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src/crypto/secp256k1/ecdsa.bend

bend-collections@1.0.0.0 · 371 lines · 65 definitions

1import Base
2import bend-collections@1.0.0.0/src/crypto/mac.bend as MAC
3import bend-collections@1.0.0.0/src/crypto/keccak/keccak.bend as K
4import bend-collections@1.0.0.0/src/crypto/secp256k1/limbs.bend as L
5import bend-collections@1.0.0.0/src/crypto/secp256k1/field.bend as F
6import bend-collections@1.0.0.0/src/crypto/secp256k1/scalar.bend as S
7import bend-collections@1.0.0.0/src/crypto/secp256k1/point.bend as P
8import bend-collections@1.0.0.0/src/crypto/secp256k1/bytes.bend as B
9
10# ECDSA over secp256k1 (SEC 1 v2 section 4.1) with deterministic nonces
11# (RFC 6979 section 3.2, HMAC-SHA256) and low-S signatures (BIP 62 /
12# Ethereum's homestead rule, libsecp256k1's secp256k1_ecdsa_sign), public
13# key recovery (SEC 1 section 4.1.6, Ethereum's ecrecover) and the Ethereum
14# address of a public key. The contract is spec/crypto/secp256k1/ecdsa.bend.
15#
16# Messages are 32-byte hashes (the caller hashes); keys and signatures are
17# byte lists of their SEC 1 lengths, malformed ones are rejected as values.
18
19type Signature is Data:
20  Signature{r: List<&2, U32>, s: List<&2, U32>, v: U32}
21
22# ---- scalars from bytes ----
23
24# 1 <= x < n
25def scalar_ok(+x: List<&2, Nat>) -> Bool:
26  Bool.and(Bool.not(S.is_zero(x)), S.lt_n(x))
27
28# a 32-byte string as a scalar mod n (bits2int with qlen = hlen = 256,
29# reduced: SEC 1 section 4.1.3 step 5, RFC 6979 section 2.3.2)
30def hash_scalar(+h: List<&2, U32>) -> List<&2, Nat>:
31  S.reduce(B.of_be(h))
32
33# ---- RFC 6979 section 3.2, HMAC_DRBG with SHA-256 ----
34
35type Drbg is Data:
36  Drbg{k: List<&2, U32>, v: List<&2, U32>}
37
38# (the key is looked at first, so that the proof checker keeps an unknown
39# HMAC folded)
40def hmac(key: List<&2, U32>, msg: List<&2, U32>) -> List<&2, U32>:
41  match key:
42    case Nil{}: MAC.sign(Nil{}, msg)
43    case k <> t: MAC.sign(k <> t, msg)
44
45def cat(xs: List<&2, U32>, ys: List<&2, U32>) -> List<&2, U32>:
46  List.append(&2, U32, xs, ys)
47
48def fill(n: Nat, +b: U32) -> List<&2, U32>:
49  List.replicate(U32, n, b)
50
51def drbg_v(+k: List<&2, U32>, +v: List<&2, U32>) -> Drbg:
52  Drbg{k, hmac(k, v)}
53
54# steps b-g: V = 0x01..., K = 0x00..., then two HMAC rounds over
55# V || 0x00 || int2octets(x) || bits2octets(h1) and V || 0x01 || ...
