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| Original file line number | Diff line number | Diff line change |
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| %builtins range_check | ||
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| // Source: https://github.com/rdubois-crypto/efficient-secp256r1/blob/4b74807c5e91f1ed4cb00a1c973be05c63986e61/src/secp256r1/ec.cairo | ||
| from starkware.cairo.common.cairo_secp.bigint import BigInt3, UnreducedBigInt3, nondet_bigint3 | ||
| from starkware.cairo.common.cairo_secp.ec import EcPoint | ||
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| // src.secp256r1.constants | ||
| // SECP_REM is defined by the equation: | ||
| // secp256r1_prime = 2 ** 256 - SECP_REM. | ||
| const SECP_REM = 2 ** 224 - 2 ** 192 - 2 ** 96 + 1; | ||
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| const BASE = 2 ** 86; | ||
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| // A = 0xffffffff00000001000000000000000000000000fffffffffffffffffffffffc | ||
| const A0 = 0x3ffffffffffffffffffffc; | ||
| const A1 = 0x3ff; | ||
| const A2 = 0xffffffff0000000100000; | ||
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| // Constants for unreduced_mul/sqr | ||
| const s2 = (-(2 ** 76)) - 2 ** 12; | ||
| const s1 = (-(2 ** 66)) + 4; | ||
| const s0 = 2 ** 56; | ||
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| const r2 = 2 ** 54 - 2 ** 22; | ||
| const r1 = -(2 ** 12); | ||
| const r0 = 4; | ||
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| // src.secp256r1.field | ||
| // Adapt from starkware.cairo.common.math's assert_250_bit | ||
| func assert_165_bit{range_check_ptr}(value) { | ||
| const UPPER_BOUND = 2 ** 165; | ||
| const SHIFT = 2 ** 128; | ||
| const HIGH_BOUND = UPPER_BOUND / SHIFT; | ||
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| let low = [range_check_ptr]; | ||
| let high = [range_check_ptr + 1]; | ||
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| %{ | ||
| from starkware.cairo.common.math_utils import as_int | ||
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| # Correctness check. | ||
| value = as_int(ids.value, PRIME) % PRIME | ||
| assert value < ids.UPPER_BOUND, f'{value} is outside of the range [0, 2**250).' | ||
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| # Calculation for the assertion. | ||
| ids.high, ids.low = divmod(ids.value, ids.SHIFT) | ||
| %} | ||
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| assert [range_check_ptr + 2] = HIGH_BOUND - 1 - high; | ||
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| assert value = high * SHIFT + low; | ||
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| let range_check_ptr = range_check_ptr + 3; | ||
| return (); | ||
| } | ||
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| // src.secp256r1.field | ||
| // Computes the multiplication of two big integers, given in BigInt3 representation, modulo the | ||
| // secp256r1 prime. | ||
| // | ||
| // Arguments: | ||
| // x, y - the two BigInt3 to operate on. | ||
| // | ||
| // Returns: | ||
| // x * y in an UnreducedBigInt3 representation (the returned limbs may be above 3 * BASE). | ||
| // | ||
| // This means that if unreduced_mul is called on the result of nondet_bigint3, or the difference | ||
| // between two such results, we have: | ||
| // Soundness guarantee: the limbs are in the range (). | ||
| // Completeness guarantee: the limbs are in the range (). | ||
| func unreduced_mul(a: BigInt3, b: BigInt3) -> (res_low: UnreducedBigInt3) { | ||
| tempvar twice_d2 = a.d2 * b.d2; | ||
| tempvar d1d2 = a.d2 * b.d1 + a.d1 * b.d2; | ||
| return ( | ||
| UnreducedBigInt3( | ||
| d0=a.d0 * b.d0 + s0 * twice_d2 + r0 * d1d2, | ||
| d1=a.d1 * b.d0 + a.d0 * b.d1 + s1 * twice_d2 + r1 * d1d2, | ||
| d2=a.d2 * b.d0 + a.d1 * b.d1 + a.d0 * b.d2 + s2 * twice_d2 + r2 * d1d2, | ||
| ), | ||
| ); | ||
| } | ||
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||
