/* * Copyright (c) 2023, Michiel Visser * * SPDX-License-Identifier: BSD-2-Clause */ #pragma once #include #include #include #include #include #include #include #include #include #include namespace Crypto::Curves { struct SECPxxxr1CurveParameters { StringView prime; StringView a; StringView b; StringView order; StringView generator_point; }; template class SECPxxxr1 : public EllipticCurve { private: using StorageType = AK::UFixedBigInt; using StorageTypeX2 = AK::UFixedBigInt; struct JacobianPoint { StorageType x; StorageType y; StorageType z; }; // Curve parameters static constexpr size_t KEY_BIT_SIZE = bit_size; static constexpr size_t KEY_BYTE_SIZE = KEY_BIT_SIZE / 8; static constexpr size_t POINT_BYTE_SIZE = 1 + 2 * KEY_BYTE_SIZE; static constexpr StorageType make_unsigned_fixed_big_int_from_string(StringView str) { StorageType result { 0 }; for (auto c : str) { if (c == '_') continue; result <<= 4; result |= parse_ascii_hex_digit(c); } return result; } static constexpr StorageType PRIME = make_unsigned_fixed_big_int_from_string(CURVE_PARAMETERS.prime); static constexpr StorageType A = make_unsigned_fixed_big_int_from_string(CURVE_PARAMETERS.a); static constexpr StorageType B = make_unsigned_fixed_big_int_from_string(CURVE_PARAMETERS.b); static constexpr StorageType ORDER = make_unsigned_fixed_big_int_from_string(CURVE_PARAMETERS.order); static constexpr Array make_generator_point_bytes(StringView generator_point) { Array buf_array { 0 }; auto it = generator_point.begin(); for (size_t i = 0; i < POINT_BYTE_SIZE; i++) { if (it == CURVE_PARAMETERS.generator_point.end()) break; while (*it == '_') { it++; } buf_array[i] = parse_ascii_hex_digit(*it) * 16; it++; if (it == CURVE_PARAMETERS.generator_point.end()) break; buf_array[i] += parse_ascii_hex_digit(*it); it++; } return buf_array; } static constexpr Array GENERATOR_POINT = make_generator_point_bytes(CURVE_PARAMETERS.generator_point); // Check that the generator point starts with 0x04 static_assert(GENERATOR_POINT[0] == 0x04); static constexpr StorageType calculate_modular_inverse_mod_r(StorageType value) { // Calculate the modular multiplicative inverse of value mod 2^bit_size using the extended euclidean algorithm using StorageTypeP1 = AK::UFixedBigInt; StorageTypeP1 old_r = value; StorageTypeP1 r = static_cast(1u) << KEY_BIT_SIZE; StorageTypeP1 old_s = 1u; StorageTypeP1 s = 0u; while (!r.is_zero_constant_time()) { StorageTypeP1 r_save = r; StorageTypeP1 quotient = old_r.div_mod(r, r); old_r = r_save; StorageTypeP1 s_save = s; s = old_s - quotient * s; old_s = s_save; } return static_cast(old_s); } static constexpr StorageType calculate_r2_mod(StorageType modulus) { // Calculate the value of R^2 mod modulus, where R = 2^bit_size using StorageTypeX2P1 = AK::UFixedBigInt; StorageTypeX2P1 r2 = static_cast(1u) << (2 * KEY_BIT_SIZE); return r2 % modulus; } // Verify that A = -3 mod p, which is required for some optimizations static_assert(A == PRIME - 3); // Precomputed helper values for reduction and Montgomery multiplication static constexpr StorageType REDUCE_PRIME = StorageType { 0 } - PRIME; static constexpr StorageType REDUCE_ORDER = StorageType { 0 } - ORDER; static constexpr StorageType PRIME_INVERSE_MOD_R = StorageType { 0 } - calculate_modular_inverse_mod_r(PRIME); static constexpr StorageType ORDER_INVERSE_MOD_R = StorageType { 0 } - calculate_modular_inverse_mod_r(ORDER); static constexpr StorageType