//------------------------------------------------------------------------------
//
// ed25519_fpga_curve_abstract.cpp
// ------------------------------------------------
// Elliptic curve arithmetic procedures for Ed25519
//
// Authors: Pavel Shatov
//
// Copyright (c) 2018 NORDUnet A/S
//
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are met:
//
// - Redistributions of source code must retain the above copyright notice,
// this list of conditions and the following disclaimer.
//
// - Redistributions in binary form+ must reproduce the above copyright notice,
// this list of conditions and the following disclaimer in the documentation
// and/or other materials provided with the distribution.
//
// - Neither the name of the NORDUnet nor the names of its contributors may be
// used to endorse or promote products derived from this software without
// specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
// AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
// ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
// LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
// CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
// SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
// INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
// CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
// ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
// POSSIBILITY OF SUCH DAMAGE.
//
//------------------------------------------------------------------------------
#include <stdio.h>
//------------------------------------------------------------------------------
// Headers
//------------------------------------------------------------------------------
#include "ed25519_fpga_model.h"
//------------------------------------------------------------------------------
// Globals
//------------------------------------------------------------------------------
FPGA_BUFFER ED25519_G_X; // x-coordinate of the base point
FPGA_BUFFER ED25519_G_Y; // y-coordinate of the base point
FPGA_BUFFER ED25519_G_Z; // z-coordinate of the base point
FPGA_BUFFER ED25519_G_T; // y-coordinate of the base point
//------------------------------------------------------------------------------
void fpga_curve_ed25519_init()
//------------------------------------------------------------------------------
{
int w_src, w_dst; // word counters
FPGA_WORD TMP_G_X[FPGA_OPERAND_NUM_WORDS] = ED25519_G_X_INIT;
FPGA_WORD TMP_G_Y[FPGA_OPERAND_NUM_WORDS] = ED25519_G_Y_INIT;
FPGA_WORD TMP_G_Z[FPGA_OPERAND_NUM_WORDS] = ED25519_G_Z_INIT;
FPGA_WORD TMP_G_T[FPGA_OPERAND_NUM_WORDS] = ED25519_G_T_INIT;
/* fill buffers for large multi-word integers */
for ( w_src = 0, w_dst = FPGA_OPERAND_NUM_WORDS - 1;
w_src < FPGA_OPERAND_NUM_WORDS;
w_src++, w_dst--)
{
ED25519_G_X.words[w_dst] = TMP_G_X[w_src];
ED25519_G_Y.words[w_dst] = TMP_G_Y[w_src];
ED25519_G_Z.words[w_dst] = TMP_G_Z[w_src];
ED25519_G_T.words[w_dst] = TMP_G_T[w_src];
}
}
//------------------------------------------------------------------------------
//
// Elliptic curve base point scalar multiplication routine.
//
// This uses Algorithm 4 ("Joye double-and-add") from "Fast and Regular
// Algorithms for Scalar Multiplication over Elliptic Curves"
// https://eprint.iacr.org/2011/338.pdf
//
//------------------------------------------------------------------------------
void fpga_curve_ed25519_base_scalar_multiply_abstract(const FPGA_BUFFER *K, FPGA_BUFFER *Q_Y)
//------------------------------------------------------------------------------
{
int word_count, bit_count; // counters
// temporary buffers
FPGA_BUFFER R0_X;
FPGA_BUFFER R0_Y;
FPGA_BUFFER R0_Z;
FPGA_BUFFER R0_T;
FPGA_BUFFER R1_X;
FPGA_BUFFER R1_Y;
FPGA_BUFFER R1_Z;
FPGA_BUFFER R1_T;
FPGA_BUFFER T_X;
FPGA_BUFFER T_Y;
FPGA_BUFFER T_Z;
FPGA_BUFFER T_T;
// initialization
fpga_multiword_copy(&CURVE25519_ZERO, &R0_X);
fpga_multiword_copy(&CURVE25519_ONE, &R0_Y);
fpga_multiword_copy(&CURVE25519_ONE, &R0_Z);
fpga_multiword_copy(&CURVE25519_ZERO, &R0_T);