| // src.secp256r1.field | ||
| // Computes the square of a big integer, given in BigInt3 representation, modulo the | ||
| // secp256r1 prime. | ||
| // | ||
| // Has the same guarantees as in unreduced_mul(a, a). | ||
| func unreduced_sqr(a: BigInt3) -> (res_low: UnreducedBigInt3) { | ||
| tempvar twice_d2 = a.d2 * a.d2; | ||
| tempvar twice_d1d2 = a.d2 * a.d1 + a.d1 * a.d2; | ||
| tempvar d1d0 = a.d1 * a.d0; | ||
| return ( | ||
| UnreducedBigInt3( | ||
| d0=a.d0 * a.d0 + s0 * twice_d2 + r0 * twice_d1d2, | ||
| d1=d1d0 + d1d0 + s1 * twice_d2 + r1 * twice_d1d2, | ||
| d2=a.d2 * a.d0 + a.d1 * a.d1 + a.d0 * a.d2 + s2 * twice_d2 + r2 * twice_d1d2, | ||
| ), | ||
| ); | ||
| } | ||
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||
| // src.secp256r1.field | ||
| // Verifies that the given unreduced value is equal to zero modulo the secp256r1 prime. | ||
| // | ||
| // Completeness assumption: val's limbs are in the range (-2**210.99, 2**210.99). | ||
| // Soundness assumption: val's limbs are in the range (-2**250, 2**250). | ||
| func verify_zero{range_check_ptr}(val: UnreducedBigInt3) { | ||
| alloc_locals; | ||
| local q; | ||
| // local q_sign; | ||
| let q_sign = 1; | ||
| // original: | ||
| // %{ from starkware.cairo.common.cairo_secp.secp_utils import SECP256R1_P as SECP_P %} | ||
| // %{ | ||
| // from starkware.cairo.common.cairo_secp.secp_utils import pack | ||
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| // q, r = divmod(pack(ids.val, PRIME), SECP_P) | ||
| // assert r == 0, f"verify_zero: Invalid input {ids.val.d0, ids.val.d1, ids.val.d2}." | ||
| // if q >= 0: | ||
| // ids.q = q % PRIME | ||
| // ids.q_sign = 1 | ||
| // else: | ||
| // ids.q = (0-q) % PRIME | ||
| // ids.q_sign = -1 % PRIME | ||
| // %} | ||
| %{ from starkware.cairo.common.cairo_secp.secp256r1_utils import SECP256R1_P as SECP_P %} | ||
| %{ | ||
| from starkware.cairo.common.cairo_secp.secp_utils import pack | ||
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| q, r = divmod(pack(ids.val, PRIME), SECP_P) | ||
| assert r == 0, f"verify_zero: Invalid input {ids.val.d0, ids.val.d1, ids.val.d2}." | ||
| ids.q = q % PRIME | ||
| %} | ||
| // assert_250_bit(q); // 256K steps | ||
| // assert_le_felt(q, 2**165); // 275K steps | ||
| assert_165_bit(q); | ||
| assert q_sign * (val.d2 + val.d1 / BASE + val.d0 / BASE ** 2) = q * ( | ||
| (BASE / 4) - SECP_REM / BASE ** 2 | ||
| ); | ||
| // Multiply by BASE**2 both sides: | ||
| // (q_sign) * val = q * (BASE**3 / 4 - SECP_REM) | ||
| // = q * (2**256 - SECP_REM) = q * secp256r1_prime = 0 mod secp256r1_prime | ||
| return (); | ||
| } | ||
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| // Computes the slope of the elliptic curve at a given point. | ||
| // The slope is used to compute point + point. | ||
| // | ||
| // Arguments: | ||
| // point - the point to operate on. | ||
| // | ||
| // Returns: | ||
| // slope - the slope of the curve at point, in BigInt3 representation. | ||
| // | ||
| // Assumption: point != 0. | ||
| func compute_doubling_slope{range_check_ptr}(point: EcPoint) -> (slope: BigInt3) { | ||
| // Note that y cannot be zero: assume that it is, then point = -point, so 2 * point = 0, which | ||
| // contradicts the fact that the size of the curve is odd. | ||
| // originals: | ||
| // %{ from starkware.cairo.common.cairo_secp.secp_utils import SECP256R1_P as SECP_P %} | ||
| // %{ from starkware.cairo.common.cairo_secp.secp_utils import SECP256R1_ALPHA as ALPHA %} | ||
| %{ from starkware.cairo.common.cairo_secp.secp256r1_utils import SECP256R1_P as SECP_P %} | ||
| %{ from starkware.cairo.common.cairo_secp.secp256r1_utils import SECP256R1_ALPHA as ALPHA %} | ||
| %{ | ||
| from starkware.cairo.common.cairo_secp.secp_utils import pack | ||
| from starkware.python.math_utils import ec_double_slope | ||
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| # Compute the slope. | ||
