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/********************************************************************************************
* SIDH: an efficient supersingular isogeny cryptography library
*
* Abstract: core functions over GF(p) and GF(p^2)
*********************************************************************************************/
// Conversion of GF(p^2) element from Montgomery to standard representation, and encoding by removing leading 0 bytes
void fp2_encode(const f2elm_t *x, unsigned char *enc) {
unsigned int i;
f2elm_t t;
from_fp2mont(x, &t);
for (i = 0; i < FP2_ENCODED_BYTES / 2; i++) {
enc[i] = ((unsigned char *) t.e)[i];
enc[i + FP2_ENCODED_BYTES / 2] = ((unsigned char *) t.e)[i + MAXBITS_FIELD / 8];
}
}
// Parse byte sequence back into GF(p^2) element, and conversion to Montgomery representation
void fp2_decode(const unsigned char *enc, f2elm_t *x) {
unsigned int i;
for (i = 0; i < 2 * (MAXBITS_FIELD / 8); i++)
((unsigned char *) x->e)[i] = 0;
for (i = 0; i < FP2_ENCODED_BYTES / 2; i++) {
((unsigned char *) x->e)[i] = enc[i];
((unsigned char *) x->e)[i + MAXBITS_FIELD / 8] = enc[i + FP2_ENCODED_BYTES / 2];
}
to_fp2mont(x, x);
}
// Copy a field element, c = a.
__inline void fpcopy(const felm_t a, felm_t c) {
unsigned int i;
for (i = 0; i < NWORDS_FIELD; i++)
c[i] = a[i];
}
// Zero a field element, a = 0.
__inline void fpzero(felm_t a) {
unsigned int i;
for (i = 0; i < NWORDS_FIELD; i++)
a[i] = 0;
}
// Conversion to Montgomery representation,
// mc = a*R^2*R^(-1) mod p = a*R mod p, where a in [0, p-1].
// The Montgomery constant R^2 mod p is the global value "Montgomery_R2".
void to_mont(const felm_t a, felm_t mc) {
fpmul_mont(a, (const digit_t *) &Montgomery_R2, mc);
}
// Conversion from Montgomery representation to standard representation,
// c = ma*R^(-1) mod p = a mod p, where ma in [0, p-1].
void from_mont(const felm_t ma, felm_t c) {
digit_t one[NWORDS_FIELD] = {0};
one[0] = 1;
fpmul_mont(ma, one, c);
fpcorrection(c);
}
// Copy wordsize digits, c = a, where lng(a) = nwords.
void copy_words(const digit_t *a, digit_t *c, const unsigned int nwords) {
unsigned int i;
for (i = 0; i < nwords; i++)
c[i] = a[i];
}
// Multiprecision multiplication, c = a*b mod p.
void fpmul_mont(const felm_t ma, const felm_t mb, felm_t mc) {
dfelm_t temp = {0};
mp_mul(ma, mb, temp, NWORDS_FIELD);
rdc_mont(temp, mc);
}
// Multiprecision squaring, c = a^2 mod p.
void fpsqr_mont(const felm_t ma, felm_t mc) {
dfelm_t temp = {0};
mp_mul(ma, ma, temp, NWORDS_FIELD);
rdc_mont(temp, mc);
}
// Field inversion using Montgomery arithmetic, a = a^(-1)*R mod p.
void fpinv_mont(felm_t a) {
felm_t tt;
fpcopy(a, tt);
fpinv_chain_mont(tt);
fpsqr_mont(tt, tt);
fpsqr_mont(tt, tt);
fpmul_mont(a, tt, a);
}
// Copy a GF(p^2) element, c = a.
void fp2copy(const f2elm_t *a, f2elm_t *c) {
fpcopy(a->e[0], c->e[0]);
fpcopy(a->e[1], c->e[1]);
}
// Zero a GF(p^2) element, a = 0.
void fp2zero(f2elm_t *a) {
fpzero(a->e[0]);
fpzero(a->e[1]);
}
// GF(p^2) negation, a = -a in GF(p^2).
void fp2neg(f2elm_t *a) {
fpneg(a->e[0]);
fpneg(a->e[1]);
}
// GF(p^2) addition, c = a+b in GF(p^2).
__inline void fp2add(const f2elm_t *a, const f2elm_t *b, f2elm_t *c) {
fpadd(a->e[0], b->e[0], c->e[0]);
fpadd(a->e[1], b->e[1], c->e[1]);
}
// GF(p^2) subtraction, c = a-b in GF(p^2).
__inline void fp2sub(const f2elm_t *a, const f2elm_t *b, f2elm_t *c) {
fpsub(a->e[0], b->e[0], c->e[0]);
fpsub(a->e[1], b->e[1], c->e[1]);
}
// GF(p^2) division by two, c = a/2 in GF(p^2).
void fp2div2(const f2elm_t *a, f2elm_t *c) {
fpdiv2(a->e[0], c->e[0]);
fpdiv2(a->e[1], c->e[1]);
}
// Modular correction, a = a in GF(p^2).
void fp2correction(f2elm_t *a) {
fpcorrection(a->e[0]);
fpcorrection(a->e[1]);
}
// Multiprecision addition, c = a+b.