fpga_multiword_copy(&ED25519_G_X, &R1_X);
fpga_multiword_copy(&ED25519_G_Y, &R1_Y);
fpga_multiword_copy(&ED25519_G_Z, &R1_Z);
fpga_multiword_copy(&ED25519_G_T, &R1_T);
// force zero bits
FPGA_BUFFER K_INT;
fpga_multiword_copy(K, &K_INT);
K_INT.words[0] &= 0xFFFFFFF8;
K_INT.words[FPGA_OPERAND_NUM_WORDS-1] &= 0x3FFFFFFF;
K_INT.words[FPGA_OPERAND_NUM_WORDS-1] |= 0x40000000;
// handy vars
FPGA_WORD k_word;
bool k_bit = false;
// multiply
for (word_count=0; word_count<FPGA_OPERAND_NUM_WORDS; word_count++)
{
for (bit_count=0; bit_count<FPGA_WORD_WIDTH; bit_count++)
{
// get current bit of K
k_word = K_INT.words[word_count] >> bit_count;
k_bit = (k_word & (FPGA_WORD)1) == 1;
// symmetric processing scheme regardless of the current private bit value
if (k_bit)
{
// T = double(R0)
fpga_curve_ed25519_double( &R0_X, &R0_Y, &R0_Z, &R0_T,
&T_X, &T_Y, &T_Z, &T_T);
// R0 = add(T, R1)
fpga_curve_ed25519_add( &T_X, &T_Y, &T_Z, &T_T,
&R1_X, &R1_Y, &R1_Z, &R1_T,
&R0_X, &R0_Y, &R0_Z, &R0_T);
}
else
{
// T = double(R1)
fpga_curve_ed25519_double( &R1_X, &R1_Y, &R1_Z, &R1_T,
&T_X, &T_Y, &T_Z, &T_T);
// R1 = add(T, R0)
fpga_curve_ed25519_add( &T_X, &T_Y, &T_Z, &T_T,
&R0_X, &R0_Y, &R0_Z, &R0_T,
&R1_X, &R1_Y, &R1_Z, &R1_T);
}
}
}
// now conversion to affine coordinates
fpga_curve_ed25519_to_affine(&R0_X, &R0_Y, &R0_Z, &R1_X, &R1_Y);
// so far we've done everything modulo 2*P, we now need
// to do final reduction modulo P, this can be done using
// our modular adder this way:
fpga_modular_add(&R1_X, &CURVE25519_ZERO, &R0_X, &CURVE25519_1P);
fpga_modular_add(&R1_Y, &CURVE25519_ZERO, &R0_Y, &CURVE25519_1P);
// process "sign" of x, see this Cryptography Stack Exchange
// answer for more details:
//
// https://crypto.stackexchange.com/questions/58921/decoding-a-ed25519-key-per-rfc8032
//
// the short story is that odd values of x are negative, so we
// just copy the lsb of x into msb of y
R0_Y.words[FPGA_OPERAND_NUM_WORDS-1] |= (R0_X.words[0] & (FPGA_WORD)1) << 31;
// store result
fpga_multiword_copy(&R0_Y, Q_Y);
}
//------------------------------------------------------------------------------
//
// Elliptic curve point doubling routine.
//
// This implements the "dbl-2008-hwcd" formulae set from
// https://hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html
//
// The only difference is that E, F, G and H have opposite signs, this is
// equivalent to the original algorithm, because the end result depends on
// (E * F) and (G * H). If both variables have opposite signs, then the
// sign of the product doesn't change.
//
//------------------------------------------------------------------------------
void fpga_curve_ed25519_double( const FPGA_BUFFER *P_X, const FPGA_BUFFER *P_Y, const FPGA_BUFFER *P_Z, const FPGA_BUFFER *P_T,
FPGA_BUFFER *Q_X, FPGA_BUFFER *Q_Y, FPGA_BUFFER *Q_Z, FPGA_BUFFER *Q_T)
{
FPGA_BUFFER A, B, C, D, E, F, G, H, I;
fpga_modular_mul(P_X, P_X, &A, &CURVE25519_2P); // A = (qx * qx) % mod
fpga_modular_mul(P_Y, P_Y, &B, &CURVE25519_2P); // B = (qy * qy) % mod
fpga_modular_mul(P_Z, P_Z, &I, &CURVE25519_2P); // I = (qz * qz) % mod
fpga_modular_add( &I, &I, &C, &CURVE25519_2P); // C = ( I + I) % mod
fpga_modular_add(P_X, P_Y, &I, &CURVE25519_2P); // I = (qx + qy) % mod
fpga_modular_mul( &I, &I, &D, &CURVE25519_2P); // D = ( I * I) % mod
fpga_modular_add( &A, &B, &H, &CURVE25519_2P); // H = ( A + B) % mod
fpga_modular_sub( &H, &D, &E, &CURVE25519_2P); // E = ( H - D) % mod
fpga_modular_sub( &A, &B, &G, &CURVE25519_2P); // G = ( A - B) % mod
fpga_modular_add( &C, &G, &F, &CURVE25519_2P); // F = ( C + G) % mod
fpga_modular_mul( &E, &F, Q_X, &CURVE25519_2P); // rx = ( E * F) % mod
fpga_modular_mul( &G, &H, Q_Y, &CURVE25519_2P); // ry = ( G * H) % mod
fpga_modular_mul( &E, &H, Q_T, &CURVE25519_2P); // rt = ( E * H) % mod
fpga_modular_mul( &F, &G, Q_Z, &CURVE25519_2P); // rz = ( F * G) % mod
}
//------------------------------------------------------------------------------
//
// Elliptic curve point addition routine.
