void gmersennemod( int n, giant g ) /* g := g (mod 2^n - 1) */ { int the_sign; giant scratch3 = borrowGiant(g->capacity); giant scratch4 = borrowGiant(1); if ((the_sign = gsign(g)) < 0) absg(g); while (bitlen(g) > n) { gtog(g,scratch3); gshiftright(n,scratch3); addg(scratch3,g); gshiftleft(n,scratch3); subg(scratch3,g); } if(isZero(g)) goto out; int_to_giant(1,scratch3); gshiftleft(n,scratch3); int_to_giant(1,scratch4); subg(scratch4,scratch3); if(gcompg(g,scratch3) >= 0) subg(scratch3,g); if (the_sign < 0) { g->sign = -g->sign; addg(scratch3,g); } out: returnGiant(scratch3); returnGiant(scratch4); }
static void numer_plus(giant x1, giant x2, giant res, curveParams *par) /* Numerator algebra. res = (x1 x2 + a)(x1 + x2) + 2(c x1 x2 + b). */ { giant t1; giant t2; PROF_START; t1 = borrowGiant(par->maxDigits); t2 = borrowGiant(par->maxDigits); gtog(x1, t1); mulg(x2, t1); feemod(par, t1); gtog(x2, t2); addg(x1, t2); feemod(par, t2); gtog(t1, res); if(!isZero(par->a)) addg(par->a, res); mulg(t2, res); feemod(par, res); if(par->curveType == FCT_Weierstrass) { // i.e., isZero(par->c) int_to_giant(0, t1); } else { mulg(par->c, t1); feemod(par, t1); } if(!isZero(par->b)) addg(par->b, t1); gshiftleft(1, t1); addg(t1, res); feemod(par, res); returnGiant(t1); returnGiant(t2); PROF_END(numerPlusTime); }
/* * Completely rewritten in CryptKit-18, 13 Jan 1997, for new IEEE-style * curveParameters. */ int which_curve(giant x, curveParams *par) /* Returns (+-1) depending on whether x is on curve (+-)y^2 = x^3 + c x^2 + a x + b. */ { giant t1; giant t2; giant t3; int result; PROF_START; t1 = borrowGiant(par->maxDigits); t2 = borrowGiant(par->maxDigits); t3 = borrowGiant(par->maxDigits); /* First, set t2:= x^3 + c x^2 + a x + b. */ gtog(x, t2); addg(par->c, t2); mulg(x, t2); addg(par->a, t2); /* t2 := x^2 + c x + a. */ feemod(par, t2); mulg(x, t2); addg(par->b, t2); feemod(par, t2); /* Next, test whether t2 is a square. */ gtog(t2, t1); make_base(par, t3); iaddg(1, t3); gshiftright(1, t3); /* t3 = (p+1)/2. */ feepowermodg(par, t1, t3); /* t1 := t2^((p+1)/2) (mod p). */ if(gcompg(t1, t2) == 0) result = CURVE_PLUS; else result = CURVE_MINUS; returnGiant(t1); returnGiant(t2); returnGiant(t3); PROF_END(whichCurveTime); return result; }
int binvaux(giant p, giant x) /* Binary inverse method. Returns zero if no inverse exists, in which case x becomes GCD(x,p). */ { giant scratch7; giant u0; giant u1; giant v0; giant v1; int result = 1; int giantSize; PROF_START; if(isone(x)) return(result); giantSize = 4 * abs(p->sign); scratch7 = borrowGiant(giantSize); u0 = borrowGiant(giantSize); u1 = borrowGiant(giantSize); v0 = borrowGiant(giantSize); v1 = borrowGiant(giantSize); int_to_giant(1, v0); gtog(x, v1); int_to_giant(0,x); gtog(p, u1); while(!isZero(v1)) { gtog(u1, u0); bdivg(v1, u0); gtog(x, scratch7); gtog(v0, x); mulg(u0, v0); subg(v0,scratch7); gtog(scratch7, v0); gtog(u1, scratch7); gtog(v1, u1); mulg(u0, v1); subg(v1,scratch7); gtog(scratch7, v1); } if (!isone(u1)) { gtog(u1,x); if(x->sign<0) addg(p, x); result = 0; goto done; } if (x->sign<0) addg(p, x); done: returnGiant(scratch7); returnGiant(u0); returnGiant(u1); returnGiant(v0); returnGiant(v1); PROF_END(binvauxTime); return(result); }
/* * Elliptic multiply: x := n * {x, 1} */ void elliptic_simple(giant x, giant n, curveParams *par) { giant ztmp = borrowGiant(par->maxDigits); giant cur_n = borrowGiant(par->maxDigits); START_ELL_MEASURE(n); int_to_giant(1, ztmp); elliptic(x, ztmp, n, par); binvg_cp(par, ztmp); mulg(ztmp, x); feemod(par, x); END_ELL_MEASURE; returnGiant(cur_n); returnGiant(ztmp); }