__inline static void mp_addfast(const digit_t *a, const digit_t *b, digit_t *c) {
#if defined(S2N_SIKEP434R2_ASM)
if (s2n_sikep434r2_asm_is_enabled()) {
mp_add_asm(a, b, c);
return;
}
#endif
mp_add(a, b, c, NWORDS_FIELD);
}
// GF(p^2) squaring using Montgomery arithmetic, c = a^2 in GF(p^2).
// Inputs: a = a0+a1*i, where a0, a1 are in [0, 2*p-1]
// Output: c = c0+c1*i, where c0, c1 are in [0, 2*p-1]
void fp2sqr_mont(const f2elm_t *a, f2elm_t *c) {
felm_t t1, t2, t3;
mp_addfast(a->e[0], a->e[1], t1); // t1 = a0+a1
fpsub(a->e[0], a->e[1], t2); // t2 = a0-a1
mp_addfast(a->e[0], a->e[0], t3); // t3 = 2a0
fpmul_mont(t1, t2, c->e[0]); // c0 = (a0+a1)(a0-a1)
fpmul_mont(t3, a->e[1], c->e[1]); // c1 = 2a0*a1
}
// Multiprecision subtraction, c = a-b, where lng(a) = lng(b) = nwords. Returns the borrow bit.
unsigned int mp_sub(const digit_t *a, const digit_t *b, digit_t *c, const unsigned int nwords) {
unsigned int i, borrow = 0;
for (i = 0; i < nwords; i++)
SUBC(borrow, a[i], b[i], borrow, c[i]);
return borrow;
}
// Multiprecision subtraction followed by addition with p*2^MAXBITS_FIELD, c = a-b+(p*2^MAXBITS_FIELD) if a-b < 0, otherwise c=a-b.
__inline static void mp_subaddfast(const digit_t *a, const digit_t *b, digit_t *c) {
#if defined(S2N_SIKEP434R2_ASM)
if (s2n_sikep434r2_asm_is_enabled()) {
mp_subaddx2_asm(a, b, c);
return;
}
#endif
felm_t t1;
digit_t mask = 0 - (digit_t) mp_sub(a, b, c, 2 * NWORDS_FIELD);
for (int i = 0; i < NWORDS_FIELD; i++)
t1[i] = ((const digit_t *) PRIME)[i] & mask;
mp_addfast((digit_t *) &c[NWORDS_FIELD], t1, (digit_t *) &c[NWORDS_FIELD]);
}
// Multiprecision subtraction, c = c-a-b, where lng(a) = lng(b) = 2*NWORDS_FIELD.
__inline static void mp_dblsubfast(const digit_t *a, const digit_t *b, digit_t *c) {
#if defined(S2N_SIKEP434R2_ASM)
if (s2n_sikep434r2_asm_is_enabled()) {
mp_dblsubx2_asm(a, b, c);
return;
}
#endif
mp_sub(c, a, c, 2 * NWORDS_FIELD);
mp_sub(c, b, c, 2 * NWORDS_FIELD);
}
// GF(p^2) multiplication using Montgomery arithmetic, c = a*b in GF(p^2).
// Inputs: a = a0+a1*i and b = b0+b1*i, where a0, a1, b0, b1 are in [0, 2*p-1]
// Output: c = c0+c1*i, where c0, c1 are in [0, 2*p-1]
void fp2mul_mont(const f2elm_t *a, const f2elm_t *b, f2elm_t *c) {
felm_t t1, t2;
dfelm_t tt1, tt2, tt3;
mp_addfast(a->e[0], a->e[1], t1); // t1 = a0+a1
mp_addfast(b->e[0], b->e[1], t2); // t2 = b0+b1
mp_mul(a->e[0], b->e[0], tt1, NWORDS_FIELD); // tt1 = a0*b0
mp_mul(a->e[1], b->e[1], tt2, NWORDS_FIELD); // tt2 = a1*b1
mp_mul(t1, t2, tt3, NWORDS_FIELD); // tt3 = (a0+a1)*(b0+b1)
mp_dblsubfast(tt1, tt2, tt3); // tt3 = (a0+a1)*(b0+b1) - a0*b0 - a1*b1
mp_subaddfast(tt1, tt2, tt1); // tt1 = a0*b0 - a1*b1 + p*2^MAXBITS_FIELD if a0*b0 - a1*b1 < 0, else tt1 = a0*b0 - a1*b1
rdc_mont(tt3, c->e[1]); // c[1] = (a0+a1)*(b0+b1) - a0*b0 - a1*b1
rdc_mont(tt1, c->e[0]); // c[0] = a0*b0 - a1*b1
}
// Chain to compute a^(p-3)/4 using Montgomery arithmetic.