56def drbg_step(+k: List<&2, U32>, +v: List<&2, U32>, +tag: U32, +seed: List<&2, U32>) -> Drbg:
57  drbg_v(hmac(k, cat(v, tag <> seed)), v)
58
59def drbg_second(d: Drbg, +seed: List<&2, U32>) -> Drbg:
60  match d:
61    case Drbg{k, v}: drbg_step(k, v, 1, seed)
62
63def drbg_init(+seed: List<&2, U32>) -> Drbg:
64  drbg_second(drbg_step(fill(32n, 0), fill(32n, 1), 0, seed), seed)
65
66# step h.3: K = HMAC_K(V || 0x00), V = HMAC_K(V)
67def drbg_next(+k: List<&2, U32>, +v: List<&2, U32>) -> Drbg:
68  drbg_v(hmac(k, cat(v, [0])), v)
69
70# ---- signing ----
71
72def b2n(b: Bool) -> Nat:
73  L.b2n(b)
74
75type Attempt is Data:
76  ARetry{}
77  ADone{sig: Signature}
78
79def finish_if(bad: Bool, +sig: Signature) -> Attempt:
80  match bad:
81    case True{}: ARetry{}
82    case False{}: ADone{sig}
83
84# r = x(R) mod n, s = k^-1 (z + r d) mod n, made low (s > n / 2 is replaced
85# by n - s, which flips the parity bit of the recovery id); the recovery
86# id is the parity of y(R), plus 2 when x(R) >= n
87def finish(+z: List<&2, Nat>, +d: List<&2, Nat>, +k: List<&2, Nat>, +x: List<&2, Nat>, +y: List<&2, Nat>) -> Attempt:
88  +r = S.reduce(x)
89  +s0 = S.mul(S.inv(k), S.add(z, S.mul(r, d)))
90  +high = b2n(S.is_high(s0))
91  +par = F.parity(y)
92  +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)))
93  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)})
94
95def attempt_aff(+z: List<&2, Nat>, +d: List<&2, Nat>, +k: List<&2, Nat>, a: P.Affine) -> Attempt:
96  match a:
97    case P.Affine{x, y}: finish(z, d, k, x, y)
98
99def attempt_ok(+z: List<&2, Nat>, +d: List<&2, Nat>, +k: List<&2, Nat>, ok: Bool) -> Attempt:
100  match ok:
101    case True{}: attempt_aff(z, d, k, P.to_affine(P.mul(k, P.g())))
102    case False{}: ARetry{}
103
104# the candidate k = bits2int(V) (RFC 6979 step h.3): used when 1 <= k < n
105def attempt(+z: List<&2, Nat>, +d: List<&2, Nat>, +v: List<&2, U32>) -> Attempt:
106  +k = B.of_be(v)
107  attempt_ok(z, d, k, scalar_ok(k))
108
109# step h: the candidate V; a k outside [1, n - 1], or one giving r = 0 or
110# s = 0 (SEC 1 section 4.1.3 steps 3 and 6), is replaced by the next
111# candidate: K = HMAC_K(V || 0x00), V = HMAC_K(V), V = HMAC_K(V). The loop
112# stops after `fuel` candidates (the chance that even 2 are needed is below
113# 2^-127).
114def 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>:
115  match fuel att:
116    case 0n ARetry{}: None{}
117    case 0n ADone{sig}: Some{sig}
118    case 1n+f ARetry{}:
119      +k2 = hmac(k, cat(v, [0]))
120      +v3 = hmac(k2, hmac(k2, v))
121      sign_loop(f, z, d, k2, v3, attempt(z, d, v3))
122    case 1n+f ADone{sig}: Some{sig}
123
124def sign_drbg(+z: List<&2, Nat>, +d: List<&2, Nat>, g: Drbg) -> Maybe<&2, Signature>:
125  match g:
126    case Drbg{+k, +v}:
127      +v1 = hmac(k, v)
128      sign_loop(16n, z, d, k, v1, attempt(z, d, v1))
129
130def secret_if(+d: List<&2, Nat>, ok: Bool) -> Maybe<&2, List<&2, Nat>>:
131  match ok:
132    case True{}: Some{d}
133    case False{}: None{}
134
135# the secret key d, when 1 <= d < n
