src/crypto/secp256k1/ecdsa.bend source

src/crypto/secp256k1/ecdsa.bend on the hub · documented module

import Baseimport ../mac.bend as MACimport ../keccak/keccak.bend as Kimport ./limbs.bend as Limport ./field.bend as Fimport ./scalar.bend as Simport ./point.bend as Pimport ./bytes.bend as B# ECDSA over secp256k1 (SEC 1 v2 section 4.1) with deterministic nonces# (RFC 6979 section 3.2, HMAC-SHA256) and low-S signatures (BIP 62 /# Ethereum's homestead rule, libsecp256k1's secp256k1_ecdsa_sign), public# key recovery (SEC 1 section 4.1.6, Ethereum's ecrecover) and the Ethereum# address of a public key. The contract is spec/crypto/secp256k1/ecdsa.bend.## Messages are 32-byte hashes (the caller hashes); keys and signatures are# byte lists of their SEC 1 lengths, malformed ones are rejected as values.type Signature is Data:  Signature{r: List<&2, U32>, s: List<&2, U32>, v: U32}# ---- scalars from bytes ----# 1 <= x < ndef scalar_ok(+x: List<&2, Nat>) -> Bool:  Bool.and(Bool.not(S.is_zero(x)), S.lt_n(x))# a 32-byte string as a scalar mod n (bits2int with qlen = hlen = 256,# reduced: SEC 1 section 4.1.3 step 5, RFC 6979 section 2.3.2)def hash_scalar(+h: List<&2, U32>) -> List<&2, Nat>:  S.reduce(B.of_be(h))# ---- RFC 6979 section 3.2, HMAC_DRBG with SHA-256 ----type Drbg is Data:  Drbg{k: List<&2, U32>, v: List<&2, U32>}# (the key is looked at first, so that the proof checker keeps an unknown# HMAC folded)def hmac(key: List<&2, U32>, msg: List<&2, U32>) -> List<&2, U32>:  match key:    case Nil{}: MAC.sign(Nil{}, msg)    case k <> t: MAC.sign(k <> t, msg)def cat(xs: List<&2, U32>, ys: List<&2, U32>) -> List<&2, U32>:  List.append(&2, U32, xs, ys)def fill(n: Nat, +b: U32) -> List<&2, U32>:  List.replicate(U32, n, b)def drbg_v(+k: List<&2, U32>, +v: List<&2, U32>) -> Drbg:  Drbg{k, hmac(k, v)}# steps b-g: V = 0x01..., K = 0x00..., then two HMAC rounds over# V || 0x00 || int2octets(x) || bits2octets(h1) and V || 0x01 || ...def drbg_step(+k: List<&2, U32>, +v: List<&2, U32>, +tag: U32, +seed: List<&2, U32>) -> Drbg:  drbg_v(hmac(k, cat(v, tag <> seed)), v)def drbg_second(d: Drbg, +seed: List<&2, U32>) -> Drbg:  match d:    case Drbg{k, v}: drbg_step(k, v, 1, seed)def drbg_init(+seed: List<&2, U32>) -> Drbg:  drbg_second(drbg_step(fill(32n, 0), fill(32n, 1), 0, seed), seed)# step h.3: K = HMAC_K(V || 0x00), V = HMAC_K(V)def drbg_next(+k: List<&2, U32>, +v: List<&2, U32>) -> Drbg:  drbg_v(hmac(k, cat(v, [0])), v)# ---- signing ----def b2n(b: Bool) -> Nat:  L.b2n(b)type Attempt is Data:  ARetry{}  ADone{sig: Signature}def finish_if(bad: Bool, +sig: Signature) -> Attempt:  match bad:    case True{}: ARetry{}    case False{}: ADone{sig}# r = x(R) mod n, s = k^-1 (z + r d) mod n, made low (s > n / 2 is replaced# by n - s, which flips the parity bit of the recovery id); the recovery# id is the parity of y(R), plus 2 when x(R) >= ndef finish(+z: List<&2, Nat>, +d: List<&2, Nat>, +k: List<&2, Nat>, +x: List<&2, Nat>, +y: List<&2, Nat>) -> Attempt:  +r = S.reduce(x)  +s0 = S.mul(S.inv(k), S.add(z, S.mul(r, d)))  +high = b2n(S.is_high(s0))  +par = F.parity(y)  +id = Nat.add(Nat.mul(b2n(Bool.not(S.lt_n(x))), 2n), Nat.sub(Nat.add(par, high), Nat.mul(Nat.mul(par, high), 2n)))  finish_if(Bool.or(S.is_zero(r), S.is_zero(s0)), Signature{B.to_be(r), B.to_be(S.select(high, S.neg(s0), s0)), U32.from_nat(id)})def attempt_aff(+z: List<&2, Nat>, +d: List<&2, Nat>, +k: List<&2, Nat>, a: P.Affine) -> Attempt:  match a:    case P.Affine{x, y}: finish(z, d, k, x, y)def attempt_ok(+z: List<&2, Nat>, +d: List<&2, Nat>, +k: List<&2, Nat>, ok: Bool) -> Attempt:  match ok:    case True{}: attempt_aff(z, d, k, P.to_affine(P.mul(k, P.g())))    case False{}: ARetry{}# the candidate k = bits2int(V) (RFC 6979 step h.3): used when 1 <= k < ndef attempt(+z: List<&2, Nat>, +d: List<&2, Nat>, +v: List<&2, U32>) -> Attempt:  +k = B.of_be(v)  attempt_ok(z, d, k, scalar_ok(k))# step h: the candidate V; a k outside [1, n - 1], or one giving r = 0 or# s = 0 (SEC 1 section 4.1.3 steps 3 and 6), is replaced by the next# candidate: K = HMAC_K(V || 0x00), V = HMAC_K(V), V = HMAC_K(V). The loop# stops after `fuel` candidates (the chance that even 2 are needed is below# 2^-127).def sign_loop(fuel: Nat, +z: List<&2, Nat>, +d: List<&2, Nat>, +k: List<&2, U32>, +v: List<&2, U32>, att: Attempt) -> Maybe<&2, Signature>:  match fuel att:    case 0n ARetry{}: None{}    case 0n ADone{sig}: Some{sig}    case 1n+f ARetry{}:      +k2 = hmac(k, cat(v, [0]))      +v3 = hmac(k2, hmac(k2, v))      sign_loop(f, z, d, k2, v3, attempt(z, d, v3))    case 1n+f ADone{sig}: Some{sig}def sign_drbg(+z: List<&2, Nat>, +d: List<&2, Nat>, g: Drbg) -> Maybe<&2, Signature>:  match g:    case Drbg{+k, +v}:      +v1 = hmac(k, v)      sign_loop(16n, z, d, k, v1, attempt(z, d, v1))def secret_if(+d: List<&2, Nat>, ok: Bool) -> Maybe<&2, List<&2, Nat>>:  match ok:    case True{}: Some{d}    case False{}: None{}# the secret key d, when 1 <= d < ndef secret(+sk: List<&2, U32>) -> Maybe<&2, List<&2, Nat>>:  +d = B.of_be(sk)  secret_if(d, Bool.and(B.has_len(32n, sk), scalar_ok(d)))# sign with a valid secret scalar d: x = int2octets(d),# h1' = bits2octets(h) = int2octets(bits2int(h) mod n)def sign_d(+h: List<&2, U32>, m: Maybe<&2, List<&2, Nat>>) -> Maybe<&2, Signature>:  match m:    case None{}: None{}    case Some{+d}:      +z = hash_scalar(h)      sign_drbg(z, d, drbg_init(cat(B.to_be(d), B.to_be(z))))def sign_h(+sk: List<&2, U32>, +h: List<&2, U32>, ok: Bool) -> Maybe<&2, Signature>:  match ok:    case True{}: sign_d(h, secret(sk))    case False{}: None{}# the low-S signature of a 32-byte hash under a 32-byte secret keydef sign(+sk: List<&2, U32>, +h: List<&2, U32>) -> Maybe<&2, Signature>:  sign_h(sk, h, B.has_len(32n, h))# ---- public keys ----def