void fpinv_chain_mont(felm_t a) {
unsigned int i, j;
felm_t t[31], tt;
// Precomputed table
fpsqr_mont(a, tt);
fpmul_mont(a, tt, t[0]);
for (i = 0; i <= 29; i++)
fpmul_mont(t[i], tt, t[i + 1]);
fpcopy(a, tt);
for (i = 0; i < 7; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[5], tt, tt);
for (i = 0; i < 10; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[14], tt, tt);
for (i = 0; i < 6; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[3], tt, tt);
for (i = 0; i < 6; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[23], tt, tt);
for (i = 0; i < 6; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[13], tt, tt);
for (i = 0; i < 6; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[24], tt, tt);
for (i = 0; i < 6; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[7], tt, tt);
for (i = 0; i < 8; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[12], tt, tt);
for (i = 0; i < 8; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[30], tt, tt);
for (i = 0; i < 6; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[1], tt, tt);
for (i = 0; i < 6; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[30], tt, tt);
for (i = 0; i < 7; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[21], tt, tt);
for (i = 0; i < 9; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[2], tt, tt);
for (i = 0; i < 9; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[19], tt, tt);
for (i = 0; i < 9; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[1], tt, tt);
for (i = 0; i < 7; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[24], tt, tt);
for (i = 0; i < 6; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[26], tt, tt);
for (i = 0; i < 6; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[16], tt, tt);
for (i = 0; i < 7; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[10], tt, tt);
for (i = 0; i < 7; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[6], tt, tt);
for (i = 0; i < 7; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[0], tt, tt);
for (i = 0; i < 9; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[20], tt, tt);
for (i = 0; i < 8; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[9], tt, tt);
for (i = 0; i < 6; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[25], tt, tt);
for (i = 0; i < 9; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[30], tt, tt);
for (i = 0; i < 6; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[26], tt, tt);
for (i = 0; i < 6; i++)
fpsqr_mont(tt, tt);
fpmul_mont(a, tt, tt);
for (i = 0; i < 7; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[28], tt, tt);
for (i = 0; i < 6; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[6], tt, tt);
for (i = 0; i < 6; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[10], tt, tt);
for (i = 0; i < 9; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[22], tt, tt);
for (j = 0; j < 35; j++) {
for (i = 0; i < 6; i++)
fpsqr_mont(tt, tt);
fpmul_mont(t[30], tt, tt);
}
fpcopy(tt, a);
}
// GF(p^2) inversion using Montgomery arithmetic, a = (a0-i*a1)/(a0^2+a1^2).
void fp2inv_mont(f2elm_t *a) {
f2elm_t t1;
fpsqr_mont(a->e[0], t1.e[0]); // t10 = a0^2
fpsqr_mont(a->e[1], t1.e[1]); // t11 = a1^2
fpadd(t1.e[0], t1.e[1], t1.e[0]); // t10 = a0^2+a1^2
fpinv_mont(t1.e[0]); // t10 = (a0^2+a1^2)^-1
fpneg(a->e[1]); // a = a0-i*a1
fpmul_mont(a->e[0], t1.e[0], a->e[0]);
fpmul_mont(a->e[1], t1.e[0], a->e[1]); // a = (a0-i*a1)*(a0^2+a1^2)^-1
}
// Conversion of a GF(p^2) element to Montgomery representation,
// mc_i = a_i*R^2*R^(-1) = a_i*R in GF(p^2).
void to_fp2mont(const f2elm_t *a, f2elm_t *mc) {
to_mont(a->e[0], mc->e[0]);
to_mont(a->e[1], mc->e[1]);
}
// Conversion of a GF(p^2) element from Montgomery representation to standard representation,
// c_i = ma_i*R^(-1) = a_i in GF(p^2).
void from_fp2mont(const f2elm_t *ma, f2elm_t *c) {
from_mont(ma->e[0], c->e[0]);
from_mont(ma->e[1], c->e[1]);
}
// Multiprecision addition, c = a+b, where lng(a) = lng(b) = nwords. Returns the carry bit.
unsigned int mp_add(const digit_t *a, const digit_t *b, digit_t *c, const unsigned int nwords) {
unsigned int i, carry = 0;
for (i = 0; i < nwords; i++) {
/* cppcheck-suppress shiftTooManyBits */
/* cppcheck-suppress unmatchedSuppression */
ADDC(carry, a[i], b[i], carry, c[i]);
}
return carry;
}
// Multiprecision right shift by one.
void mp_shiftr1(digit_t *x, const unsigned int nwords) {
unsigned int i;
for (i = 0; i < nwords - 1; i++) {
SHIFTR(x[i + 1], x[i], 1, x[i], RADIX);
}
x[nwords - 1] >>= 1;
}
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