136def secret(+sk: List<&2, U32>) -> Maybe<&2, List<&2, Nat>>:
137  +d = B.of_be(sk)
138  secret_if(d, Bool.and(B.has_len(32n, sk), scalar_ok(d)))
139
140# sign with a valid secret scalar d: x = int2octets(d),
141# h1' = bits2octets(h) = int2octets(bits2int(h) mod n)
142def sign_d(+h: List<&2, U32>, m: Maybe<&2, List<&2, Nat>>) -> Maybe<&2, Signature>:
143  match m:
144    case None{}: None{}
145    case Some{+d}:
146      +z = hash_scalar(h)
147      sign_drbg(z, d, drbg_init(cat(B.to_be(d), B.to_be(z))))
148
149def sign_h(+sk: List<&2, U32>, +h: List<&2, U32>, ok: Bool) -> Maybe<&2, Signature>:
150  match ok:
151    case True{}: sign_d(h, secret(sk))
152    case False{}: None{}
153
154# the low-S signature of a 32-byte hash under a 32-byte secret key
155def sign(+sk: List<&2, U32>, +h: List<&2, U32>) -> Maybe<&2, Signature>:
156  sign_h(sk, h, B.has_len(32n, h))
157
158# ---- public keys ----
159
160def public_point(+d: List<&2, Nat>) -> P.Point:
161  P.mul(d, P.g())
162
163def pk_of(compressed: Bool, +q: P.Point) -> List<&2, U32>:
164  match compressed:
165    case True{}: P.encode_compressed(q)
166    case False{}: P.encode_uncompressed(q)
167
168def public_key_d(compressed: Bool, m: Maybe<&2, List<&2, Nat>>) -> Maybe<&2, List<&2, U32>>:
169  match m:
170    case None{}: None{}
171    case Some{d}: Some{pk_of(compressed, public_point(d))}
172
173# the SEC 1 encoding of [d] G (33 bytes compressed, 65 uncompressed)
174def public_key(+sk: List<&2, U32>, compressed: Bool) -> Maybe<&2, List<&2, U32>>:
175  public_key_d(compressed, secret(sk))
176
177# ---- verification (SEC 1 section 4.1.4) ----
178
179# u1 = z w, u2 = r w with w = s^-1; R = [u1] G + [u2] Q must not be the
180# point at infinity and x(R) mod n must be r
181def verify_rs(+q: P.Point, +z: List<&2, Nat>, +r: List<&2, Nat>, +s: List<&2, Nat>) -> Bool:
182  +w = S.inv(s)
183  +rr = P.add(P.mul(S.mul(z, w), P.g()), P.mul(S.mul(r, w), q))
184  Bool.and(Bool.not(P.is_inf(rr)), S.eq(S.reduce(P.aff_x(P.to_affine(rr))), r))
185
186def verify_ok(+q: P.Point, +z: List<&2, Nat>, +r: List<&2, Nat>, +s: List<&2, Nat>, ok: Bool) -> Bool:
187  match ok:
188    case True{}: verify_rs(q, z, r, s)
189    case False{}: False{}
190
191def verify_q(+h: List<&2, U32>, +sig: List<&2, U32>, strict: Bool, m: Maybe<&2, P.Point>) -> Bool:
192  match m:
193    case None{}: False{}
194    case Some{+q}:
195      +r = B.of_be(B.prefix(32n, sig))
196      +s = B.of_be(B.suffix(32n, sig))
197      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)))))
198
199def verify_len(+pk: List<&2, U32>, +h: List<&2, U32>, +sig: List<&2, U32>, strict: Bool, ok: Bool) -> Bool:
200  match ok:
201    case True{}: verify_q(h, sig, strict, P.decode(pk))
202    case False{}: False{}
203
204def verify_with(+pk: List<&2, U32>, +h: List<&2, U32>, +sig: List<&2, U32>, strict: Bool) -> Bool:
205  verify_len(pk, h, sig, strict, Bool.and(B.has_len(32n, h), B.has_len(64n, sig)))
206
207# SEC 1 verification of a 64-byte r || s against a SEC 1 public key (33 or
208# 65 bytes); high s is accepted, as SEC 1 and OpenSSL do