| x = pack(ids.point.x, PRIME) | ||
| y = pack(ids.point.y, PRIME) | ||
| value = slope = ec_double_slope(point=(x, y), alpha=ALPHA, p=SECP_P) | ||
| %} | ||
| let (slope: BigInt3) = nondet_bigint3(); | ||
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| let (x_sqr: UnreducedBigInt3) = unreduced_sqr(point.x); | ||
| let (slope_y: UnreducedBigInt3) = unreduced_mul(slope, point.y); | ||
| verify_zero( | ||
| UnreducedBigInt3( | ||
| d0=3 * x_sqr.d0 + A0 - 2 * slope_y.d0, | ||
| d1=3 * x_sqr.d1 + A1 - 2 * slope_y.d1, | ||
| d2=3 * x_sqr.d2 + A2 - 2 * slope_y.d2, | ||
| ), | ||
| ); | ||
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| return (slope=slope); | ||
| } | ||
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| func test_doubling_slope{range_check_ptr}() { | ||
| let point = EcPoint(BigInt3(614323, 5456867, 101208), BigInt3(773712524, 77371252, 5298795)); | ||
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| let (slope) = compute_doubling_slope(point); | ||
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| assert slope = BigInt3( | ||
| 64081873649130491683833713, 34843994309543177837008178, 16548672716077616016846383 | ||
| ); | ||
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| let point = EcPoint( | ||
| BigInt3(51215, 36848548548458, 634734734), BigInt3(26362, 263724839599, 901297012) | ||
| ); | ||
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| let (slope) = compute_doubling_slope(point); | ||
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| assert slope = BigInt3( | ||
| 71848883893335852660776740, 75644451964360469099209675, 547087410329256463669633 | ||
| ); | ||
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| return (); | ||
| } | ||
|
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| func main{range_check_ptr}() { | ||
| test_doubling_slope(); | ||
| return (); | ||
| } |
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,45 @@ | ||
| %builtins range_check | ||
|
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| from starkware.cairo.common.cairo_secp.bigint import BigInt3, BASE | ||
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| // Hint arguments: value. | ||
| func nondet_bigint3{range_check_ptr}() -> (res: BigInt3) { | ||
| // The result should be at the end of the stack after the function returns. | ||
| let res: BigInt3 = [cast(ap + 5, BigInt3*)]; | ||
| %{ | ||
| from starkware.cairo.common.cairo_secp.secp_utils import split | ||
| segments.write_arg(ids.res.address_, split(value)) | ||
| %} | ||
| // The maximal possible sum of the limbs, assuming each of them is in the range [0, BASE). | ||
| const MAX_SUM = 3 * (BASE - 1); | ||
| assert [range_check_ptr] = MAX_SUM - (res.d0 + res.d1 + res.d2); | ||
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| // Prepare the result at the end of the stack. | ||
| tempvar range_check_ptr = range_check_ptr + 4; | ||
| [range_check_ptr - 3] = res.d0, ap++; | ||
| [range_check_ptr - 2] = res.d1, ap++; | ||
| [range_check_ptr - 1] = res.d2, ap++; | ||
| static_assert &res + BigInt3.SIZE == ap; | ||
| return (res=res); | ||
| } | ||
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| func main{range_check_ptr}() { | ||
| alloc_locals; | ||
| // Take these hints from div_mod_n just to have a value in scope | ||
| local a: BigInt3 = BigInt3(1,2,3); | ||
| local b: BigInt3 = BigInt3(4,5,6); | ||
| %{ | ||
| from starkware.cairo.common.cairo_secp.secp_utils import N, pack | ||
| from starkware.python.math_utils import div_mod, safe_div | ||
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| a = pack(ids.a, PRIME) | ||
| b = pack(ids.b, PRIME) | ||
| value = res = div_mod(a, b, N) | ||
| %} | ||
| let (r) = nondet_bigint3(); | ||
| assert r.d0 = 46511138620617205537268188; | ||
| assert r.d1 = 26286155657000021849694253; | ||
| assert r.d2 = 3102515549921694024741409; | ||
| return(); | ||
| } | ||
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