void millertate(element_t z, element_t P, element_t Q) { element_t Z; element_t z0; element_init_same_as(Z, P); element_init_same_as(z0, z); element_set(Z, P); do_tangent(z, Z, Q); element_double(Z, Z); do_vert(z0, Z, Q); element_div(z, z, z0); element_printf("presquare: z = %B\n", z); element_square(z, z); element_printf("square: z = %B\n", z); do_tangent(z0, Z, Q); element_mul(z, z, z0); element_clear(z0); element_clear(Z); }
void miller(element_t z, element_t PR, element_t R, element_t P, element_t Q) { int m = mpz_sizeinbase(order, 2) - 2; element_t Z; element_t z1; element_t x1; element_init_same_as(Z, PR); element_set(Z, P); element_set1(z); element_init_same_as(z1, z); element_init_same_as(x1, z); do_vert(x1, PR, Q); element_printf("vert(P+R) %B\n", x1); do_line(z1, P, R, Q); element_printf("line(P,R) %B\n", z1); element_div(x1, x1, z1); element_printf("x1 %B\n", x1); element_set(z, x1); for (;;) { printf("iteration %d: %d\n", m, mpz_tstbit(order,m)); element_square(z, z); element_printf("squared: %B\n", z); do_tangent(z1, Z, Q); element_mul(z, z, z1); element_double(Z, Z); do_vert(z1, Z, Q); element_div(z, z, z1); element_printf("pre-if: %B\n", z); if (mpz_tstbit(order, m)) { element_mul(z, z, x1); do_vert(z1, P, Q); element_mul(z, z, z1); element_printf("done %B\n", z); /* do_line(z1, Z, P, Q); element_mul(z, z, z1); element_add(Z, Z, P); do_vert(z1, Z, Q); element_div(z, z, z1); */ } if (!m) break; m--; } element_clear(x1); element_clear(z1); }
static void miller(element_t res, element_t P, element_ptr QR, element_ptr R, int n) { // Collate divisions. mp_bitcnt_t m; element_t v, vd; element_t Z; element_t a, b, c; const element_ptr cca = curve_a_coeff(P); const element_ptr Px = curve_x_coord(P); const element_ptr Py = curve_y_coord(P); element_t e0, e1; mpz_t q; element_ptr Zx, Zy; const element_ptr numx = curve_x_coord(QR); const element_ptr numy = curve_y_coord(QR); const element_ptr denomx = curve_x_coord(R); const element_ptr denomy = curve_y_coord(R); #define do_vertical(e, edenom) { \ element_sub(e0, numx, Zx); \ element_mul((e), (e), e0); \ \ element_sub(e0, denomx, Zx); \ element_mul((edenom), (edenom), e0); \ } #define do_tangent(e, edenom) { \ /*a = -slope_tangent(A.x, A.y); \ b = 1; \ c = -(A.y + a * A.x); \ but we multiply by 2*A.y to avoid division*/ \ \ /*a = -Ax * (Ax + Ax + Ax + twicea_2) - a_4; \ Common curves: a2 = 0 (and cc->a is a_4), so \ a = -(3 Ax^2 + cc->a) \ b = 2 * Ay \ c = -(2 Ay^2 + a Ax); */ \ \ if (element_is0(Zy)) { \ do_vertical((e), (edenom)); \ } else { \ element_square(a, Zx); \ element_mul_si(a, a, 3); \ element_add(a, a, cca); \ element_neg(a, a); \ \ element_add(b, Zy, Zy); \ \ element_mul(e0, b, Zy); \ element_mul(c, a, Zx); \ element_add(c, c, e0); \ element_neg(c, c); \ \ element_mul(e0, a, numx); \ element_mul(e1, b, numy); \ element_add(e0, e0, e1); \ element_add(e0, e0, c); \ element_mul((e), (e), e0); \ \ element_mul(e0, a, denomx); \ element_mul(e1, b, denomy); \ element_add(e0, e0, e1); \ element_add(e0, e0, c); \ element_mul((edenom), (edenom), e0); \ } \ } #define do_line(e, edenom) { \ if (!element_cmp(Zx, Px)) { \ if (!element_cmp(Zy, Py)) { \ do_tangent(e, edenom); \ } else { \ do_vertical(e, edenom); \ } \ } else { \ element_sub(b, Px, Zx); \ element_sub(a, Zy, Py); \ element_mul(c, Zx, Py); \ element_mul(e0, Zy, Px); \ element_sub(c, c, e0); \ \ element_mul(e0, a, numx); \ element_mul(e1, b, numy); \ element_add(e0, e0, e1); \ element_add(e0, e0, c); \ element_mul(e, e, e0); \ \ element_mul(e0, a, denomx); \ element_mul(e1, b, denomy); \ element_add(e0, e0, e1); \ element_add(e0, e0, c); \ element_mul(edenom, edenom, e0); \ } \ } element_init(a, res->field); element_init(b, res->field); element_init(c, res->field); element_init(e0, res->field); element_init(e1, res->field); element_init(v, res->field); element_init(vd, res->field); element_init(Z, P->field); element_set(Z, P); Zx = curve_x_coord(Z); Zy = curve_y_coord(Z); element_set1(v); element_set1(vd); mpz_init(q); mpz_set_ui(q, n); m = (mp_bitcnt_t)mpz_sizeinbase(q, 2); m = (m > 2 ? m - 2 : 0); for (;;) { element_square(v, v); element_square(vd, vd); do_tangent(v, vd); element_double(Z, Z); do_vertical(vd, v); if (mpz_tstbit(q, m)) { do_line(v, vd); element_add(Z, Z, P); if (m) { do_vertical(vd, v); } } if (!m) break; m--; } mpz_clear(q); element_invert(vd, vd); element_mul(res, v, vd); element_clear(v); element_clear(vd); element_clear(Z); element_clear(a); element_clear(b); element_clear(c); element_clear(e0); element_clear(e1); #undef do_vertical #undef do_tangent #undef do_line }
//TODO: the following code is useless as the Tate pairing is degenerate on singular curves static void sn_miller(element_t res, mpz_t q, element_t P, element_ptr Qx, element_ptr Qy) { //collate divisions int m; element_t v, vd; element_t Z; element_t a, b, c; element_t e0, e1; element_ptr Zx; element_ptr Zy; const element_ptr Px = curve_x_coord(P); const element_ptr Py = curve_y_coord(P); #define do_vertical(e) \ element_sub(e0, Qx, Zx); \ element_mul(e, e, e0); //a = -slope_tangent(Z.x, Z.y); //b = 1; //c = -(Z.y + a * Z.x); //but we multiply by 2*Z.y to avoid division //a = -Zx * (Zx + Zx + Zx + 2) //b = 2 * Zy //c = -(2 Zy^2 + a Zx); #define do_tangent(e) \ element_double(e0, Zx); \ element_add(a, Zx, e0); \ element_set_si(e0, 2); \ element_add(a, a, e0); \ element_mul(a, a, Zx); \ element_neg(a, a); \ element_add(b, Zy, Zy); \ element_mul(e0, b, Zy); \ element_mul(c, a, Zx); \ element_add(c, c, e0); \ element_neg(c, c); \ element_mul(e0, a, Qx); \ element_mul(e1, b, Qy); \ element_add(e0, e0, e1); \ element_add(e0, e0, c); \ element_mul(e, e, e0); //a = -(B.y - A.y) / (B.x - A.x); //b = 1; //c = -(A.y + a * A.x); //but we'll multiply by B.x - A.x to avoid division #define do_line(e) \ element_sub(b, Px, Zx); \ element_sub(a, Zy, Py); \ element_mul(e0, b, Zy); \ element_mul(c, a, Zx); \ element_add(c, c, e0); \ element_neg(c, c); \ element_mul(e0, a, Qx); \ element_mul(e1, b, Qy); \ element_add(e0, e0, e1); \ element_add(e0, e0, c); \ element_mul(e, e, e0); element_init(a, Px->field); element_init(b, Px->field); element_init(c, Px->field); element_init(e0, res->field); element_init(e1, res->field); element_init(v, res->field); element_init(vd, res->field); element_init(Z, P->field); element_set(Z, P); Zx = curve_x_coord(Z); Zy = curve_y_coord(Z); element_set1(v); element_set1(vd); m = mpz_sizeinbase(q, 2) - 2; while(m >= 0) { element_mul(v, v, v); element_mul(vd, vd, vd); do_tangent(v); element_double(Z, Z); do_vertical(vd); if (mpz_tstbit(q, m)) { do_line(v); element_add(Z, Z, P); do_vertical(vd); } m--; } #undef do_tangent #undef do_vertical #undef do_line element_invert(vd, vd); element_mul(res, v, vd); element_clear(v); element_clear(vd); element_clear(Z); element_clear(a); element_clear(b); element_clear(c); element_clear(e0); element_clear(e1); }
static void e_miller_proj(element_t res, element_t P, element_ptr QR, element_ptr R, e_pairing_data_ptr p) { //collate divisions int n; element_t v, vd; element_t v1, vd1; element_t Z, Z1; element_t a, b, c; const element_ptr cca = curve_a_coeff(P); element_t e0, e1; const element_ptr e2 = a, e3 = b; element_t z, z2; int i; element_ptr Zx, Zy; const element_ptr Px = curve_x_coord(P); const element_ptr numx = curve_x_coord(QR); const element_ptr numy = curve_y_coord(QR); const element_ptr denomx = curve_x_coord(R); const element_ptr denomy = curve_y_coord(R); //convert Z from weighted projective (Jacobian) to affine //i.e. (X, Y, Z) --> (X/Z^2, Y/Z^3) //also sets z to 1 #define to_affine() { \ element_invert(z, z); \ element_square(e0, z); \ element_mul(Zx, Zx, e0); \ element_mul(e0, e0, z); \ element_mul(Zy, Zy, e0); \ element_set1(z); \ element_set1(z2); \ } #define proj_double() { \ const element_ptr x = Zx; \ const element_ptr y = Zy; \ /* e0 = 3x^2 + (cc->a) z^4 */ \ element_square(e0, x); \ /* element_mul_si(e0, e0, 3); */ \ element_double(e1, e0); \ element_add(e0, e0, e1); \ element_square(e1, z2); \ element_mul(e1, e1, cca); \ element_add(e0, e0, e1); \ \ /* z_out = 2 y z */ \ element_mul(z, y, z); \ /* element_mul_si(z, z, 2); */ \ element_double(z, z); \ element_square(z2, z); \ \ /* e1 = 4 x y^2 */ \ element_square(e2, y); \ element_mul(e1, x, e2); \ /* element_mul_si(e1, e1, 4); */ \ element_double(e1, e1); \ element_double(e1, e1); \ \ /* x_out = e0^2 - 2 e1 */ \ /* element_mul_si(e3, e1, 2); */ \ element_double(e3, e1); \ element_square(x, e0); \ element_sub(x, x, e3); \ \ /* e2 = 8y^4 */ \ element_square(e2, e2); \ /* element_mul_si(e2, e2, 8); */ \ element_double(e2, e2); \ element_double(e2, e2); \ element_double(e2, e2); \ \ /* y_out = e0(e1 - x_out) - e2 */ \ element_sub(e1, e1, x); \ element_mul(e0, e0, e1); \ element_sub(y, e0, e2); \ } #define do_tangent(e, edenom) { \ /* a = -(3x^2 + cca z^4) */ \ /* b = 2 y z^3 */ \ /* c = -(2 y^2 + x a) */ \ /* a = z^2 a */ \ element_square(a, z2); \ element_mul(a, a, cca); \ element_square(b, Zx); \ /* element_mul_si(b, b, 3); */ \ element_double(e0, b); \ element_add(b, b, e0); \ element_add(a, a, b); \ element_neg(a, a); \ \ /* element_mul_si(e0, Zy, 2); */ \ element_double(e0, Zy); \ element_mul(b, e0, z2); \ element_mul(b, b, z); \ \ element_mul(c, Zx, a); \ element_mul(a, a, z2); \ element_mul(e0, e0, Zy); \ element_add(c, c, e0); \ element_neg(c, c); \ \ element_mul(e0, a, numx); \ element_mul(e1, b, numy); \ element_add(e0, e0, e1); \ element_add(e0, e0, c); \ element_mul(e, e, e0); \ \ element_mul(e0, a, denomx); \ element_mul(e1, b, denomy); \ element_add(e0, e0, e1); \ element_add(e0, e0, c); \ element_mul(edenom, edenom, e0); \ } #define do_vertical(e, edenom, Ax) { \ element_mul(e0, numx, z2); \ element_sub(e0, e0, Ax); \ element_mul(e, e, e0); \ \ element_mul(e0, denomx, z2); \ element_sub(e0, e0, Ax); \ element_mul(edenom, edenom, e0); \ } #define do_line(e, edenom, A, B) { \ element_ptr Ax = curve_x_coord(A); \ element_ptr Ay = curve_y_coord(A); \ element_ptr Bx = curve_x_coord(B); \ element_ptr By = curve_y_coord(B); \ \ element_sub(b, Bx, Ax); \ element_sub(a, Ay, By); \ element_mul(c, Ax, By); \ element_mul(e0, Ay, Bx); \ element_sub(c, c, e0); \ \ element_mul(e0, a, numx); \ element_mul(e1, b, numy); \ element_add(e0, e0, e1); \ element_add(e0, e0, c); \ element_mul(e, e, e0); \ \ element_mul(e0, a, denomx); \ element_mul(e1, b, denomy); \ element_add(e0, e0, e1); \ element_add(e0, e0, c); \ element_mul(edenom, edenom, e0); \ } element_init(a, res->field); element_init(b, res->field); element_init(c, res->field); element_init(e0, res->field); element_init(e1, res->field); element_init(z, res->field); element_init(z2, res->field); element_set1(z); element_set1(z2); element_init(v, res->field); element_init(vd, res->field); element_init(v1, res->field); element_init(vd1, res->field); element_init(Z, P->field); element_init(Z1, P->field); element_set(Z, P); Zx = curve_x_coord(Z); Zy = curve_y_coord(Z); element_set1(v); element_set1(vd); element_set1(v1); element_set1(vd1); n = p->exp1; for (i=0; i<n; i++) { element_square(v, v); element_square(vd, vd); do_tangent(v, vd); proj_double(); do_vertical(vd, v, Zx); } to_affine(); if (p->sign1 < 0) { element_set(v1, vd); element_set(vd1, v); do_vertical(vd1, v1, Zx); element_neg(Z1, Z); } else { element_set(v1, v); element_set(vd1, vd); element_set(Z1, Z); } n = p->exp2; for (; i<n; i++) { element_square(v, v); element_square(vd, vd); do_tangent(v, vd); proj_double(); do_vertical(vd, v, Zx); } to_affine(); element_mul(v, v, v1); element_mul(vd, vd, vd1); do_line(v, vd, Z, Z1); element_add(Z, Z, Z1); do_vertical(vd, v, Zx); if (p->sign0 > 0) { do_vertical(v, vd, Px); } element_invert(vd, vd); element_mul(res, v, vd); element_clear(v); element_clear(vd); element_clear(v1); element_clear(vd1); element_clear(z); element_clear(z2); element_clear(Z); element_clear(Z1); element_clear(a); element_clear(b); element_clear(c); element_clear(e0); element_clear(e1); #undef to_affine #undef proj_double #undef do_tangent #undef do_vertical #undef do_line }
static void e_miller_affine(element_t res, element_t P, element_ptr QR, element_ptr R, e_pairing_data_ptr p) { //collate divisions int n; element_t v, vd; element_t v1, vd1; element_t Z, Z1; element_t a, b, c; element_t e0, e1; const element_ptr Px = curve_x_coord(P); const element_ptr cca = curve_a_coeff(P); element_ptr Zx, Zy; int i; const element_ptr numx = curve_x_coord(QR); const element_ptr numy = curve_y_coord(QR); const element_ptr denomx = curve_x_coord(R); const element_ptr denomy = curve_y_coord(R); #define do_vertical(e, edenom, Ax) { \ element_sub(e0, numx, Ax); \ element_mul(e, e, e0); \ \ element_sub(e0, denomx, Ax); \ element_mul(edenom, edenom, e0); \ } #define do_tangent(e, edenom) { \ /* a = -slope_tangent(A.x, A.y); */ \ /* b = 1; */ \ /* c = -(A.y + a * A.x); */ \ /* but we multiply by 2*A.y to avoid division */ \ \ /* a = -Ax * (Ax + Ax + Ax + twicea_2) - a_4; */ \ /* Common curves: a2 = 0 (and cc->a is a_4), so */ \ /* a = -(3 Ax^2 + cc->a) */ \ /* b = 2 * Ay */ \ /* c = -(2 Ay^2 + a Ax); */ \ \ element_square(a, Zx); \ element_mul_si(a, a, 3); \ element_add(a, a, cca); \ element_neg(a, a); \ \ element_add(b, Zy, Zy); \ \ element_mul(e0, b, Zy); \ element_mul(c, a, Zx); \ element_add(c, c, e0); \ element_neg(c, c); \ \ element_mul(e0, a, numx); \ element_mul(e1, b, numy); \ element_add(e0, e0, e1); \ element_add(e0, e0, c); \ element_mul(e, e, e0); \ \ element_mul(e0, a, denomx); \ element_mul(e1, b, denomy); \ element_add(e0, e0, e1); \ element_add(e0, e0, c); \ element_mul(edenom, edenom, e0); \ } #define do_line(e, edenom, A, B) { \ element_ptr Ax = curve_x_coord(A); \ element_ptr Ay = curve_y_coord(A); \ element_ptr Bx = curve_x_coord(B); \ element_ptr By = curve_y_coord(B); \ \ element_sub(b, Bx, Ax); \ element_sub(a, Ay, By); \ element_mul(c, Ax, By); \ element_mul(e0, Ay, Bx); \ element_sub(c, c, e0); \ \ element_mul(e0, a, numx); \ element_mul(e1, b, numy); \ element_add(e0, e0, e1); \ element_add(e0, e0, c); \ element_mul(e, e, e0); \ \ element_mul(e0, a, denomx); \ element_mul(e1, b, denomy); \ element_add(e0, e0, e1); \ element_add(e0, e0, c); \ element_mul(edenom, edenom, e0); \ } element_init(a, res->field); element_init(b, res->field); element_init(c, res->field); element_init(e0, res->field); element_init(e1, res->field); element_init(v, res->field); element_init(vd, res->field); element_init(v1, res->field); element_init(vd1, res->field); element_init(Z, P->field); element_init(Z1, P->field); element_set(Z, P); Zx = curve_x_coord(Z); Zy = curve_y_coord(Z); element_set1(v); element_set1(vd); element_set1(v1); element_set1(vd1); n = p->exp1; for (i=0; i<n; i++) { element_square(v, v); element_square(vd, vd); do_tangent(v, vd); element_double(Z, Z); do_vertical(vd, v, Zx); } if (p->sign1 < 0) { element_set(v1, vd); element_set(vd1, v); do_vertical(vd1, v1, Zx); element_neg(Z1, Z); } else { element_set(v1, v); element_set(vd1, vd); element_set(Z1, Z); } n = p->exp2; for (; i<n; i++) { element_square(v, v); element_square(vd, vd); do_tangent(v, vd); element_double(Z, Z); do_vertical(vd, v, Zx); } element_mul(v, v, v1); element_mul(vd, vd, vd1); do_line(v, vd, Z, Z1); element_add(Z, Z, Z1); do_vertical(vd, v, Zx); if (p->sign0 > 0) { do_vertical(v, vd, Px); } element_invert(vd, vd); element_mul(res, v, vd); element_clear(v); element_clear(vd); element_clear(v1); element_clear(vd1); element_clear(Z); element_clear(Z1); element_clear(a); element_clear(b); element_clear(c); element_clear(e0); element_clear(e1); #undef do_vertical #undef do_tangent #undef do_line }