static void numer_times(giant x1, giant z1, giant x2, giant z2, giant res, curveParams *par) /* Numerator algebra. res := (x1 x2 - a z1 z2)^2 - 4 b(x1 z2 + x2 z1 + c z1 z2) z1 z2 */ { giant t1; giant t2; giant t3; giant t4; PROF_START; t1 = borrowGiant(par->maxDigits); t2 = borrowGiant(par->maxDigits); t3 = borrowGiant(par->maxDigits); t4 = borrowGiant(par->maxDigits); gtog(x1, t1); mulg(x2, t1); feemod(par, t1); gtog(z1, t2); mulg(z2, t2); feemod(par, t2); gtog(t1, res); if(!isZero(par->a)) { gtog(par->a, t3); mulg(t2, t3); feemod(par, t3); subg(t3, res); } gsquare(res); feemod(par, res); if(isZero(par->b)) goto done; if(par->curveType != FCT_Weierstrass) { // i.e., !isZero(par->c) gtog(par->c, t3); mulg(t2, t3); feemod(par, t3); } else int_to_giant(0, t3); gtog(z1, t4); mulg(x2, t4); feemod(par, t4); addg(t4, t3); gtog(x1, t4); mulg(z2, t4); feemod(par, t4); addg(t4, t3); mulg(par->b, t3); feemod(par, t3); mulg(t2, t3); gshiftleft(2, t3); feemod(par, t3); subg(t3, res); feemod(par, res); done: returnGiant(t1); returnGiant(t2); returnGiant(t3); returnGiant(t4); PROF_END(numerTimesTime); }
int signature_compare(giant p0x, giant p1x, giant p2x, curveParams *par) /* Returns non-zero iff p0x cannot be the x-coordinate of the sum of two points whose respective x-coordinates are p1x, p2x. */ { int ret = 0; giant t1; giant t2; giant t3; giant t4; giant t5; PROF_START; t1 = borrowGiant(par->maxDigits); t2 = borrowGiant(par->maxDigits); t3 = borrowGiant(par->maxDigits); t4 = borrowGiant(par->maxDigits); t5 = borrowGiant(par->maxDigits); if(gcompg(p1x, p2x) == 0) { int_to_giant(1, t1); numer_double(p1x, t1, t2, par); denom_double(p1x, t1, t3, par); mulg(p0x, t3); subg(t3, t2); feemod(par, t2); } else { numer_plus(p1x, p2x, t1, par); gshiftleft(1, t1); feemod(par, t1); int_to_giant(1, t3); numer_times(p1x, t3, p2x, t3, t2, par); int_to_giant(1, t4); int_to_giant(1, t5); denom_times(p1x, t4 , p2x, t5, t3, par); /* Now we require t3 x0^2 - t1 x0 + t2 == 0. */ mulg(p0x, t3); feemod(par, t3); subg(t1, t3); mulg(p0x, t3); feemod(par, t3); addg(t3, t2); feemod(par, t2); } if(!isZero(t2)) ret = SIGNATURE_INVALID; returnGiant(t1); returnGiant(t2); returnGiant(t3); returnGiant(t4); returnGiant(t5); PROF_END(sigCompTime); return(ret); }
static void powermodg( giant x, giant n, curveParams *cp ) /* x becomes x^n (mod basePrime). */ { int len, pos; giant scratch2 = borrowGiant(cp->maxDigits); gtog(x, scratch2); int_to_giant(1, x); len = bitlen(n); pos = 0; while (1) { if (bitval(n, pos++)) { mulg(scratch2, x); feemod(cp, x); } if (pos>=len) break; gsquare(scratch2); feemod(cp, scratch2); } returnGiant(scratch2); }
/* * New, 13 Jan 1997. */ static void feepowermodg(curveParams *par, giant x, giant n) /* Power ladder. x := x^n (mod 2^q-k) */ { int len, pos; giant t1; PROF_START; t1 = borrowGiant(par->maxDigits); gtog(x, t1); int_to_giant(1, x); len = bitlen(n); pos = 0; while(1) { if(bitval(n, pos++)) { mulg(t1, x); feemod(par, x); } if(pos>=len) break; gsquare(t1); feemod(par, t1); } returnGiant(t1); PROF_END(powerModTime); }
/* * g *= (int n) * * FIXME - we can improve this... */ void imulg(unsigned n, giant g) { giant tmp = borrowGiant(abs(g->sign) + sizeof(int)); int_to_giant(n, tmp); mulg(tmp, g); returnGiant(tmp); }