R2_MOD_PRIME = calculate_r2_mod(PRIME); static constexpr StorageType R2_MOD_ORDER = calculate_r2_mod(ORDER); public: size_t key_size() override { return POINT_BYTE_SIZE; } ErrorOr generate_private_key() override { auto buffer = TRY(ByteBuffer::create_uninitialized(KEY_BYTE_SIZE)); fill_with_random(buffer); return buffer; } ErrorOr generate_public_key(ReadonlyBytes a) override { return compute_coordinate(a, GENERATOR_POINT); } ErrorOr compute_coordinate(ReadonlyBytes scalar_bytes, ReadonlyBytes point_bytes) override { AK::FixedMemoryStream scalar_stream { scalar_bytes }; AK::FixedMemoryStream point_stream { point_bytes }; StorageType scalar = TRY(scalar_stream.read_value>()); JacobianPoint point = TRY(read_uncompressed_point(point_stream)); JacobianPoint result = TRY(compute_coordinate_internal(scalar, point)); // Export the values into an output buffer auto buf = TRY(ByteBuffer::create_uninitialized(POINT_BYTE_SIZE)); AK::FixedMemoryStream buf_stream { buf.bytes() }; TRY(buf_stream.write_value(0x04)); TRY(buf_stream.write_value>(result.x)); TRY(buf_stream.write_value>(result.y)); return buf; } ErrorOr derive_premaster_key(ReadonlyBytes shared_point) override { VERIFY(shared_point.size() == POINT_BYTE_SIZE); VERIFY(shared_point[0] == 0x04); ByteBuffer premaster_key = TRY(ByteBuffer::create_uninitialized(KEY_BYTE_SIZE)); premaster_key.overwrite(0, shared_point.data() + 1, KEY_BYTE_SIZE); return premaster_key; } ErrorOr verify(ReadonlyBytes hash, ReadonlyBytes pubkey, ReadonlyBytes signature) { Crypto::ASN1::Decoder asn1_decoder(signature); TRY(asn1_decoder.enter()); auto r_bigint = TRY(asn1_decoder.read(Crypto::ASN1::Class::Universal, Crypto::ASN1::Kind::Integer)); auto s_bigint = TRY(asn1_decoder.read(Crypto::ASN1::Class::Universal, Crypto::ASN1::Kind::Integer)); StorageType r = 0u; StorageType s = 0u; for (size_t i = 0; i < (KEY_BIT_SIZE / 32); i++) { StorageType rr = r_bigint.words()[i]; StorageType ss = s_bigint.words()[i]; r |= (rr << (i * 32)); s |= (ss << (i * 32)); } // z is the hash StorageType z = 0u; for (uint8_t byte : hash) { z <<= 8; z |= byte; } AK::FixedMemoryStream pubkey_stream { pubkey }; JacobianPoint pubkey_point = TRY(read_uncompressed_point(pubkey_stream)); StorageType r_mo = to_montgomery_order(r); StorageType s_mo = to_montgomery_order(s); StorageType z_mo = to_montgomery_order(z); StorageType s_inv = modular_inverse_order(s_mo); StorageType u1 = modular_multiply_order(z_mo, s_inv); StorageType u2 = modular_multiply_order(r_mo, s_inv); u1 = from_montgomery_order(u1); u2 = from_montgomery_order(u2); JacobianPoint point1 = TRY(generate_public_key_internal(u1)); JacobianPoint point2 = TRY(compute_coordinate_internal(u2, pubkey_point)); // Convert the input point into Montgomery form point1.x = to_montgomery(point1.x); point1.y = to_montgomery(point1.y); point1.z = to_montgomery(point1.z); VERIFY(is_point_on_curve(point1)); // Convert the input point into Montgomery form point2.x = to_montgomery(point2.x); point2.y = to_montgomery(point2.y); point2.z = to_montgomery(point2.z); VERIFY(is_point_on_curve(point2)); JacobianPoint result = point_add(point1, point2); // Convert from Jacobian coordinates back to Affine coordinates convert_jacobian_to_affine(result); // Make sure the resulting point is on the curve VERIFY(is_point_on_curve(result)); // Convert the result back from Montgomery form result.x = from_montgomery(result.x); result.y = from_montgomery(result.y); // Final modular reduction on the coordinates result.x = modular_reduce(result.x); result.y = modular_reduce(result.y); return r.is_equal_to_constant_time(result.x); } private: ErrorOr generate_public_key_internal(StorageType a) { AK::FixedMemoryStream generator_point_stream { GENERATOR_POINT }; JacobianPoint point = TRY(read_uncompressed_point(generator_point_stream)); return compute_coordinate_internal(a, point); } ErrorOr compute_coordinate_internal(StorageType scalar, JacobianPoint point) { // FIXME: This will slightly bias the distribution of client secrets scalar = modular_reduce_order(scalar); if (scalar.is_zero_constant_time()) return Error::from_string_literal("SECPxxxr1: scalar is zero"); // Convert the input point into Montgomery form point.x = to_montgomery(point.x); point.y = to_montgomery(point.y); point.z = to_montgomery(point.z); // Check that the point is on the curve if (!is_point_on_curve(point)) return Error::from_string_literal("SECPxxxr1: point is not on the curve"); JacobianPoint result { 0, 0, 0 }; JacobianPoint temp_result { 0, 0, 0 }; // Calculate the scalar times point multiplication in constant time for (size_t i = 0; i < KEY_BIT_SIZE; i++) { temp_result = point_add(result, point); auto condition = (scalar & 1u) == 1u; result.x = select(result.x, temp_result.x, condition); result.y = select(result.y, temp_result.y, condition); result.z = select(result.z, temp_result.z, condition); point = point_double(point); scalar >>= 1u; } // Convert from Jacobian coordinates back to Affine coordinates convert_jacobian_to_affine(result); // Make sure the resulting point is on the curve VERIFY(is_point_on_curve(result)); // Convert the result back from Montgomery form result.x = from_montgomery(result.x); result.y = from_montgomery(result.y); result.z = from_montgomery(result.z); // Final modular reduction on the coordinates result.x = modular_reduce(result.x); result.y = modular_reduce(result.y); result.z = modular_reduce(result.z); return result; } static ErrorOr read_uncompressed_point(Stream& stream) { // Make sure the point is uncompressed if (TRY(stream.read_value()) != 0x04) return Error::from_string_literal("SECPxxxr1: point is not uncompressed format"); JacobianPoint point { TRY(stream.read_value>()), TRY(stream.read_value>()), 1u, }; return point; } constexpr StorageType select(StorageType const& left, StorageType const& right, bool condition) { // If condition = 0 return left else right StorageType mask = static_cast(condition) - 1; AK::taint_for_optimizer(mask); return (left & mask) | (right & ~mask); } constexpr StorageType modular_reduce(StorageType const& value) { // Add -prime % 2^KEY_BIT_SIZE bool carry = false; StorageType other = value.addc(REDUCE_PRIME, carry); // Check for overflow return select(value, other, carry); } constexpr StorageType modular_reduce_order(StorageType const& value) { // Add -order % 2^KEY_BIT_SIZE bool carry = false; StorageType other = value.addc(REDUCE_ORDER, carry); // Check for overflow return select(value, other, carry); } constexpr StorageType modular_add(StorageType const& left, StorageType const& right, bool carry_in = false) { bool carry = carry_in; StorageType output = left.addc(right, carry); // If there is a carry, subtract p by adding 2^KEY_BIT_SIZE - p StorageType addend = select(0u, REDUCE_PRIME, carry); carry = false; output = output.addc(addend, carry); // If there is