209def verify(+pk: List<&2, U32>, +h: List<&2, U32>, +sig: List<&2, U32>) -> Bool:
210  verify_with(pk, h, sig, False{})
211
212# the same, rejecting s > n / 2 (Bitcoin's BIP 62/146 LOW_S rule,
213# Ethereum's homestead rule for transactions)
214def verify_strict(+pk: List<&2, U32>, +h: List<&2, U32>, +sig: List<&2, U32>) -> Bool:
215  verify_with(pk, h, sig, True{})
216
217# ---- recovery (SEC 1 section 4.1.6, Ethereum ecrecover) ----
218
219def recover_q(+q: P.Point, inf: Bool) -> Maybe<&2, List<&2, U32>>:
220  match inf:
221    case True{}: None{}
222    case False{}: Some{P.encode_uncompressed(q)}
223
224# Q = r^-1 (s R - z G), with R the point of x-coordinate r + n j (j = id / 2)
225# and y parity id mod 2
226def recover_r(+z: List<&2, Nat>, +r: List<&2, Nat>, +s: List<&2, Nat>, m: Maybe<&2, P.Point>) -> Maybe<&2, List<&2, U32>>:
227  match m:
228    case None{}: None{}
229    case Some{+pr}:
230      +ri = S.inv(r)
231      +q = P.add(P.mul(S.mul(S.neg(z), ri), P.g()), P.mul(S.mul(s, ri), pr))
232      recover_q(q, P.is_inf(q))
233
234def 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>>:
235  match ok:
236    case True{}: recover_r(z, r, s, P.decompress(x, Nat.mod(id, 2n)))
237    case False{}: None{}
238
239# c_n = 2^256 - n and c_p = 2^256 - p as 16-limb field elements
240def cn_fe() -> List<&2, Nat>:
241  [48831n, 12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n, 0n, 0n, 0n, 0n, 0n, 0n, 0n]
242
243def cp_fe() -> List<&2, Nat>:
244  [977n, 0n, 1n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n]
245
246# (r + n) mod p = (r - c_n + 2^256) mod p = (r - c_n + c_p) mod p
247def rn_fe(+r: List<&2, Nat>) -> List<&2, Nat>:
248  F.add(F.sub(r, cn_fe()), cp_fe())
249
250# x = r + j n with j = id / 2. For j = 1, r + n must be below p: then
251# (r + n) mod p = r + n >= n, while a wrapped r + n - p is below r < n
252def recover_ok(+z: List<&2, Nat>, +r: List<&2, Nat>, +s: List<&2, Nat>, +id: Nat, ok: Bool) -> Maybe<&2, List<&2, U32>>:
253  match ok:
254    case True{}:
255      +j = Nat.div(id, 2n)
256      +x1 = rn_fe(r)
257      recover_x(z, r, s, id, F.select(j, x1, r), Bool.or(Nat.is_eq(j, 0n), Bool.not(S.lt_n(x1))))
258    case False{}: None{}
259
260def recover_sig(+h: List<&2, U32>, +sig: List<&2, U32>) -> Maybe<&2, List<&2, U32>>:
261  +r = B.of_be(B.prefix(32n, sig))
262  +s = B.of_be(B.prefix(32n, B.suffix(32n, sig)))
263  +id = U32.to_nat(B.head(B.suffix(64n, sig)))
264  recover_ok(hash_scalar(h), r, s, id, Bool.and(Bool.and(scalar_ok(r), scalar_ok(s)), Nat.is_lt(id, 4n)))
265
266def recover_len(+h: List<&2, U32>, +sig: List<&2, U32>, ok: Bool) -> Maybe<&2, List<&2, U32>>:
267  match ok:
268    case True{}: recover_sig(h, sig)
269    case False{}: None{}
270
271# the 65-byte uncompressed public key that signed the 32-byte hash, from a
272# 65-byte r || s || id with id in 0..3 (go-ethereum's crypto.Ecrecover;
273# Ethereum's precompile passes id = v - 27); None when r or s is not in
274# [1, n - 1], x(R) is not a field element on the curve, or Q is infinity.
275# High s is accepted, as the precompile does.