static void numer_double(giant x, giant z, giant res, curveParams *par) /* Numerator algebra. res := (x^2 - a z^2)^2 - 4 b (2 x + c z) z^3. */ { giant t1; giant t2; PROF_START; t1 = borrowGiant(par->maxDigits); t2 = borrowGiant(par->maxDigits); gtog(x, t1); gsquare(t1); feemod(par, t1); gtog(z, res); gsquare(res); feemod(par, res); gtog(res, t2); if(!isZero(par->a) ) { if(!isone(par->a)) { /* Speedup - REC 17 Jan 1997. */ mulg(par->a, res); feemod(par, res); } subg(res, t1); feemod(par, t1); } gsquare(t1); feemod(par, t1); /* t1 := (x^2 - a z^2)^2. */ if(isZero(par->b)) { /* Speedup - REC 17 Jan 1997. */ gtog(t1, res); goto done; } if(par->curveType != FCT_Weierstrass) { // i.e., !isZero(par->c) // Speedup - REC 17 Jan 1997. gtog(z, res); mulg(par->c, res); feemod(par, res); } else { int_to_giant(0, res); } addg(x, res); addg(x, res); mulg(par->b, res); feemod(par, res); gshiftleft(2, res); mulg(z, res); feemod(par, res); mulg(t2, res); feemod(par, res); negg(res); addg(t1, res); feemod(par, res); done: returnGiant(t1); returnGiant(t2); PROF_END(numerDoubleTime); }
static void powFp2(giant a, giant b, giant w2, giant n, curveParams *cp) /* Perform powering in the field F_p^2: a + b w := (a + b w)^n (mod p), where parameter w2 is a quadratic nonresidue (formally equal to w^2). */ { int j; giant t6; giant t7; giant t8; giant t9; if(isZero(n)) { int_to_giant(1,a); int_to_giant(0,b); return; } t6 = borrowGiant(cp->maxDigits); t7 = borrowGiant(cp->maxDigits); t8 = borrowGiant(cp->maxDigits); t9 = borrowGiant(cp->maxDigits); gtog(a, t8); gtog(b, t9); for(j = bitlen(n)-2; j >= 0; j--) { gtog(b, t6); mulg(a, b); addg(b,b); feemod(cp, b); /* b := 2 a b. */ gsquare(t6); feemod(cp, t6); mulg(w2, t6); feemod(cp, t6); gsquare(a); addg(t6, a); feemod(cp, a); /* a := a^2 + b^2 w2. */ if(bitval(n, j)) { gtog(b, t6); mulg(t8, b); feemod(cp, b); gtog(a, t7); mulg(t9, a); addg(a, b); feemod(cp, b); mulg(t9, t6); feemod(cp, t6); mulg(w2, t6); feemod(cp, t6); mulg(t8, a); addg(t6, a); feemod(cp, a); } } returnGiant(t6); returnGiant(t7); returnGiant(t8); returnGiant(t9); return; }
void ellDoubleProj(pointProj pt, curveParams *cp) /* pt := 2 pt on the curve. */ { giant x = pt->x, y = pt->y, z = pt->z; giant t1; giant t2; giant t3; if(isZero(y) || isZero(z)) { int_to_giant(1,x); int_to_giant(1,y); int_to_giant(0,z); return; } t1 = borrowGiant(cp->maxDigits); t2 = borrowGiant(cp->maxDigits); t3 = borrowGiant(cp->maxDigits); if((cp->a->sign >= 0) || (cp->a->n[0] != 3)) { /* Path prior to Apr2001. */ gtog(z,t1); gsquare(t1); feemod(cp, t1); gsquare(t1); feemod(cp, t1); mulg(cp->a, t1); feemod(cp, t1); /* t1 := a z^4. */ gtog(x, t2); gsquare(t2); feemod(cp, t2); smulg(3, t2); /* t2 := 3x^2. */ addg(t2, t1); feemod(cp, t1); /* t1 := slope m. */ } else { /* New optimization for a = -3 (post Apr 2001). */ gtog(z, t1); gsquare(t1); feemod(cp, t1); /* t1 := z^2. */ gtog(x, t2); subg(t1, t2); /* t2 := x-z^2. */ addg(x, t1); smulg(3, t1); /* t1 := 3(x+z^2). */ mulg(t2, t1); feemod(cp, t1); /* t1 := slope m. */ } mulg(y, z); addg(z,z); feemod(cp, z); /* z := 2 y z. */ gtog(y, t2); gsquare(t2); feemod(cp, t2); /* t2 := y^2. */ gtog(t2, t3); gsquare(t3); feemod(cp, t3); /* t3 := y^4. */ gshiftleft(3, t3); /* t3 := 8 y^4. */ mulg(x, t2); gshiftleft(2, t2); feemod(cp, t2); /* t2 := 4xy^2. */ gtog(t1, x); gsquare(x); feemod(cp, x); subg(t2, x); subg(t2, x); feemod(cp, x); /* x done. */ gtog(t1, y); subg(x, t2); mulg(t2, y); subg(t3, y); feemod(cp, y); returnGiant(t1); returnGiant(t2); returnGiant(t3); }