//
// This implements the "add-2008-hwcd-4" formulae set from
// https://hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html
//
//------------------------------------------------------------------------------
void fpga_curve_ed25519_add( const FPGA_BUFFER *P_X, const FPGA_BUFFER *P_Y, const FPGA_BUFFER *P_Z, const FPGA_BUFFER *P_T,
const FPGA_BUFFER *Q_X, const FPGA_BUFFER *Q_Y, const FPGA_BUFFER *Q_Z, const FPGA_BUFFER *Q_T,
FPGA_BUFFER *R_X, FPGA_BUFFER *R_Y, FPGA_BUFFER *R_Z, FPGA_BUFFER *R_T)
{
FPGA_BUFFER A, B, C, D, E, F, G, H, I, J;
fpga_modular_sub(P_Y, P_X, &I, &CURVE25519_2P); // I = (qy - qx) % mod
fpga_modular_add(Q_Y, Q_X, &J, &CURVE25519_2P); // J = (py + px) % mod
fpga_modular_mul( &I, &J, &A, &CURVE25519_2P); // A = ( I * J) % mod
fpga_modular_add(P_Y, P_X, &I, &CURVE25519_2P); // I = (qy + qx) % mod
fpga_modular_sub(Q_Y, Q_X, &J, &CURVE25519_2P); // J = (py - px) % mod
fpga_modular_mul( &I, &J, &B, &CURVE25519_2P); // B = ( I * J) % mod
fpga_modular_mul(P_Z, Q_T, &I, &CURVE25519_2P); // I = (qz * pt) % mod
fpga_modular_add( &I, &I, &C, &CURVE25519_2P); // C = ( I + I) % mod
fpga_modular_mul(P_T, Q_Z, &I, &CURVE25519_2P); // I = (qt * pz) % mod
fpga_modular_add( &I, &I, &D, &CURVE25519_2P); // D = ( I + I) % mod
fpga_modular_add( &D, &C, &E, &CURVE25519_2P); // E = (D + C) % mod
fpga_modular_sub( &B, &A, &F, &CURVE25519_2P); // F = (B - A) % mod
fpga_modular_add( &B, &A, &G, &CURVE25519_2P); // G = (B + A) % mod
fpga_modular_sub( &D, &C, &H, &CURVE25519_2P); // H = (D - C) % mod
fpga_modular_mul( &E, &F, R_X, &CURVE25519_2P); // rx = (E * F) % mod
fpga_modular_mul( &G, &H, R_Y, &CURVE25519_2P); // ry = (G * H) % mod
fpga_modular_mul( &E, &H, R_T, &CURVE25519_2P); // rt = (E * H) % mod
fpga_modular_mul( &F, &G, R_Z, &CURVE25519_2P); // rz = (F * G) % mod
}
//------------------------------------------------------------------------------
//
// Conversion to affine coordinates.
//
// Q_X = P_X / P_Z = P_X * P_Z ^ -1
// Q_Y = P_Y / P_Z = P_Y * P_Z ^ -1
//
//------------------------------------------------------------------------------
void fpga_curve_ed25519_to_affine (const FPGA_BUFFER *P_X, const FPGA_BUFFER *P_Y,
const FPGA_BUFFER *P_Z,
FPGA_BUFFER *Q_X, FPGA_BUFFER *Q_Y)
//------------------------------------------------------------------------------
{
FPGA_BUFFER P_Z_1;
fpga_modular_inv_abstract(P_Z, &P_Z_1, &CURVE25519_2P);
fpga_modular_mul(P_X, &P_Z_1, Q_X, &CURVE25519_2P);
fpga_modular_mul(P_Y, &P_Z_1, Q_Y, &CURVE25519_2P);
}
//------------------------------------------------------------------------------
// End-of-File
//------------------------------------------------------------------------------
