serenity/Userland/Libraries/LibCrypto/Curves/SECP256r1.cpp

/*
 * Copyright (c) 2022, Michiel Visser <opensource@webmichiel.nl>
 *
 * SPDX-License-Identifier: BSD-2-Clause
 */

#include <AK/ByteReader.h>
#include <AK/Endian.h>
#include <AK/Random.h>
#include <AK/StringBuilder.h>
#include <AK/UFixedBigInt.h>
#include <AK/UFixedBigIntDivision.h>
#include <LibCrypto/ASN1/DER.h>
#include <LibCrypto/Curves/SECP256r1.h>

namespace Crypto::Curves {

struct JacobianPoint {
    u256 x { 0u };
    u256 y { 0u };
    u256 z { 0u };
};

static constexpr u256 calculate_modular_inverse_mod_r(u256 value)
{
    // Calculate the modular multiplicative inverse of value mod 2^256 using the extended euclidean algorithm
    u512 old_r = value;
    u512 r = static_cast<u512>(1u) << 256u;
    u512 old_s = 1u;
    u512 s = 0u;

    while (!r.is_zero_constant_time()) {
        u512 quotient = old_r / r;
        u512 temp = r;
        r = old_r - quotient * r;
        old_r = temp;

        temp = s;
        s = old_s - quotient * s;
        old_s = temp;
    }

    return old_s.low();
}

static constexpr u256 calculate_r2_mod(u256 modulus)
{
    // Calculate the value of R^2 mod modulus, where R = 2^256
    u1024 r = static_cast<u1024>(1u) << 256u;
    u1024 r2 = r * r;
    u1024 result = r2 % static_cast<u1024>(modulus);
    return result.low().low();
}

// SECP256r1 curve parameters
static constexpr u256 PRIME { { 0xffffffffffffffffull, 0x00000000ffffffffull, 0x0000000000000000ull, 0xffffffff00000001ull } };
static constexpr u256 A { { 0xfffffffffffffffcull, 0x00000000ffffffffull, 0x0000000000000000ull, 0xffffffff00000001ull } };
static constexpr u256 B { { 0x3bce3c3e27d2604bull, 0x651d06b0cc53b0f6ull, 0xb3ebbd55769886bcull, 0x5ac635d8aa3a93e7ull } };
static constexpr u256 ORDER { { 0xf3b9cac2fc632551ull, 0xbce6faada7179e84ull, 0xffffffffffffffffull, 0xffffffff00000000ull } };

// 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 u256 REDUCE_PRIME = u256 { 0 } - PRIME;
static constexpr u256 REDUCE_ORDER = u256 { 0 } - ORDER;
static constexpr u256 PRIME_INVERSE_MOD_R = u256 { 0 } - calculate_modular_inverse_mod_r(PRIME);
static constexpr u256 ORDER_INVERSE_MOD_R = u256 { 0 } - calculate_modular_inverse_mod_r(ORDER);
static constexpr u256 R2_MOD_PRIME = calculate_r2_mod(PRIME);
static constexpr u256 R2_MOD_ORDER = calculate_r2_mod(ORDER);

static u256 import_big_endian(ReadonlyBytes data)
{
    VERIFY(data.size() == 32);

    u64 d = AK::convert_between_host_and_big_endian(ByteReader::load64(data.offset_pointer(0 * sizeof(u64))));
    u64 c = AK::convert_between_host_and_big_endian(ByteReader::load64(data.offset_pointer(1 * sizeof(u64))));
    u64 b = AK::convert_between_host_and_big_endian(ByteReader::load64(data.offset_pointer(2 * sizeof(u64))));
    u64 a = AK::convert_between_host_and_big_endian(ByteReader::load64(data.offset_pointer(3 * sizeof(u64))));

    return u256 { { a, b, c, d } };
}

static void export_big_endian(u256 const& value, Bytes data)
{
    u64 a = AK::convert_between_host_and_big_endian(value.low().low());
    u64 b = AK::convert_between_host_and_big_endian(value.low().high());
    u64 c = AK::convert_between_host_and_big_endian(value.high().low());
    u64 d = AK::convert_between_host_and_big_endian(value.high().high());