/* * Simple projective multiply. * * pt := pt * k, result normalized. */ void ellMulProjSimple(pointProj pt0, giant k, curveParams *cp) { pointProjStruct pt1; // local, giants borrowed CKASSERT(isone(pt0->z)); CKASSERT(cp->curveType == FCT_Weierstrass); /* ellMulProj assumes constant pt0, can't pass as src and dst */ pt1.x = borrowGiant(cp->maxDigits); pt1.y = borrowGiant(cp->maxDigits); pt1.z = borrowGiant(cp->maxDigits); ellMulProj(pt0, &pt1, k, cp); normalizeProj(&pt1, cp); CKASSERT(isone(pt1.z)); ptopProj(&pt1, pt0); returnGiant(pt1.x); returnGiant(pt1.y); returnGiant(pt1.z); }
/* * This version is not normally used; it's for when the reciprocal of * curveOrder is not known and won't be used again. */ void curveOrderJustify(giant g, giant curveOrder) { giant recip; CKASSERT(!isZero(curveOrder)); recip = borrowGiant(2 * abs(g->sign)); make_recip(curveOrder, recip); curveOrderJustifyWithRecip(g, curveOrder, recip); returnGiant(recip); }
void make_base_prim(curveParams *cp) /* Jams cp->basePrime with 2^q-k. Assumes valid maxDigits, q, k. */ { giant tmp = borrowGiant(cp->maxDigits); CKASSERT(cp->primeType != FPT_General); int_to_giant(1, cp->basePrime); gshiftleft((int)cp->q, cp->basePrime); int_to_giant(cp->k, tmp); subg(tmp, cp->basePrime); returnGiant(tmp); }
/* * General elliptic multiply. * * {xx, zz} := k * {xx, zz} */ void elliptic(giant xx, giant zz, giant k, curveParams *par) { int len = bitlen(k); int pos = len - 2; giant xs; giant zs; giant xorg; giant zorg; PROF_START; if(sequalg(1,k)) return; if(sequalg(2,k)) { ell_even(xx, zz, xx, zz, par); goto out; } zs = borrowGiant(par->maxDigits); xs = borrowGiant(par->maxDigits); zorg = borrowGiant(par->maxDigits); xorg = borrowGiant(par->maxDigits); gtog(xx, xorg); gtog(zz, zorg); ell_even(xx, zz, xs, zs, par); do { if(bitval(k, pos--)) { ell_odd(xs, zs, xx, zz, xorg, zorg, par); ell_even(xs, zs, xs, zs, par); } else { ell_odd(xx, zz, xs, zs, xorg, zorg, par); ell_even(xx, zz, xx, zz, par); } } while (pos >= 0); // REC fix 9/23/94 returnGiant(xs); returnGiant(zs); returnGiant(xorg); returnGiant(zorg); out: PROF_END(ellipticTime); }
static void denom_double(giant x, giant z, giant res, curveParams *par) /* Denominator algebra. res = 4 z (x^3 + c x^2 z + a x z^2 + b z^3). */ { giant t1; giant t2; PROF_START; t1 = borrowGiant(par->maxDigits); t2 = borrowGiant(par->maxDigits); gtog(x, res); gtog(z, t1); if(par->curveType != FCT_Weierstrass) { // i.e., !isZero(par->c) gtog(par->c, t2); mulg(t1, t2); feemod(par, t2); addg(t2, res); } mulg(x, res); feemod(par, res); gsquare(t1); feemod(par, t1); if(!isZero(par->a)) { gtog(t1, t2); mulg(par->a, t2); feemod(par, t2); addg(t2, res); } mulg(x, res); feemod(par, res); if(!isZero(par->b)) { mulg(z, t1); feemod(par, t1); mulg(par->b, t1); feemod(par, t1); addg(t1, res); } mulg(z, res); gshiftleft(2, res); feemod(par, res); returnGiant(t1); returnGiant(t2); PROF_END(denomDoubleTime); }
static int jacobi_symbol(giant a, curveParams *cp) /* Standard Jacobi symbol (a/cp->basePrime). basePrime must be odd, positive. */ { int t = 1, u; giant t5 = borrowGiant(cp->maxDigits); giant t6 = borrowGiant(cp->maxDigits); giant t7 = borrowGiant(cp->maxDigits); int rtn; gtog(a, t5); feemod(cp, t5); gtog(cp->basePrime, t6); while(!isZero(t5)) { u = (t6->n[0]) & 7; while((t5->n[0] & 1) == 0) { gshiftright(1, t5); if((u==3) || (u==5)) t = -t; } gtog(t5, t7); gtog(t6, t5); gtog(t7, t6); u = (t6->n[0]) & 3; if(((t5->n[0] & 3) == 3) && ((u & 3) == 3)) t = -t; modg(t6, t5); } if(isone(t6)) { rtn = t; } else { rtn = 0; } returnGiant(t5); returnGiant(t6); returnGiant(t7); return rtn; }
