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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,112 @@ | ||
| %builtins range_check | ||
|
|
||
| from starkware.cairo.common.cairo_secp.bigint import ( | ||
| BigInt3, | ||
| UnreducedBigInt3, | ||
| nondet_bigint3, | ||
| bigint_to_uint256, | ||
| uint256_to_bigint, | ||
| ) | ||
|
|
||
| struct EcPoint { | ||
| x: BigInt3, | ||
| y: BigInt3, | ||
| } | ||
|
|
||
| const BASE = 2 ** 86; | ||
| const SECP_REM = 19; | ||
|
|
||
| func compute_slope{range_check_ptr}(point0: EcPoint, point1: EcPoint) -> (slope: BigInt3) { | ||
| alloc_locals; | ||
| %{ | ||
| from starkware.python.math_utils import line_slope | ||
| from starkware.cairo.common.cairo_secp.secp_utils import pack | ||
| SECP_P = 2**255-19 | ||
| # Compute the slope. | ||
| x0 = pack(ids.point0.x, PRIME) | ||
| y0 = pack(ids.point0.y, PRIME) | ||
| x1 = pack(ids.point1.x, PRIME) | ||
| y1 = pack(ids.point1.y, PRIME) | ||
| value = slope = line_slope(point1=(x0, y0), point2=(x1, y1), p=SECP_P) | ||
| %} | ||
| let (slope) = nondet_bigint3(); | ||
|
|
||
| let x_diff = BigInt3( | ||
| d0=point0.x.d0 - point1.x.d0, d1=point0.x.d1 - point1.x.d1, d2=point0.x.d2 - point1.x.d2 | ||
| ); | ||
| let (x_diff_slope: UnreducedBigInt3) = unreduced_mul(x_diff, slope); | ||
|
|
||
| verify_zero( | ||
| UnreducedBigInt3( | ||
| d0=x_diff_slope.d0 - point0.y.d0 + point1.y.d0, | ||
| d1=x_diff_slope.d1 - point0.y.d1 + point1.y.d1, | ||
| d2=x_diff_slope.d2 - point0.y.d2 + point1.y.d2), | ||
| ); | ||
|
|
||
| return (slope=slope); | ||
| } | ||
|
|
||
| func unreduced_mul(a: BigInt3, b: BigInt3) -> (res_low: UnreducedBigInt3) { | ||
| // The result of the product is: | ||
| // sum_{i, j} a.d_i * b.d_j * BASE**(i + j) | ||
| // Since we are computing it mod secp256k1_prime, we replace the term | ||
| // a.d_i * b.d_j * BASE**(i + j) | ||
| // where i + j >= 3 with | ||
| // a.d_i * b.d_j * BASE**(i + j - 3) * 4 * SECP_REM | ||
| // since BASE ** 3 = 4 * SECP_REM (mod secp256k1_prime). | ||
| return ( | ||
| UnreducedBigInt3( | ||
| d0=a.d0 * b.d0 + (a.d1 * b.d2 + a.d2 * b.d1) * (8 * SECP_REM), | ||
| d1=a.d0 * b.d1 + a.d1 * b.d0 + (a.d2 * b.d2) * (8 * SECP_REM), | ||
| d2=a.d0 * b.d2 + a.d1 * b.d1 + a.d2 * b.d0), | ||
| ); | ||
| } | ||
|
|
||
| func verify_zero{range_check_ptr}(val: UnreducedBigInt3) { | ||
| let q = [ap]; | ||
| %{ | ||
| from starkware.cairo.common.cairo_secp.secp_utils import pack | ||
| SECP_P = 2**255-19 | ||
| to_assert = pack(ids.val, PRIME) | ||
| 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 | ||
| %} | ||
| let q_biased = [ap + 1]; | ||
| q_biased = q + 2 ** 127, ap++; | ||
| [range_check_ptr] = q_biased, ap++; | ||
| // This implies that q is in the range [-2**127, 2**127). | ||
|
|
||
| tempvar r1 = (val.d0 + q * SECP_REM) / BASE; | ||
| assert [range_check_ptr + 1] = r1 + 2 ** 127; | ||
| // This implies that r1 is in the range [-2**127, 2**127). | ||
| // Therefore, r1 * BASE is in the range [-2**213, 2**213). | ||
| // By the soundness assumption, val.d0 is in the range (-2**250, 2**250). | ||
| // This implies that r1 * BASE = val.d0 + q * SECP_REM (as integers). | ||
|
|
||
| tempvar r2 = (val.d1 + r1) / BASE; | ||
| assert [range_check_ptr + 2] = r2 + 2 ** 127; | ||
| // Similarly, this implies that r2 * BASE = val.d1 + r1 (as integers). | ||
| // Therefore, r2 * BASE**2 = val.d1 * BASE + r1 * BASE. | ||
|
|
||
| assert val.d2 = q * (BASE / 8) - r2; | ||
| // Similarly, this implies that q * BASE / 4 = val.d2 + r2 (as integers). | ||
| // Therefore, | ||
| // q * BASE**3 / 4 = val.d2 * BASE**2 + r2 * BASE ** 2 = | ||
| // val.d2 * BASE**2 + val.d1 * BASE + r1 * BASE = | ||
| // val.d2 * BASE**2 + val.d1 * BASE + val.d0 + q * SECP_REM = | ||
| // val + q * SECP_REM. | ||
| // Hence, val = q * (BASE**3 / 4 - SECP_REM) = q * (2**256 - SECP_REM) = q * secp256k1_prime. | ||
|
|
||
| let range_check_ptr = range_check_ptr + 3; | ||
| return (); | ||
| } | ||
|
|
||
| func main{range_check_ptr}() { | ||
| let point_1 = EcPoint(BigInt3(512,2412,133), BigInt3(64,0,6546)); | ||
| let point_2 = EcPoint(BigInt3(7,8,123), BigInt3(1,7,465)); | ||
|
|
||
| let (slope) = compute_slope(point_1, point_2); | ||
| assert slope = BigInt3(32565103718045841981942279,60662980405630750722698303,6577829329490861459174478); | ||
| return (); | ||
| } |
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