    ByteReader::store(data.offset_pointer(3 * sizeof(u64)), a);
    ByteReader::store(data.offset_pointer(2 * sizeof(u64)), b);
    ByteReader::store(data.offset_pointer(1 * sizeof(u64)), c);
    ByteReader::store(data.offset_pointer(0 * sizeof(u64)), d);
}

static constexpr u256 select(u256 const& left, u256 const& right, bool condition)
{
    // If condition = 0 return left else right
    u256 mask = (u256)condition - 1;

    return (left & mask) | (right & ~mask);
}

static constexpr u512 multiply(u256 const& left, u256 const& right)
{
    return left.wide_multiply(right);
}

static constexpr u256 modular_reduce(u256 const& value)
{
    // Add -prime % 2^256 = 2^224-2^192-2^96+1
    bool carry = false;
    u256 other = value.addc(REDUCE_PRIME, carry);

    // Check for overflow
    return select(value, other, carry);
}

static constexpr u256 modular_reduce_order(u256 const& value)
{
    // Add -order % 2^256
    bool carry = false;
    u256 other = value.addc(REDUCE_ORDER, carry);

    // Check for overflow
    return select(value, other, carry);
}

static constexpr u256 modular_add(u256 const& left, u256 const& right, bool carry_in = false)
{
    bool carry = carry_in;
    u256 output = left.addc(right, carry);

    // If there is a carry, subtract p by adding 2^256 - p
    u256 addend = select(0u, REDUCE_PRIME, carry);
    carry = false;
    output = output.addc(addend, carry);

    // If there is still a carry, subtract p by adding 2^256 - p
    addend = select(0u, REDUCE_PRIME, carry);
    return output + addend;
}

static constexpr u256 modular_sub(u256 const& left, u256 const& right)
{
    bool borrow = false;
    u256 output = left.subc(right, borrow);

    // If there is a borrow, add p by subtracting 2^256 - p
    u256 sub = select(0u, REDUCE_PRIME, borrow);
    borrow = false;
    output = output.subc(sub, borrow);

    // If there is still a borrow, add p by subtracting 2^256 - p
    sub = select(0u, REDUCE_PRIME, borrow);
    return output - sub;
}

static constexpr u256 modular_multiply(u256 const& left, u256 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
    u512 mult = multiply(left, right);

    // m = ((T mod R) * curve_p')
    u512 m = multiply(mult.low(), PRIME_INVERSE_MOD_R);

    // mp = (m mod R) * curve_p
    u512 mp = multiply(m.low(), PRIME);

    // t = (T + mp)
    bool carry = false;
    mult.low().addc(mp.low(), carry);

    // output = t / R
    return modular_add(mult.high(), mp.high(), carry);
}

static constexpr u256 modular_square(u256 const& value)
{
    return modular_multiply(value, value);
}

static constexpr u256 to_montgomery(u256 const& value)
{
    return modular_multiply(value, R2_MOD_PRIME);
}

static constexpr u256 from_montgomery(u256 const& value)
{
    return modular_multiply(value, 1u);
}

static constexpr u256 modular_inverse(u256 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.
    u256 base = value;
    u256 result = to_montgomery(1u);
    u256 prime_minus_2 = PRIME - 2u;

    for (size_t i = 0; i < 256; i++) {
        if ((prime_minus_2 & 1u) == 1u) {
            result = modular_multiply(result, base);
        }
        base = modular_square(base);
        prime_minus_2 >>= 1u;
    }

    return result;
}

static constexpr u256 modular_add_order(u256 const& left, u256 const& right, bool carry_in = false)
{
    bool carry = carry_in;
    u256 output = left.addc(right, carry);

    // If there is a carry, subtract n by adding 2^256 - n
    u256 addend = select(0u, REDUCE_ORDER, carry);
    carry = false;
    output = output.addc(addend, carry);

    // If there is still a carry, subtract n by adding 2^256 - n
    addend = select(0u, REDUCE_ORDER, carry);
    return output + addend;
}

static constexpr u256 modular_multiply_order(u256 const& left, u256 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
    u512 mult = multiply(left, right);

    // m = ((T mod R) * curve_n')
    u512 m = multiply(mult.low(), ORDER_INVERSE_MOD_R);

    // mp = (m mod R) * curve_n
    u512 mp = multiply(m.low(), ORDER);