static void cc_miller_no_denom(element_t res, mpz_t q, element_t P, element_ptr Qx, element_ptr Qy, element_t negalpha) { mp_bitcnt_t m; element_t v; element_t Z; element_t a, b, c; element_t t0; element_t e0, e1; element_ptr Zx, Zy; const element_ptr Px = curve_x_coord(P); const element_ptr Py = curve_y_coord(P); #define do_term(i, j, k, flag) { \ element_ptr e2; \ e2 = element_item(e0, i); \ element_mul(e1, element_item(v, j), Qx); \ if (flag == 1) element_mul(e1, e1, negalpha); \ element_mul(element_x(e1), element_x(e1), a); \ element_mul(element_y(e1), element_y(e1), a); \ element_mul(e2, element_item(v, k), Qy); \ element_mul(element_x(e2), element_x(e2), b); \ element_mul(element_y(e2), element_y(e2), b); \ element_add(e2, e2, e1); \ if (flag == 2) element_mul(e2, e2, negalpha); \ element_mul(element_x(e1), element_x(element_item(v, i)), c); \ element_mul(element_y(e1), element_y(element_item(v, i)), c); \ element_add(e2, e2, e1); \ } // a, b, c lie in Fq // Qx, Qy lie in Fq^2 // Qx is coefficient of x^4 // Qy is coefficient of x^3 // // computes v *= (a Qx x^4 + b Qy x^3 + c) // // recall x^6 = -alpha thus // x^4 (u0 + u1 x^1 + ... + u5 x^5) = // u0 x^4 + u1 x^5 // - alpha u2 - alpha u3 x - alpha u4 x^2 - alpha u5 x^3 // and // x^4 (u0 + u1 x^1 + ... + u5 x^5) = // u0 x^3 + u1 x^4 + u2 x^5 // - alpha u3 - alpha u4 x - alpha u5 x^2 #define f_miller_evalfn() { \ do_term(0, 2, 3, 2); \ do_term(1, 3, 4, 2); \ do_term(2, 4, 5, 2); \ do_term(3, 5, 0, 1); \ do_term(4, 0, 1, 0); \ do_term(5, 1, 2, 0); \ element_set(v, e0); \ } /* element_ptr e1; e1 = element_item(e0, 4); element_mul(element_x(e1), element_x(Qx), a); element_mul(element_y(e1), element_y(Qx), a); e1 = element_item(e0, 3); element_mul(element_x(e1), element_x(Qy), b); element_mul(element_y(e1), element_y(Qy), b); element_set(element_x(element_item(e0, 0)), c); element_mul(v, v, e0); */ //a = -3 Zx^2 since cc->a is 0 for D = 3 //b = 2 * Zy //c = -(2 Zy^2 + a Zx); #define do_tangent() { \ element_square(a, Zx); \ element_mul_si(a, a, 3); \ element_neg(a, a); \ \ element_add(b, Zy, Zy); \ \ element_mul(t0, b, Zy); \ element_mul(c, a, Zx); \ element_add(c, c, t0); \ element_neg(c, c); \ \ f_miller_evalfn(); \ } //a = -(B.y - A.y) / (B.x - A.x); //b = 1; //c = -(A.y + a * A.x); //but we'll multiply by B.x - A.x to avoid division #define do_line() { \ element_sub(b, Px, Zx); \ element_sub(a, Zy, Py); \ element_mul(t0, b, Zy); \ element_mul(c, a, Zx); \ element_add(c, c, t0); \ element_neg(c, c); \ \ f_miller_evalfn(); \ } element_init(a, Px->field); element_init(b, a->field); element_init(c, a->field); element_init(t0, a->field); element_init(e0, res->field); element_init(e1, Qx->field); element_init(v, res->field); element_init(Z, P->field); element_set(Z, P); Zx = curve_x_coord(Z); Zy = curve_y_coord(Z); element_set1(v); m = (mp_bitcnt_t)mpz_sizeinbase(q, 2); m = (m > 2 ? m - 2 : 0); //TODO: sliding NAF for(;;) { do_tangent(); if (!m) break; element_double(Z, Z); if (mpz_tstbit(q, m)) { do_line(); element_add(Z, Z, P); } m--; element_square(v, v); } element_set(res, v); element_clear(v); element_clear(Z); element_clear(a); element_clear(b); element_clear(c); element_clear(t0); element_clear(e0); element_clear(e1); #undef do_term #undef f_miller_evalfn #undef do_tangent #undef do_line }