public_point(+d: List<&2, Nat>) -> P.Point:  P.mul(d, P.g())def pk_of(compressed: Bool, +q: P.Point) -> List<&2, U32>:  match compressed:    case True{}: P.encode_compressed(q)    case False{}: P.encode_uncompressed(q)def public_key_d(compressed: Bool, m: Maybe<&2, List<&2, Nat>>) -> Maybe<&2, List<&2, U32>>:  match m:    case None{}: None{}    case Some{d}: Some{pk_of(compressed, public_point(d))}# the SEC 1 encoding of [d] G (33 bytes compressed, 65 uncompressed)def public_key(+sk: List<&2, U32>, compressed: Bool) -> Maybe<&2, List<&2, U32>>:  public_key_d(compressed, secret(sk))# ---- verification (SEC 1 section 4.1.4) ----# u1 = z w, u2 = r w with w = s^-1; R = [u1] G + [u2] Q must not be the# point at infinity and x(R) mod n must be rdef verify_rs(+q: P.Point, +z: List<&2, Nat>, +r: List<&2, Nat>, +s: List<&2, Nat>) -> Bool:  +w = S.inv(s)  +rr = P.add(P.mul(S.mul(z, w), P.g()), P.mul(S.mul(r, w), q))  Bool.and(Bool.not(P.is_inf(rr)), S.eq(S.reduce(P.aff_x(P.to_affine(rr))), r))def verify_ok(+q: P.Point, +z: List<&2, Nat>, +r: List<&2, Nat>, +s: List<&2, Nat>, ok: Bool) -> Bool:  match ok:    case True{}: verify_rs(q, z, r, s)    case False{}: False{}def verify_q(+h: List<&2, U32>, +sig: List<&2, U32>, strict: Bool, m: Maybe<&2, P.Point>) -> Bool:  match m:    case None{}: False{}    case Some{+q}:      +r = B.of_be(B.prefix(32n, sig))      +s = B.of_be(B.suffix(32n, sig))      verify_ok(q, hash_scalar(h), r, s, Bool.and(Bool.and(scalar_ok(r), scalar_ok(s)), Bool.or(Bool.not(strict), Bool.not(S.is_high(s)))))def verify_len(+pk: List<&2, U32>, +h: List<&2, U32>, +sig: List<&2, U32>, strict: Bool, ok: Bool) -> Bool:  match ok:    case True{}: verify_q(h, sig, strict, P.decode(pk))    case False{}: False{}def verify_with(+pk: List<&2, U32>, +h: List<&2, U32>, +sig: List<&2, U32>, strict: Bool) -> Bool:  verify_len(pk, h, sig, strict, Bool.and(B.has_len(32n, h), B.has_len(64n, sig)))# SEC 1 verification of a 64-byte r || s against a SEC 1 public key (33 or# 65 bytes); high s is accepted, as SEC 1 and OpenSSL dodef verify(+pk: List<&2, U32>, +h: List<&2, U32>, +sig: List<&2, U32>) -> Bool:  verify_with(pk, h, sig, False{})# the same, rejecting s > n / 2 (Bitcoin's BIP 62/146 LOW_S rule,# Ethereum's homestead rule for transactions)def verify_strict(+pk: List<&2, U32>, +h: List<&2, U32>, +sig: List<&2, U32>) -> Bool:  verify_with(pk, h, sig, True{})# ---- recovery (SEC 1 section 4.1.6, Ethereum ecrecover) ----def recover_q(+q: P.Point, inf: Bool) -> Maybe<&2, List<&2, U32>>:  match inf:    case True{}: None{}    case False{}: Some{P.encode_uncompressed(q)}# Q = r^-1 (s R - z G), with R the point of x-coordinate r + n j (j = id / 2)# and y parity id mod 2def recover_r(+z: List<&2, Nat>, +r: List<&2, Nat>, +s: List<&2, Nat>, m: Maybe<&2, P.Point>) -> Maybe<&2, List<&2, U32>>:  match m:    case None{}: None{}    case Some{+pr}:      +ri = S.inv(r)      +q = P.add(P.mul(S.mul(S.neg(z), ri), P.g()), P.mul(S.mul(s, ri), pr))      recover_q(q, P.is_inf(q))def recover_x(+z: List<&2, Nat>, +r: List<&2, Nat>, +s: List<&2, Nat>, +id: Nat, +x: List<&2, Nat>, ok: Bool) -> Maybe<&2, List<&2, U32>>:  match ok:    case True{}: recover_r(z, r, s, P.decompress(x, Nat.mod(id, 2n)))    case False{}: None{}# c_n = 2^256 - n and c_p = 2^256 - p as 16-limb field elementsdef cn_fe() -> List<&2, Nat>:  [48831n, 12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n, 0n, 0n, 0n, 0n, 0n, 0n, 0n]def cp_fe() -> List<&2, Nat>:  [977n, 0n, 1n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n]# (r + n) mod p = (r - c_n + 2^256) mod p = (r - c_n + c_p) mod pdef rn_fe(+r: List<&2, Nat>) -> List<&2, Nat>:  F.add(F.sub(r, cn_fe()), cp_fe())# x = r + j n with j = id / 2. For j = 1, r + n must be below p: then# (r + n) mod p = r + n >= n, while a wrapped r + n - p is below r < ndef recover_ok(+z: List<&2, Nat>, +r: List<&2, Nat>, +s: List<&2, Nat>, +id: Nat, ok: Bool) -> Maybe<&2, List<&2, U32>>:  match ok:    case True{}:      +j = Nat.div(id, 2n)      +x1 = rn_fe(r)      recover_x(z, r, s, id, F.select(j, x1, r), Bool.or(Nat.is_eq(j, 0n), Bool.not(S.lt_n(x1))))    case False{}: None{}def recover_sig(+h: List<&2, U32>, +sig: List<&2, U32>) -> Maybe<&2, List<&2, U32>>:  +r = B.of_be(B.prefix(32n, sig))  +s = B.of_be(B.prefix(32n, B.suffix(32n, sig)))  +id = U32.to_nat(B.head(B.suffix(64n, sig)))  recover_ok(hash_scalar(h), r, s, id, Bool.and(Bool.and(scalar_ok(r), scalar_ok(s)), Nat.is_lt(id, 4n)))def recover_len(+h: List<&2, U32>, +sig: List<&2, U32>, ok: Bool) -> Maybe<&2, List<&2, U32>>:  match ok:    case True{}: recover_sig(h, sig)    case False{}: None{}# the 65-byte uncompressed public key that signed the 32-byte hash, from a# 65-byte r || s || id with id in 0..3 (go-ethereum's crypto.Ecrecover;# Ethereum's precompile passes id = v - 27); None when r or s is not in# [1, n - 1], x(R) is not a field element on the curve, or Q is infinity.# High s is accepted, as the precompile does.def recover(+h: List<&2, U32>, +sig: List<&2, U32>) -> Maybe<&2, List<&2, U32>>:  recover_len(h, sig, Bool.and(B.has_len(32n, h), B.has_len(65n, sig)))# ---- Ethereum addresses ----# little-endian U32 words of bytes (a missing byte reads as 0)def word(+b0: U32, +b1: U32, +b2: U32, +b3: U32) -> U32:  U32.or(U32.or(b0, U32.shln(b1, 8n)), U32.or(U32.shln(b2, 16n), U32.shln(b3, 24n)))def words(n: Nat, +bs: List<&2, U32>) -> List<&2, U32>:  match n:    case 0n: Nil{}    case 1n+k: word(B.head(bs), B.head(B.tail(bs)), B.head(B.suffix(2n, bs)), B.head(B.suffix(3n, bs))) <> words(k, B.suffix(4n, bs))def pack(ws: List<&2, U32>, +i: U32, a: Array<U32>) -> Array<U32>:  match ws:    case Nil{}: a    case w <> t: pack(t, U32.inc(i), Array.set(U32, a, i, w))def unpack(ws: List<&1, U32>) -> List<&2, U32>:  match