    // t = (T + mp)
    bool carry = false;
    mult.low().addc(mp.low(), carry);

    // output = t / R
    return modular_add_order(mult.high(), mp.high(), carry);
}

static constexpr u256 modular_square_order(u256 const& value)
{
    return modular_multiply_order(value, value);
}

static constexpr u256 to_montgomery_order(u256 const& value)
{
    return modular_multiply_order(value, R2_MOD_ORDER);
}

static constexpr u256 from_montgomery_order(u256 const& value)
{
    return modular_multiply_order(value, 1u);
}

static constexpr u256 modular_inverse_order(u256 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.
    u256 base = value;
    u256 result = to_montgomery_order(1u);
    u256 order_minus_2 = ORDER - 2u;

    for (size_t i = 0; i < 256; i++) {
        if ((order_minus_2 & 1u) == 1u) {
            result = modular_multiply_order(result, base);
        }
        base = modular_square_order(base);
        order_minus_2 >>= 1u;
    }

    return result;
}

static void point_double(JacobianPoint& output_point, 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();
    }

    u256 temp;

    // Y2 = Y^2
    u256 y2 = modular_square(point.y);

    // S = 4*X*Y2
    u256 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);
    u256 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
    u256 xp = modular_square(m);
    xp = modular_sub(xp, s);
    xp = modular_sub(xp, s);

    // Y' = M*(S - X') - 8*Y2^2
    u256 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
    u256 zp = modular_multiply(point.y, point.z);
    zp = modular_add(zp, zp);

    // return (X', Y', Z')
    output_point.x = xp;
    output_point.y = yp;
    output_point.z = zp;
}

static void point_add(JacobianPoint& output_point, 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()) {
        output_point.x = point_b.x;
        output_point.y = point_b.y;
        output_point.z = point_b.z;
        return;
    }

    u256 temp;

    temp = modular_square(point_b.z);
    // U1 = X1*Z2^2
    u256 u1 = modular_multiply(point_a.x, temp);
    // S1 = Y1*Z2^3
    u256 s1 = modular_multiply(point_a.y, temp);
    s1 = modular_multiply(s1, point_b.z);

    temp = modular_square(point_a.z);
    // U2 = X2*Z1^2
    u256 u2 = modular_multiply(point_b.x, temp);
    // S2 = Y2*Z1^3
    u256 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)) {
            point_double(output_point, point_a);
            return;
        } else {
            VERIFY_NOT_REACHED();
        }
    }

    // H = U2 - U1
    u256 h = modular_sub(u2, u1);
    u256 h2 = modular_square(h);
    u256 h3 = modular_multiply(h2, h);
    // R = S2 - S1
    u256 r = modular_sub(s2, s1);
    // X3 = R^2 - H^3 - 2*U1*H^2
    u256 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
    u256 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
    u256 z3 = modular_multiply(h, point_a.z);
    z3 = modular_multiply(z3, point_b.z);
    // return (X3, Y3, Z3)
    output_point.x = x3;
    output_point.y = y3;
    output_point.z = z3;
}

static void convert_jacobian_to_affine(JacobianPoint& point)
{
    u256 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);
}

static bool is_point_on_curve(JacobianPoint const& point)
{
    // This check requires the point to be in Montgomery form, with Z=1
    u256 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));
}

ErrorOr<ByteBuffer> SECP256r1::generate_private_key()
{
    auto buffer = TRY(ByteBuffer::create_uninitialized(32));
    fill_with_random(buffer);
    return buffer;
}

ErrorOr<ByteBuffer> SECP256r1::generate_public_key(ReadonlyBytes a)
{
    // clang-format off
    u8 generator_bytes[65] {
        0x04,
        0x6B, 0x17, 0xD1, 0xF2, 0xE1, 0x2C, 0x42, 0x47, 0xF8, 0xBC, 0xE6, 0xE5, 0x63, 0xA4, 0x40, 0xF2,
        0x77, 0x03, 0x7D, 0x81, 0x2D, 0xEB, 0x33, 0xA0, 0xF4, 0xA1, 0x39, 0x45, 0xD8, 0x98, 0xC2, 0x96,
        0x4F, 0xE3, 0x42, 0xE2, 0xFE, 0x1A, 0x7F, 0x9B, 0x8E, 0xE7, 0xEB, 0x4A, 0x7C, 0x0F, 0x9E, 0x16,
        0x2B, 0xCE, 0x33, 0x57, 0x6B, 0x31, 0x5E, 0xCE, 0xCB, 0xB6, 0x40, 0x68, 0x37, 0xBF, 0x51, 0xF5,
    };
    // clang-format on
    return compute_coordinate(a, { generator_bytes, 65 });
}