void gmersennemod(int n, giant g) { /* g becomes g mod ((2^n)-1) 31 Jul 96 modified REC. 17 Jan 97 modified REC. */ unsigned bits = n & (GIANT_BITS_PER_DIGIT - 1); unsigned digits = 1 + ((n-1) >> GIANT_LOG2_BITS_PER_DIGIT); int isPositive = (g->sign > 0); int j; int b; int size; int foundzero; giantDigit mask = (bits == 0) ? GIANT_DIGIT_MASK : (giantDigit)((1<<bits)-1); giant scratch1; b = bitlen(g); if(b < n) { unsigned numDigits = (n + GIANT_BITS_PER_DIGIT - 1) >> GIANT_LOG2_BITS_PER_DIGIT; giantDigit lastWord = 0; giantDigit bits = 1; if(g->sign >= 0) return; /* * Cons up ((2**n)-1), add to g. */ scratch1 = borrowGiant(numDigits + 1); scratch1->sign = numDigits; for(j=0; j<(int)(numDigits-1); j++) { scratch1->n[j] = GIANT_DIGIT_MASK; } /* * Last word has lower (n & (GIANT_BITS_PER_DIGIT-1)) bits set. */ for(j=0; j < (int)(n & (GIANT_BITS_PER_DIGIT-1)); j++) { lastWord |= bits; bits <<= 1; } scratch1->n[numDigits-1] = lastWord; addg(g, scratch1); /* One version. */ gtog(scratch1, g); returnGiant(scratch1); return; }
static void denom_times(giant x1, giant z1, giant x2, giant z2, giant res, curveParams *par) /* Denominator algebra. res := (x1 z2 - x2 z1)^2 */ { giant t1; PROF_START; t1 = borrowGiant(par->maxDigits); gtog(x1, res); mulg(z2, res); feemod(par, res); gtog(z1, t1); mulg(x2, t1); feemod(par, t1); subg(t1, res); gsquare(res); feemod(par, res); returnGiant(t1); PROF_END(denomTimesTime); }
void ellMulProj(pointProj pt0, pointProj pt1, giant k, curveParams *cp) /* General elliptic multiplication; pt1 := k*pt0 on the curve, with k an arbitrary integer. */ { giant x = pt0->x, y = pt0->y, z = pt0->z, xx = pt1->x, yy = pt1->y, zz = pt1->z; int ksign, hlen, klen, b, hb, kb; giant t0; CKASSERT(cp->curveType == FCT_Weierstrass); if(isZero(k)) { int_to_giant(1, xx); int_to_giant(1, yy); int_to_giant(0, zz); return; } t0 = borrowGiant(cp->maxDigits); ksign = k->sign; if(ksign < 0) negg(k); gtog(x,xx); gtog(y,yy); gtog(z,zz); gtog(k, t0); addg(t0, t0); addg(k, t0); /* t0 := 3k. */ hlen = bitlen(t0); klen = bitlen(k); for(b = hlen-2; b > 0; b--) { ellDoubleProj(pt1,cp); hb = bitval(t0, b); if(b < klen) kb = bitval(k, b); else kb = 0; if((hb != 0) && (kb == 0)) ellAddProj(pt1, pt0, cp); else if((hb == 0) && (kb !=0)) ellSubProj(pt1, pt0, cp); } if(ksign < 0) { ellNegProj(pt1, cp); k->sign = -k->sign; } returnGiant(t0); }
void normalizeProj(pointProj pt, curveParams *cp) /* Obtain actual x,y coords via normalization: {x,y,z} := {x/z^2, y/z^3, 1}. */ { giant x = pt->x, y = pt->y, z = pt->z; giant t1; CKASSERT(cp->curveType == FCT_Weierstrass); if(isZero(z)) { int_to_giant(1,x); int_to_giant(1,y); return; } t1 = borrowGiant(cp->maxDigits); binvg_cp(cp, z); // was binvaux(p, z); gtog(z, t1); gsquare(z); feemod(cp, z); mulg(z, x); feemod(cp, x); mulg(t1, z); mulg(z, y); feemod(cp, y); int_to_giant(1, z); returnGiant(t1); }
/* * New optimzation of curveOrderJustify using known reciprocal, 11 June 1997. * g is set to be within [2, curveOrder-2]. */ static void curveOrderJustifyWithRecip(giant g, giant curveOrder, giant recip) { giant tmp; CKASSERT(!isZero(curveOrder)); modg_via_recip(curveOrder, recip, g); // g now in [0, curveOrder-1] if(isZero(g)) { /* * First degenerate case - (g == 0) : set g := 2 */ dbgLog(("curveOrderJustify: case 1\n")); int_to_giant(2, g); return; } if(isone(g)) { /* * Second case - (g == 1) : set g := 2 */ dbgLog(("curveOrderJustify: case 2\n")); int_to_giant(2, g); return; } tmp = borrowGiant(g->capacity); gtog(g, tmp); iaddg(1, tmp); if(gcompg(tmp, curveOrder) == 0) { /* * Third degenerate case - (g == (curveOrder-1)) : set g -= 1 */ dbgLog(("curveOrderJustify: case 3\n")); int_to_giant(1, tmp); subg(tmp, g); } returnGiant(tmp); return; }