still a carry, subtract p by adding 2^KEY_BIT_SIZE - p addend = select(0u, REDUCE_PRIME, carry); return output + addend; } constexpr StorageType modular_sub(StorageType const& left, StorageType const& right) { bool borrow = false; StorageType output = left.subc(right, borrow); // If there is a borrow, add p by subtracting 2^KEY_BIT_SIZE - p StorageType sub = select(0u, REDUCE_PRIME, borrow); borrow = false; output = output.subc(sub, borrow); // If there is still a borrow, add p by subtracting 2^KEY_BIT_SIZE - p sub = select(0u, REDUCE_PRIME, borrow); return output - sub; } constexpr StorageType modular_multiply(StorageType const& left, StorageType const& right) { // Modular multiplication using the Montgomery method: https://en.wikipedia.org/wiki/Montgomery_modular_multiplication // This requires that the inputs to this function are in Montgomery form. // T = left * right StorageTypeX2 mult = left.wide_multiply(right); StorageType mult_mod_r = static_cast(mult); // m = ((T mod R) * curve_p') StorageType m = mult_mod_r * PRIME_INVERSE_MOD_R; // mp = (m mod R) * curve_p StorageTypeX2 mp = m.wide_multiply(PRIME); // t = (T + mp) bool carry = false; mult_mod_r.addc(static_cast(mp), carry); // output = t / R StorageType mult_high = static_cast(mult >> KEY_BIT_SIZE); StorageType mp_high = static_cast(mp >> KEY_BIT_SIZE); return modular_add(mult_high, mp_high, carry); } constexpr StorageType modular_square(StorageType const& value) { return modular_multiply(value, value); } constexpr StorageType to_montgomery(StorageType const& value) { return modular_multiply(value, R2_MOD_PRIME); } constexpr StorageType from_montgomery(StorageType const& value) { return modular_multiply(value, 1u); } constexpr StorageType modular_inverse(StorageType const& value) { // Modular inverse modulo the curve prime can be computed using Fermat's little theorem: a^(p-2) mod p = a^-1 mod p. // Calculating a^(p-2) mod p can be done using the square-and-multiply exponentiation method, as p-2 is constant. StorageType base = value; StorageType result = to_montgomery(1u); StorageType prime_minus_2 = PRIME - 2u; for (size_t i = 0; i < KEY_BIT_SIZE; i++) { if ((prime_minus_2 & 1u) == 1u) { result = modular_multiply(result, base); } base = modular_square(base); prime_minus_2 >>= 1u; } return result; } constexpr StorageType modular_add_order(StorageType const& left, StorageType const& right, bool carry_in = false) { bool carry = carry_in; StorageType output = left.addc(right, carry); // If there is a carry, subtract n by adding 2^KEY_BIT_SIZE - n StorageType addend = select(0u, REDUCE_ORDER, carry); carry = false; output = output.addc(addend, carry); // If there is still a carry, subtract n by adding 2^KEY_BIT_SIZE - n addend = select(0u, REDUCE_ORDER, carry); return output + addend; } constexpr StorageType modular_multiply_order(StorageType const& left, StorageType const& right) { // Modular multiplication using the Montgomery method: https://en.wikipedia.org/wiki/Montgomery_modular_multiplication // This requires that the inputs to this function are in Montgomery form. // T = left * right StorageTypeX2 mult = left.wide_multiply(right); StorageType mult_mod_r = static_cast(mult); // m = ((T mod R) * curve_n') StorageType m = mult_mod_r * ORDER_INVERSE_MOD_R; // mp = (m mod R) * curve_n StorageTypeX2 mp = m.wide_multiply(ORDER); // t = (T + mp) bool carry = false; mult_mod_r.addc(static_cast(mp), carry); // output = t / R StorageType mult_high = static_cast(mult >> KEY_BIT_SIZE); StorageType mp_high = static_cast(mp >> KEY_BIT_SIZE); return modular_add_order(mult_high, mp_high, carry); } constexpr StorageType modular_square_order(StorageType const& value) { return modular_multiply_order(value, value); } constexpr StorageType to_montgomery_order(StorageType const& value) { return modular_multiply_order(value, R2_MOD_ORDER); } constexpr StorageType from_montgomery_order(StorageType const& value) { return modular_multiply_order(value, 1u); } constexpr StorageType modular_inverse_order(StorageType const& value) { // Modular inverse modulo the curve order can be computed using Fermat's little theorem: a^(n-2) mod n = a^-1 mod n. // Calculating a^(n-2) mod n can be done using the square-and-multiply exponentiation method, as n-2 is constant. StorageType base = value; StorageType result = to_montgomery_order(1u); StorageType order_minus_2 = ORDER - 2u; for (size_t i = 0; i < KEY_BIT_SIZE; i++) { if ((order_minus_2 & 1u) == 1u) { result = modular_multiply_order(result, base); } base = modular_square_order(base); order_minus_2 >>= 1u; } return result; } JacobianPoint point_double(JacobianPoint const& point) { // Based on "Point Doubling" from http://point-at-infinity.org/ecc/Prime_Curve_Jacobian_Coordinates.html // if (Y == 0) // return POINT_AT_INFINITY if (point.y.is_zero_constant_time()) { VERIFY_NOT_REACHED(); } StorageType temp; // Y2 = Y^2 StorageType y2 = modular_square(point.y); // S = 4*X*Y2 StorageType s = modular_multiply(point.x, y2); s = modular_add(s, s); s = modular_add(s, s); // M = 3*X^2 + a*Z^4 = 3*(X + Z^2)*(X - Z^2) // This specific equation from https://github.com/earlephilhower/bearssl-esp8266/blob/6105635531027f5b298aa656d44be2289b2d434f/src/ec/ec_p256_m64.c#L811-L816 // This simplification only works because a = -3 mod p temp = modular_square(point.z); StorageType m = modular_add(point.x, temp); temp = modular_sub(point.x, temp); m = modular_multiply(m, temp); temp = modular_add(m, m); m = modular_add(m, temp); // X' = M^2 - 2*S StorageType xp = modular_square(m); xp = modular_sub(xp, s); xp = modular_sub(xp, s); // Y' = M*(S - X') - 8*Y2^2 StorageType yp = modular_sub(s, xp); yp = modular_multiply(yp, m); temp = modular_square(y2); temp = modular_add(temp, temp); temp = modular_add(temp, temp); temp = modular_add(temp, temp); yp = modular_sub(yp, temp); // Z' = 2*Y*Z StorageType zp = modular_multiply(point.y, point.z); zp = modular_add(zp, zp); // return (X', Y', Z') return JacobianPoint { xp, yp, zp }; } JacobianPoint point_add(JacobianPoint const& point_a, JacobianPoint const& point_b) { // Based on "Point Addition" from http://point-at-infinity.org/ecc/Prime_Curve_Jacobian_Coordinates.html if (point_a.x.is_zero_constant_time() && point_a.y.is_zero_constant_time() && point_a.z.is_zero_constant_time()) { return point_b; } StorageType temp; temp = modular_square(point_b.z); // U1 = X1*Z2^2 StorageType u1 = modular_multiply(point_a.x, temp); // S1 = Y1*Z2^3 StorageType s1 = modular_multiply(point_a.y, temp); s1 = modular_multiply(s1, point_b.z); temp = modular_square(point_a.z); // U2 = X2*Z1^2 StorageType u2 = modular_multiply(point_b.x, temp); // S2 = Y2*Z1^3 StorageType s2 = modular_multiply(point_b.y, temp); s2 = modular_multiply(s2, point_a.z); // if (U1 == U2) // if (S1 != S2) // return POINT_AT_INFINITY // else // return POINT_DOUBLE(X1, Y1, Z1) if (u1.is_equal_to_constant_time(u2)) { if (s1.is_equal_to_constant_time(s2)) { return point_double(point_a); } else { VERIFY_NOT_REACHED(); } } // H = U2 - U1 StorageType h = modular_sub(u2, u1); StorageType h2 = modular_square(h); StorageType h3 = modular_multiply(h2, h); // R = S2 - S1 StorageType r = modular_sub(s2, s1); // X3 = R^2 - H^3 - 2*U1*H^2 StorageType x3 = modular_square(r); x3 = modular_sub(x3, h3); temp = modular_multiply(u1, h2); temp = modular_add(temp, temp); x3 = modular_sub(x3, temp); // Y3 = R*(U1*H^2 - X3) - S1*H^3 StorageType y3 = modular_multiply(u1, h2); y3 = modular_sub(y3, x3); y3 = modular_multiply(y3, r); temp = modular_multiply(s1, h3); y3 = modular_sub(y3, temp); // Z3 = H*Z1*Z2 StorageType z3 = modular_multiply(h, point_a.z); z3 = modular_multiply(z3, point_b.z); // return (X3, Y3, Z3) return JacobianPoint { x3, y3, z3 }; } void convert_jacobian_to_affine(JacobianPoint& point) { StorageType temp; // X' = X/Z^2 temp = modular_square(point.z); temp = modular_inverse(temp); point.x = modular_multiply(point.x, temp); // Y' = Y/Z^3 temp = modular_square(point.z); temp = modular_multiply(temp, point.z); temp = modular_inverse(temp); point.y = modular_multiply(point.y, temp); // Z' = 1 point.z = to_montgomery(1u); } bool is_point_on_curve(JacobianPoint const& point) { // This check requires the point to be in Montgomery form, with Z=1 StorageType temp, temp2; // Calulcate Y^2 - X^3 - a*X - b = Y^2 - X^3 + 3*X - b temp = modular_square(point.y); temp2 = modular_square(point.x); temp2 = modular_multiply(temp2, point.x); temp = modular_sub(temp, temp2); temp = modular_add(temp, point.x); temp = modular_add(temp, point.x); temp = modular_add(temp, point.x); temp = modular_sub(temp, to_montgomery(B)); temp = modular_reduce(temp); return temp.is_zero_constant_time() && point.z.is_equal_to_constant_time(to_montgomery(1u)); } }; // SECP256r1 curve static constexpr SECPxxxr1CurveParameters SECP256r1_CURVE_PARAMETERS { .prime = "FFFFFFFF_00000001_00000000_00000000_00000000_FFFFFFFF_FFFFFFFF_FFFFFFFF"sv, .a = "FFFFFFFF_00000001_00000000_00000000_00000000_FFFFFFFF_FFFFFFFF_FFFFFFFC"sv, .b = "5AC635D8_AA3A93E7_B3EBBD55_769886BC_651D06B0_CC53B0F6_3BCE3C3E_27D2604B"sv, .order = "FFFFFFFF_00000000_FFFFFFFF_FFFFFFFF_BCE6FAAD_A7179E84_F3B9CAC2_FC632551"sv, .generator_point = "04_6B17D1F2_E12C4247_F8BCE6E5_63A440F2_77037D81_2DEB33A0_F4A13945_D898C296_4FE342E2_FE1A7F9B_8EE7EB4A_7C0F9E16_2BCE3357_6B315ECE_CBB64068_37BF51F5"sv, }; using SECP256r1 = SECPxxxr1<256, SECP256r1_CURVE_PARAMETERS>; // SECP384r1 curve static constexpr SECPxxxr1CurveParameters SECP384r1_CURVE_PARAMETERS { .prime = "FFFFFFFF_FFFFFFFF_FFFFFFFF_FFFFFFFF_FFFFFFFF_FFFFFFFF_FFFFFFFF_FFFFFFFE_FFFFFFFF_00000000_00000000_FFFFFFFF"sv, .a = "FFFFFFFF_FFFFFFFF_FFFFFFFF_FFFFFFFF_FFFFFFFF_FFFFFFFF_FFFFFFFF_FFFFFFFE_FFFFFFFF_00000000_00000000_FFFFFFFC"sv, .b = "B3312FA7_E23EE7E4_988E056B_E3F82D19_181D9C6E_FE814112_0314088F_5013875A_C656398D_8A2ED19D_2A85C8ED_D3EC2AEF"sv, .order = "FFFFFFFF_FFFFFFFF_FFFFFFFF_FFFFFFFF_FFFFFFFF_FFFFFFFF_C7634D81_F4372DDF_581A0DB2_48B0A77A_ECEC196A_CCC52973"sv, .generator_point = "04_AA87CA22_BE8B0537_8EB1C71E_F320AD74_6E1D3B62_8BA79B98_59F741E0_82542A38_5502F25D_BF55296C_3A545E38_72760AB7_3617DE4A_96262C6F_5D9E98BF_9292DC29_F8F41DBD_289A147C_E9DA3113_B5F0B8C0_0A60B1CE_1D7E819D_7A431D7C_90EA0E5F"sv, }; using SECP384r1 = SECPxxxr1<384, SECP384r1_CURVE_PARAMETERS>; }