ws:    case Nil{}: Nil{}    case +w <> t: U32.and(w, 255) <> U32.and(U32.shrn(w, 8n), 255) <> U32.and(U32.shrn(w, 16n), 255) <> U32.shrn(w, 24n) <> unpack(t)def digest_bytes(m: Maybe<&1, Array<U32>>) -> List<&2, U32>:  match m:    case None{}: Nil{}    case Some{a}: unpack(Array.to_list(~U32, a))# The hash of addresses, as a value (only Keccak-256 exists), so that the# proofs can keep it foldedtype Hash is Data:  Keccak256{}def hash(+h: Hash, a: Array<U32>, length: Nat) -> Maybe<&1, Array<U32>>:  match h:    case Keccak256{}: K.keccak256(a, length)# the hash of 64 bytesdef keccak64(+h: Hash, +bs: List<&2, U32>) -> List<&2, U32>:  digest_bytes(hash(h, pack(words(16n, bs), 0, Array.new(U32, 4n, 0)), 64n))def eth_if(ok: Bool, +dg: List<&2, U32>) -> Maybe<&2, List<&2, U32>>:  match ok:    case True{}: Some{B.suffix(12n, dg)}    case False{}: None{}def eth_go(+h: Hash, +pk: List<&2, U32>) -> Maybe<&2, List<&2, U32>>:  eth_if(Bool.and(B.has_len(65n, pk), U32.is_eq(B.head(pk), 4)), keccak64(h, B.tail(pk)))# (the key is looked at first, so that the proof checker keeps an unknown# address folded)def eth_with(+h: Hash, pk: List<&2, U32>) -> Maybe<&2, List<&2, U32>>:  match pk:    case Nil{}: eth_go(h, Nil{})    case b <> t: eth_go(h, b <> t)# the 20-byte Ethereum address of a 65-byte uncompressed public key:# the last 20 bytes of Keccak-256 of its 64 coordinate bytesdef eth_address(+pk: List<&2, U32>) -> Maybe<&2, List<&2, U32>>:  eth_with(Keccak256{}, pk)# ---- Ethereum's ECRECOVER precompile (address 0x01) ----def zeros_ok(bs: List<&2, U32>) -> Bool:  match bs:    case Nil{}: True{}    case b <> t: Bool.and(U32.is_eq(b, 0), zeros_ok(t))def eth_word(m: Maybe<&2, List<&2, U32>>) -> Maybe<&2, List<&2, U32>>:  match m:    case None{}: None{}    case Some{a}: Some{cat(fill(12n, 0), a)}def addr_word(m: Maybe<&2, List<&2, U32>>) -> Maybe<&2, List<&2, U32>>:  match m:    case None{}: None{}    case Some{pk}: eth_word(eth_address(pk))def ecrecover_v(+h: List<&2, U32>, +rs: List<&2, U32>, +v: U32, ok: Bool) -> Maybe<&2, List<&2, U32>>:  match ok:    case True{}: addr_word(recover(h, cat(rs, [U32.sub(v, 27)])))    case False{}: None{}def ecrecover_in(+x: List<&2, U32>) -> Maybe<&2, List<&2, U32>>:  +vw = B.prefix(32n, B.suffix(32n, x))  +v = B.head(B.suffix(31n, vw))  ecrecover_v(B.prefix(32n, x), B.suffix(64n, x), v, Bool.and(zeros_ok(B.prefix(31n, vw)), Bool.or(U32.is_eq(v, 27), U32.is_eq(v, 28))))# The precompile's semantics (Ethereum yellow paper appendix E): the input,# zero-padded or cut to 128 bytes, is hash || v || r || s (32 bytes each,# big-endian); v must be 27 or 28. The output is the 32-byte word holding# the signer's address, or None (the precompile's empty output) when v, r# or s is invalid or no key is recovered.def ecrecover(input: List<&2, U32>) -> Maybe<&2, List<&2, U32>>:  ecrecover_in(B.prefix(128n, cat(input, fill(128n, 0))))