static void bdivg(giant v, giant u) /* u becomes greatest power of two not exceeding u/v. */ { int diff = bitlen(u) - bitlen(v); giant scratch7; if (diff<0) { int_to_giant(0,u); return; } scratch7 = borrowGiant(u->capacity); gtog(v, scratch7); gshiftleft(diff,scratch7); if(gcompg(u,scratch7) < 0) diff--; if(diff<0) { int_to_giant(0,u); returnGiant(scratch7); return; } int_to_giant(1,u); gshiftleft(diff,u); returnGiant(scratch7); }
static int sqrtmod(giant x, curveParams *cp) /* If Sqrt[x] (mod p) exists, function returns 1, else 0. In either case x is modified, but if 1 is returned, x:= Sqrt[x] (mod p). */ { int rtn; giant t0 = borrowGiant(cp->maxDigits); giant t1 = borrowGiant(cp->maxDigits); giant t2 = borrowGiant(cp->maxDigits); giant t3 = borrowGiant(cp->maxDigits); giant t4 = borrowGiant(cp->maxDigits); giant p = cp->basePrime; feemod(cp, x); /* Justify the argument. */ gtog(x, t0); /* Store x for eventual validity check on square root. */ if((p->n[0] & 3) == 3) { /* The case p = 3 (mod 4). */ gtog(p, t1); iaddg(1, t1); gshiftright(2, t1); powermodg(x, t1, cp); goto resolve; } /* Next, handle case p = 5 (mod 8). */ if((p->n[0] & 7) == 5) { gtog(p, t1); int_to_giant(1, t2); subg(t2, t1); gshiftright(2, t1); gtog(x, t2); powermodg(t2, t1, cp); /* t2 := x^((p-1)/4) % p. */ iaddg(1, t1); gshiftright(1, t1); /* t1 := (p+3)/8. */ if(isone(t2)) { powermodg(x, t1, cp); /* x^((p+3)/8) is root. */ goto resolve; } else { int_to_giant(1, t2); subg(t2, t1); /* t1 := (p-5)/8. */ gshiftleft(2,x); powermodg(x, t1, cp); mulg(t0, x); addg(x, x); feemod(cp, x); /* 2x (4x)^((p-5)/8. */ goto resolve; } } /* Next, handle tougher case: p = 1 (mod 8). */ int_to_giant(2, t1); while(1) { /* Find appropriate nonresidue. */ gtog(t1, t2); gsquare(t2); subg(x, t2); feemod(cp, t2); if(jacobi_symbol(t2, cp) == -1) break; iaddg(1, t1); } /* t2 is now w^2 in F_p^2. */ int_to_giant(1, t3); gtog(p, t4); iaddg(1, t4); gshiftright(1, t4); powFp2(t1, t3, t2, t4, cp); gtog(t1, x); resolve: gtog(x,t1); gsquare(t1); feemod(cp, t1); if(gcompg(t0, t1) == 0) { rtn = 1; /* Success. */ } else { rtn = 0; /* no square root */ } returnGiant(t0); returnGiant(t1); returnGiant(t2); returnGiant(t3); returnGiant(t4); return rtn; }
static void borrowPointProj(pointProj pt, unsigned maxDigits) { pt->x = borrowGiant(maxDigits); pt->y = borrowGiant(maxDigits); pt->z = borrowGiant(maxDigits); }
feeReturn feeECDSAVerify(const unsigned char *sigData, size_t sigDataLen, const unsigned char *data, unsigned dataLen, feePubKey pubKey, feeSigFormat format) { /* giant integers per IEEE P1363 notation */ giant h; // s^(-1) giant h1; // f h giant h2; // c times h giant littleC; // newGiant from ECDSA_decode giant littleD; // ditto giant c; // borrowed, full size giant d; // ditto giant cPrime = NULL; // i mod r pointProj h1G = NULL; // h1 'o' G pointProj h2W = NULL; // h2 'o' W key W; // i.e., their public key unsigned version; feeReturn frtn; curveParams *cp = feePubKeyCurveParams(pubKey); unsigned groupBytesLen = ((feePubKeyBitsize(pubKey)+7) / 8); int result; if(cp == NULL) { return FR_BadPubKey; } /* * First decode the byteRep string. */ frtn = ECDSA_decode( format, groupBytesLen, sigData, sigDataLen, &littleC, &littleD, &version); if(frtn) { return