, ha, hb, hk, hg, lt32_small(one, h1, Nat.add(255n, 2040n), {==}))
158  +ek = Equal.trans(Nat, FS.value(ks), K.valo(one, ks), Nat.mul(8n, 1n+pp), Equal.sym(Nat, K.valo(one, ks), FS.value(ks), K.valo_value(one, h1, ks)), hv)
159  +hS = Equal.trans(Nat, Nat.add(FS.value(F.subl(a, b, ks)), FS.value(b)), Nat.add(FS.value(a), FS.value(ks)), Nat.add(FS.value(a), Nat.mul(8n, 1n+pp)), es, M.cong_r(FS.value(a), FS.value(ks), Nat.mul(8n, 1n+pp), ek))
160  Equal.trans(Nat, Nat.mod(FS.value(F.reduce(F.subl(a, b, ks))), 1n+pp), Nat.mod(FS.value(F.subl(a, b, ks)), 1n+pp), Nat.mod(Nat.sub(Nat.add(FS.value(a), 1n+pp), Nat.mod(FS.value(b), 1n+pp)), 1n+pp), e1, sub_core(pp, FS.value(a), FS.value(b), FS.value(F.subl(a, b, ks)), hS))