def sqrt_v

bend-collections-laws-crypto@1.0.0.0 · proofs/crypto/secp256k1/fieldpow.bend:336

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/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

336def sqrt_v(+one: Nat, +h1: {one == 1n : Nat}, +a: List<&2, Nat>, +ha: {FS.reduced(one, FS.prime(one), a) == True{} : Bool}) -> {FS.value(one, F.sqrt(a)) == FS.fsqrt(FS.prime(one), FS.value(one, a)) : Nat}:
337  %Equal.sym(Nat, FS.value(one, F.pow(a, F.sqrt_bits(), F.one())), LV.ev(F.pow(a, F.sqrt_bits(), F.one())), LV.ev_value(one, h1, F.pow(a, F.sqrt_bits(), F.one()))) : {_ == Nat.mod(Nat.pow(FS.value(one, a), Nat.div(Nat.add(FS.prime(one), 1n), 4n)), FS.prime(one)) : Nat}
338  %Equal.sym(Nat, LV.ev(F.pow(a, F.sqrt_bits(), F.one())), Nat.mod(Nat.pow(LV.ev(a), BV.hv(F.sqrt_bits(), one, 0n)), 1n+K.pp(one)), pow_ev(one, h1, a, ha, F.sqrt_bits(), F.one(), one_r(one, h1), 0n, one_e(one, h1, LV.ev(a)), sqrt_bits01(one))) : {_ == Nat.mod(Nat.pow(FS.value(one, a), Nat.div(Nat.add(FS.prime(one), 1n), 4n)), FS.prime(one)) : Nat}
339  %Equal.sym(Nat, BV.hv(F.sqrt_bits(), one, 0n), Nat.div(Nat.add(1n+K.pp(one), 1n), 4n), exp_sqrt(one, h1)) : {Nat.mod(Nat.pow(LV.ev(a), _), 1n+K.pp(one)) == Nat.mod(Nat.pow(FS.value(one, a), Nat.div(Nat.add(FS.prime(one), 1n), 4n)), FS.prime(one)) : Nat}
340  %Equal.sym(Nat, FS.value(one, a), LV.ev(a), LV.ev_value(one, h1, a)) : {Nat.mod(Nat.pow(LV.ev(a), Nat.div(Nat.add(1n+K.pp(one), 1n), 4n)), 1n+K.pp(one)) == Nat.mod(Nat.pow(_, Nat.div(Nat.add(FS.prime(one), 1n), 4n)), FS.prime(one)) : Nat}
341  %Equal.sym(Nat, FS.prime(one), 1n+K.pp(one), K.prime_eq(one, h1)) : {Nat.mod(Nat.pow(LV.ev(a), Nat.div(Nat.add(1n+K.pp(one), 1n), 4n)), 1n+K.pp(one)) == Nat.mod(Nat.pow(LV.ev(a), Nat.div(Nat.add(_, 1n), 4n)), FS.prime(one)) : Nat}
342  %Equal.sym(Nat, FS.prime(one), 1n+K.pp(one), K.prime_eq(one, h1)) : {Nat.mod(Nat.pow(LV.ev(a), Nat.div(Nat.add(1n+K.pp(one), 1n), 4n)), 1n+K.pp(one)) == Nat.mod(Nat.pow(LV.ev(a), Nat.div(Nat.add(1n+K.pp(one), 1n), 4n)), _) : Nat}
343  {==}