frtn; } /* * littleC and littleD have capacity = abs(sign), probably * not big enough.... */ c = borrowGiant(cp->maxDigits); d = borrowGiant(cp->maxDigits); gtog(littleC, c); gtog(littleD, d); freeGiant(littleC); freeGiant(littleD); sigDbg(("ECDSA verify:\n")); /* * Verify that c and d are within [1,group_order-1] */ if((gcompg(cp->cOrderPlus, c) != 1) || (gcompg(cp->cOrderPlus, d) != 1) || isZero(c) || isZero(d)) { returnGiant(c); returnGiant(d); return FR_InvalidSignature; } /* * W = signer's public key */ W = feePubKeyPlusCurve(pubKey); /* * 1) Compute h = d^(-1) (mod x1OrderPlus); */ SIGPROF_START; h = borrowGiant(cp->maxDigits); gtog(d, h); binvg_x1OrderPlus(cp, h); SIGPROF_END(vfyStep1); /* * 2) h1 = digest as giant (skips assigning to 'f' in P1363) */ if(dataLen > (cp->maxDigits * GIANT_BYTES_PER_DIGIT)) { h1 = borrowGiant(BYTES_TO_GIANT_DIGITS(dataLen)); } else { h1 = borrowGiant(cp->maxDigits); } deserializeGiant(data, h1, dataLen); /* * Certicom SEC1 states that if the digest is larger than the modulus, * use the left q bits of the digest. */ unsigned hashBits = dataLen * 8; if(hashBits > cp->q) { gshiftright(hashBits - cp->q, h1); } sigLogGiant(" Wx : ", W->x); sigLogGiant(" f : ", h1); sigLogGiant(" c : ", c); sigLogGiant(" d : ", d); sigLogGiant(" s^(-1) : ", h); /* * 3) Compute h1 = f * h mod x1OrderPlus; */ SIGPROF_START; mulg(h, h1); // h1 := f * h x1OrderPlusMod(h1, cp); SIGPROF_END(vfyStep3); /* * 4) Compute h2 = c * h (mod x1OrderPlus); */ SIGPROF_START; h2 = borrowGiant(cp->maxDigits); gtog(c, h2); mulg(h, h2); // h2 := c * h x1OrderPlusMod(h2, cp); SIGPROF_END(vfyStep4); /* * 5) Compute h2W = h2 'o' W (W = theirPub) */ CKASSERT((W->y != NULL) && !isZero(W->y)); h2W = newPointProj(cp->maxDigits); gtog(W->x, h2W->x); gtog(W->y, h2W->y); int_to_giant(1, h2W->z); ellMulProjSimple(h2W, h2, cp); /* * 6) Compute h1G = h1 'o' G (G = {x1Plus, y1Plus, 1} ) */ CKASSERT((cp->y1Plus != NULL) && !isZero(cp->y1Plus)); h1G = newPointProj(cp->maxDigits); gtog(cp->x1Plus, h1G->x); gtog(cp->y1Plus, h1G->y); int_to_giant(1, h1G->z); ellMulProjSimple(h1G, h1, cp); /* * 7) h1G := (h1 'o' G) + (h2 'o' W) */ ellAddProj(h1G, h2W, cp); /* * 8) If elliptic sum is point at infinity, signature is bad; stop. */ if(isZero(h1G->z)) { dbgLog(("feeECDSAVerify: h1 * G = point at infinity\n")); result = 1; goto vfyDone; } normalizeProj(h1G, cp); /* * 9) cPrime = x coordinate of elliptic sum, mod x1OrderPlus */ cPrime = borrowGiant(cp->maxDigits); gtog(h1G->x, cPrime); x1OrderPlusMod(cPrime, cp); /* * 10) Good sig iff cPrime == c */ result = gcompg(c, cPrime); vfyDone: if(result) { frtn = FR_InvalidSignature; #if LOG_BAD_SIG printf("***yup, bad sig***\n"); #endif // LOG_BAD_SIG } else { frtn = FR_Success; } returnGiant(c); returnGiant(d); returnGiant(h); returnGiant(h1); returnGiant(h2); if(h1G != NULL) { freePointProj(h1G); } if(h2W != NULL) { freePointProj(h2W); } if(cPrime != NULL) { returnGiant(cPrime); } return frtn; }