276def recover(+h: List<&2, U32>, +sig: List<&2, U32>) -> Maybe<&2, List<&2, U32>>:
277  recover_len(h, sig, Bool.and(B.has_len(32n, h), B.has_len(65n, sig)))
278
279# ---- Ethereum addresses ----
280
281# little-endian U32 words of bytes (a missing byte reads as 0)
282def word(+b0: U32, +b1: U32, +b2: U32, +b3: U32) -> U32:
283  U32.or(U32.or(b0, U32.shln(b1, 8n)), U32.or(U32.shln(b2, 16n), U32.shln(b3, 24n)))
284
285def words(n: Nat, +bs: List<&2, U32>) -> List<&2, U32>:
286  match n:
287    case 0n: Nil{}
288    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))
289
290def pack(ws: List<&2, U32>, +i: U32, a: Array<U32>) -> Array<U32>:
291  match ws:
292    case Nil{}: a
293    case w <> t: pack(t, U32.inc(i), Array.set(U32, a, i, w))
294
295def unpack(ws: List<&1, U32>) -> List<&2, U32>:
296  match ws:
297    case Nil{}: Nil{}
298    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)
299
300def digest_bytes(m: Maybe<&1, Array<U32>>) -> List<&2, U32>:
301  match m:
302    case None{}: Nil{}
303    case Some{a}: unpack(Array.to_list(~U32, a))
304
305# The hash of addresses, as a value (only Keccak-256 exists), so that the
306# proofs can keep it folded
307type Hash is Data:
308  Keccak256{}
309
310def hash(+h: Hash, a: Array<U32>, length: Nat) -> Maybe<&1, Array<U32>>:
311  match h:
312    case Keccak256{}: K.keccak256(a, length)
313
314# the hash of 64 bytes
315def keccak64(+h: Hash, +bs: List<&2, U32>) -> List<&2, U32>:
316  digest_bytes(hash(h, pack(words(16n, bs), 0, Array.new(U32, 4n, 0)), 64n))
317
318def eth_if(ok: Bool, +dg: List<&2, U32>) -> Maybe<&2, List<&2, U32>>:
319  match ok:
320    case True{}: Some{B.suffix(12n, dg)}
321    case False{}: None{}
322
323def eth_go(+h: Hash, +pk: List<&2, U32>) -> Maybe<&2, List<&2, U32>>:
324  eth_if(Bool.and(B.has_len(65n, pk), U32.is_eq(B.head(pk), 4)), keccak64(h, B.tail(pk)))
325
326# (the key is looked at first, so that the proof checker keeps an unknown
327# address folded)
328def eth_with(+h: Hash, pk: List<&2, U32>) -> Maybe<&2, List<&2, U32>>:
329  match pk:
330    case Nil{}: eth_go(h, Nil{})
331    case b <> t: eth_go(h, b <> t)
332
333# the 20-byte Ethereum address of a 65-byte uncompressed public key:
334# the last 20 bytes of Keccak-256 of its 64 coordinate bytes
335def eth_address(+pk: List<&2, U32>) -> Maybe<&2, List<&2, U32>>:
336  eth_with(Keccak256{}, pk)
337
338# ---- Ethereum's ECRECOVER precompile (address 0x01) ----
339
340def zeros_ok(bs: List<&2, U32>) -> Bool:
341  match bs:
342    case Nil{}: True{}
343    case b <> t: Bool.and(U32.is_eq(b, 0), zeros_ok(t))
344
345def eth_word(m: Maybe<&2, List<&2, U32>>) -> Maybe<&2, List<&2, U32>>:
346  match m:
347    case None{}: None{}
348    case Some{a}: Some{cat(fill(12n, 0), a)}
349
350def addr_word(m: Maybe<&2, List<&2, U32>>) -> Maybe<&2, List<&2, U32>>:
351  match m:
352    case None{}: None{}
353    case Some{pk}: eth_word(eth_address(pk))
354
355def ecrecover_v(+h: List<&2, U32>, +rs: List<&2, U32>, +v: U32, ok: Bool) -> Maybe<&2, List<&2, U32>>:
356  match ok:
357    case True{}: addr_word(recover(h, cat(rs, [U32.sub(v, 27)])))
358    case False{}: None{}
359
360def ecrecover_in(+x: List<&2, U32>) -> Maybe<&2, List<&2, U32>>:
361  +vw = B.prefix(32n, B.suffix(32n, x))
362  +v = B.head(B.suffix(31n, vw))
363  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))))
364
365# The precompile's semantics (Ethereum yellow paper appendix E): the input,
366# zero-padded or cut to 128 bytes, is hash || v || r || s (32 bytes each,
367# big-endian); v must be 27 or 28. The output is the 32-byte word holding
368# the signer's address, or None (the precompile's empty output) when v, r
369# or s is invalid or no key is recovered.
370def ecrecover(input: List<&2, U32>) -> Maybe<&2, List<&2, U32>>:
371  ecrecover_in(B.prefix(128n, cat(input, fill(128n, 0))))