ErrorOr<ByteBuffer> SECP256r1::compute_coordinate(ReadonlyBytes scalar_bytes, ReadonlyBytes point_bytes)
{
    VERIFY(scalar_bytes.size() == 32);

    u256 scalar = import_big_endian(scalar_bytes);
    // 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("SECP256r1: scalar is zero");

    // Make sure the point is uncompressed
    if (point_bytes.size() != 65 || point_bytes[0] != 0x04)
        return Error::from_string_literal("SECP256r1: point is not uncompressed format");

    JacobianPoint point {
        import_big_endian(point_bytes.slice(1, 32)),
        import_big_endian(point_bytes.slice(33, 32)),
        1u,
    };

    // 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("SECP256r1: point is not on the curve");

    JacobianPoint result;
    JacobianPoint temp_result;

    // Calculate the scalar times point multiplication in constant time
    for (auto i = 0; i < 256; i++) {
        point_add(temp_result, 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_double(point, 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);
    // Final modular reduction on the coordinates
    result.x = modular_reduce(result.x);
    result.y = modular_reduce(result.y);

    // Export the values into an output buffer
    auto buf = TRY(ByteBuffer::create_uninitialized(65));
    buf[0] = 0x04;
    export_big_endian(result.x, buf.bytes().slice(1, 32));
    export_big_endian(result.y, buf.bytes().slice(33, 32));
    return buf;
}

ErrorOr<ByteBuffer> SECP256r1::derive_premaster_key(ReadonlyBytes shared_point)
{
    VERIFY(shared_point.size() == 65);
    VERIFY(shared_point[0] == 0x04);

    ByteBuffer premaster_key = TRY(ByteBuffer::create_uninitialized(32));
    premaster_key.overwrite(0, shared_point.data() + 1, 32);
    return premaster_key;
}

ErrorOr<bool> SECP256r1::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::UnsignedBigInteger>(Crypto::ASN1::Class::Universal, Crypto::ASN1::Kind::Integer));
    auto s_bigint = TRY(asn1_decoder.read<Crypto::UnsignedBigInteger>(Crypto::ASN1::Class::Universal, Crypto::ASN1::Kind::Integer));

    u256 r = 0u;
    u256 s = 0u;
    for (size_t i = 0; i < 8; i++) {
        u256 rr = r_bigint.words()[i];
        u256 ss = s_bigint.words()[i];
        r |= (rr << (i * 32));
        s |= (ss << (i * 32));
    }

    // z is the hash
    u256 z = import_big_endian(hash.slice(0, 32));

    u256 r_mo = to_montgomery_order(r);
    u256 s_mo = to_montgomery_order(s);
    u256 z_mo = to_montgomery_order(z);

    u256 s_inv = modular_inverse_order(s_mo);

    u256 u1 = modular_multiply_order(z_mo, s_inv);
    u256 u2 = modular_multiply_order(r_mo, s_inv);

    u1 = from_montgomery_order(u1);
    u2 = from_montgomery_order(u2);

    auto u1_buf = TRY(ByteBuffer::create_uninitialized(32));
    export_big_endian(u1, u1_buf.bytes());
    auto u2_buf = TRY(ByteBuffer::create_uninitialized(32));
    export_big_endian(u2, u2_buf.bytes());

    auto p1 = TRY(generate_public_key(u1_buf));
    auto p2 = TRY(compute_coordinate(u2_buf, pubkey));

    JacobianPoint point1 {
        import_big_endian(TRY(p1.slice(1, 32))),
        import_big_endian(TRY(p1.slice(33, 32))),
        1u,
    };

    // 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));

    JacobianPoint point2 {
        import_big_endian(TRY(p2.slice(1, 32))),
        import_big_endian(TRY(p2.slice(33, 32))),
        1u,
    };

    // 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(result, 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);
}

}