/* * Create new feeSig object, including a random large integer 'randGiant' for * possible use in salting a feeHash object, and 'PmX', equal to * randGiant 'o' P1. Note that this is not called when *verifying* a * signature, only when signing. */ feeSig feeSigNewWithKey( feePubKey pubKey, feeRandFcn randFcn, /* optional */ void *randRef) { sigInst *sinst = sinstAlloc(); feeRand frand; unsigned char *randBytes; unsigned randBytesLen; curveParams *cp; if(pubKey == NULL) { return NULL; } cp = feePubKeyCurveParams(pubKey); if(cp == NULL) { return NULL; } /* * Generate random m, a little larger than key size, save as randGiant */ randBytesLen = (feePubKeyBitsize(pubKey) / 8) + 1; randBytes = (unsigned char*) fmalloc(randBytesLen); if(randFcn) { randFcn(randRef, randBytes, randBytesLen); } else { frand = feeRandAlloc(); feeRandBytes(frand, randBytes, randBytesLen); feeRandFree(frand); } sinst->randGiant = giant_with_data(randBytes, randBytesLen); memset(randBytes, 0, randBytesLen); ffree(randBytes); #if FEE_DEBUG if(isZero(sinst->randGiant)) { printf("feeSigNewWithKey: randGiant = 0!\n"); } #endif // FEE_DEBUG /* * Justify randGiant to be in [2, x1OrderPlus] */ x1OrderPlusJustify(sinst->randGiant, cp); /* PmX := randGiant 'o' P1 */ sinst->PmX = newGiant(cp->maxDigits); #if CRYPTKIT_ELL_PROJ_ENABLE if(cp->curveType == FCT_Weierstrass) { pointProjStruct pt0; sinst->PmY = newGiant(cp->maxDigits); /* cook up pt0 as P1 */ pt0.x = sinst->PmX; pt0.y = sinst->PmY; pt0.z = borrowGiant(cp->maxDigits); gtog(cp->x1Plus, pt0.x); gtog(cp->y1Plus, pt0.y); int_to_giant(1, pt0.z); /* pt0 := P1 'o' randGiant */ ellMulProjSimple(&pt0, sinst->randGiant, cp); returnGiant(pt0.z); } else { if(SIG_CURVE == CURVE_PLUS) { gtog(cp->x1Plus, sinst->PmX); } else { gtog(cp->x1Minus, sinst->PmX); } elliptic_simple(sinst->PmX, sinst->randGiant, cp); } #else /* CRYPTKIT_ELL_PROJ_ENABLE */ if(SIG_CURVE == CURVE_PLUS) { gtog(cp->x1Plus, sinst->PmX); } else { gtog(cp->x1Minus, sinst->PmX); } elliptic_simple(sinst->PmX, sinst->randGiant, cp); #endif /* CRYPTKIT_ELL_PROJ_ENABLE */ return sinst; }
void ellAddProj(pointProj pt0, pointProj pt1, curveParams *cp) /* pt0 := pt0 + pt1 on the curve. */ { giant x0 = pt0->x, y0 = pt0->y, z0 = pt0->z, x1 = pt1->x, y1 = pt1->y, z1 = pt1->z; giant t1; giant t2; giant t3; giant t4; giant t5; giant t6; giant t7; if(isZero(z0)) { gtog(x1,x0); gtog(y1,y0); gtog(z1,z0); return; } if(isZero(z1)) return; t1 = borrowGiant(cp->maxDigits); t2 = borrowGiant(cp->maxDigits); t3 = borrowGiant(cp->maxDigits); t4 = borrowGiant(cp->maxDigits); t5 = borrowGiant(cp->maxDigits); t6 = borrowGiant(cp->maxDigits); t7 = borrowGiant(cp->maxDigits); gtog(x0, t1); gtog(y0,t2); gtog(z0, t3); gtog(x1, t4); gtog(y1, t5); if(!isone(z1)) { gtog(z1, t6); gtog(t6, t7); gsquare(t7); feemod(cp, t7); mulg(t7, t1); feemod(cp, t1); mulg(t6, t7); feemod(cp, t7); mulg(t7, t2); feemod(cp, t2); } gtog(t3, t7); gsquare(t7); feemod(cp, t7); mulg(t7, t4); feemod(cp, t4); mulg(t3, t7); feemod(cp, t7); mulg(t7, t5); feemod(cp, t5); negg(t4); addg(t1, t4); feemod(cp, t4); negg(t5); addg(t2, t5); feemod(cp, t5); if(isZero(t4)) { if(isZero(t5)) { ellDoubleProj(pt0, cp); } else { int_to_giant(1, x0); int_to_giant(1, y0); int_to_giant(0, z0); } goto out; } addg(t1, t1); subg(t4, t1); feemod(cp, t1); addg(t2, t2); subg(t5, t2); feemod(cp, t2); if(!isone(z1)) { mulg(t6, t3); feemod(cp, t3); } mulg(t4, t3); feemod(cp, t3); gtog(t4, t7); gsquare(t7); feemod(cp, t7); mulg(t7, t4); feemod(cp, t4); mulg(t1, t7); feemod(cp, t7); gtog(t5, t1); gsquare(t1); feemod(cp, t1); subg(t7, t1); feemod(cp, t1); subg(t1, t7); subg(t1, t7); feemod(cp, t7); mulg(t7, t5); feemod(cp, t5); mulg(t2, t4); feemod(cp, t4); gtog(t5, t2); subg(t4,t2); feemod(cp, t2); if(t2->n[0] & 1) { /* Test if t2 is odd. */ addg(cp->basePrime, t2); } gshiftright(1, t2); gtog(t1, x0); gtog(t2, y0); gtog(t3, z0); out: returnGiant(t1); returnGiant(t2); returnGiant(t3); returnGiant(t4); returnGiant(t5); returnGiant(t6); returnGiant(t7); }