static void d_pairing_pp_init(pairing_pp_t p, element_ptr in1, pairing_t pairing) { element_ptr P = in1; const element_ptr Px = curve_x_coord(P); const element_ptr Py = curve_y_coord(P); element_t Z; mp_bitcnt_t m; pptr info = pairing->data; element_t t0; element_t a, b, c; field_ptr Fq = info->Fq; pp_coeff_t *coeff; mpz_ptr q = pairing->r; pp_coeff_ptr pp; const element_ptr cca = curve_a_coeff(P); element_ptr Zx; element_ptr Zy; #define store_abc() { \ element_init(pp->a, Fq); \ element_init(pp->b, Fq); \ element_init(pp->c, Fq); \ element_set(pp->a, a); \ element_set(pp->b, b); \ element_set(pp->c, c); \ pp++; \ } #define do_tangent() { \ /* a = -slope_tangent(Z.x, Z.y); */ \ /* b = 1; */ \ /* c = -(Z.y + a * Z.x); */ \ /* but we multiply by 2*Z.y to avoid division. */ \ \ /* a = -Zx * (3 Zx + twicea_2) - a_4; */ \ /* Common curves: a2 = 0 (and cc->a is a_4), so */ \ /* a = -(3 Zx^2 + cc->a) */ \ /* b = 2 * Zy */ \ /* c = -(2 Zy^2 + a Zx); */ \ \ element_square(a, Zx); \ element_double(t0, a); \ element_add(a, a, t0); \ element_add(a, a, cca); \ element_neg(a, a); \ \ element_add(b, Zy, Zy); \ \ element_mul(t0, b, Zy); \ element_mul(c, a, Zx); \ element_add(c, c, t0); \ element_neg(c, c); \ \ store_abc(); \ } #define do_line() { \ /* a = -(B.y - A.y) / (B.x - A.x); */ \ /* b = 1; */ \ /* c = -(A.y + a * A.x); */ \ /* but we'll multiply by B.x - A.x to avoid division */ \ \ element_sub(b, Px, Zx); \ element_sub(a, Zy, Py); \ element_mul(t0, b, Zy); \ element_mul(c, a, Zx); \ element_add(c, c, t0); \ element_neg(c, c); \ \ store_abc(); \ } element_init(Z, P->field); element_set(Z, P); Zx = curve_x_coord(Z); Zy = curve_y_coord(Z); element_init(t0, Fq); element_init(a, Fq); element_init(b, Fq); element_init(c, Fq); m = (mp_bitcnt_t)mpz_sizeinbase(q, 2); m = (m > 2 ? m - 2 : 0); p->data = pbc_malloc(sizeof(pp_coeff_t) * 2 * m); coeff = (pp_coeff_t *) p->data; pp = coeff[0]; for(;;) { do_tangent(); if (!m) break; element_double(Z, Z); if (mpz_tstbit(q, m)) { do_line(); element_add(Z, Z, P); } m--; } element_clear(t0); element_clear(a); element_clear(b); element_clear(c); element_clear(Z); #undef store_abc #undef do_tangent #undef do_line }
// Same as above, but with affine coordinates. static void cc_miller_no_denom_affine(element_t res, mpz_t q, element_t P, element_ptr Qx, element_ptr Qy) { mp_bitcnt_t m; element_t v; element_t Z; element_t a, b, c; element_t t0; element_t e0; const element_ptr cca = curve_a_coeff(P); const element_ptr Px = curve_x_coord(P); const element_ptr Py = curve_y_coord(P); element_ptr Zx, Zy; /* TODO: when exactly is this not needed? void do_vertical() { mapbase(e0, Z->x); element_sub(e0, Qx, e0); element_mul(v, v, e0); } */ #define do_tangent() { \ /* a = -(3 Zx^2 + cc->a) */ \ /* b = 2 * Zy */ \ /* c = -(2 Zy^2 + a Zx); */ \ \ element_square(a, Zx); \ element_mul_si(a, a, 3); \ element_add(a, a, cca); \ element_neg(a, a); \ \ element_add(b, Zy, Zy); \ \ element_mul(t0, b, Zy); \ element_mul(c, a, Zx); \ element_add(c, c, t0); \ element_neg(c, c); \ \ d_miller_evalfn(e0, a, b, c, Qx, Qy); \ element_mul(v, v, e0); \ } #define do_line() { \ /* a = -(B.y - A.y) / (B.x - A.x); */ \ /* b = 1; */ \ /* c = -(A.y + a * A.x); */ \ /* but we multiply by B.x - A.x to avoid division. */ \ \ element_sub(b, Px, Zx); \ element_sub(a, Zy, Py); \ element_mul(t0, b, Zy); \ element_mul(c, a, Zx); \ element_add(c, c, t0); \ element_neg(c, c); \ \ d_miller_evalfn(e0, a, b, c, Qx, Qy); \ element_mul(v, v, e0); \ } element_init(a, Px->field); element_init(b, a->field); element_init(c, a->field); element_init(t0, a->field); element_init(e0, res->field); element_init(v, res->field); element_init(Z, P->field); element_set(Z, P); Zx = curve_x_coord(Z); Zy = curve_y_coord(Z); element_set1(v); m = (mp_bitcnt_t)mpz_sizeinbase(q, 2); m = (m > 2 ? m - 2 : 0); for(;;) { do_tangent(); if (!m) break; element_double(Z, Z); if (mpz_tstbit(q, m)) { do_line(); element_add(Z, Z, P); } m--; element_square(v, v); } element_set(res, v); element_clear(v); element_clear(Z); element_clear(a); element_clear(b); element_clear(c); element_clear(t0); element_clear(e0); #undef do_tangent #undef do_line }
// Miller's algorithm, assuming we can ignore the denominator. We can do this // with careful group selection when the embedding degree is even. See thesis. // This version uses projective coordinates, which don't seem much faster. static void cc_miller_no_denom_proj(element_t res, mpz_t q, element_t P, element_ptr Qx, element_ptr Qy) { mp_bitcnt_t m; element_t v; element_t Z; element_t a, b, c; element_t t0, t1; element_ptr t2 = a, t3 = b, t4 = c; element_t e0; element_t z, z2; element_ptr Zx, Zy; const element_ptr curve_a = curve_a_coeff(P); const element_ptr Px = curve_x_coord(P); const element_ptr Py = curve_y_coord(P); #define proj_double() { \ /* t0 = 3x^2 + (curve_a) z^4 */ \ element_square(t0, Zx); \ /* element_mul_si(t0, t0, 3); */ \ element_double(t1, t0); \ element_add(t0, t0, t1); \ element_square(t1, z2); \ element_mul(t1, t1, curve_a); \ element_add(t0, t0, t1); \ \ /* z_out = 2 y z */ \ element_mul(z, Zy, z); \ /* element_mul_si(z, z, 2); */ \ element_double(z, z); \ element_square(z2, z); \ \ /* t1 = 4 x y^2 */ \ element_square(t2, Zy); \ element_mul(t1, Zx, t2); \ /* element_mul_si(t1, t1, 4); */ \ element_double(t1, t1); \ element_double(t1, t1); \ \ /* x_out = t0^2 - 2 t1 */ \ /* element_mul_si(t3, t1, 2); */ \ element_double(t3, t1); \ element_square(Zx, t0); \ element_sub(Zx, Zx, t3); \ \ /* t2 = 8y^4 */ \ element_square(t2, t2); \ /* element_mul_si(t2, t2, 8); */ \ element_double(t2, t2); \ element_double(t2, t2); \ element_double(t2, t2); \ \ /* y_out = t0(t1 - x_out) - t2 */ \ element_sub(t1, t1, Zx); \ element_mul(t0, t0, t1); \ element_sub(Zy, t0, t2); \ } #define proj_mixin() { \ /* t2 = Px z^2 */ \ element_mul(t2, z2, Px); \ \ /* t3 = Zx - t2 */ \ element_sub(t3, Zx, t2); \ \ /* t0 = Py z^3 */ \ element_mul(t0, z2, Py); \ element_mul(t0, t0, z); \ \ /* t1 = Zy - t0 */ \ element_sub(t1, Zy, t0); \ \ /* e7 = Zx + t2, use t2 to double for e7 */ \ element_add(t2, Zx, t2); \ \ /* e8 = Zy + t0, use t0 to double for e8 */ \ element_add(t0, Zy, t0); \ \ /* z = z t3 */ \ element_mul(z, z, t3); \ element_square(z2, z); \ \ /* Zx = t1^2 - e7 t3^2 */ \ /* t3 now holds t3^3, */ \ /* t4 holds e7 t3^2. */ \ element_square(t4, t3); \ element_mul(t3, t4, t3); \ element_square(Zx, t1); \ element_mul(t4, t2, t4); \ element_sub(Zx, Zx, t4); \ \ /* t4 = e7 t3^2 - 2 Zx */ \ element_sub(t4, t4, Zx); \ element_sub(t4, t4, Zx); \ \ /* Zy = (t4 t1 - e8 t3^3)/2 */ \ element_mul(t4, t4, t1); \ element_mul(t0, t0, t3); \ element_sub(t4, t4, t0); \ element_halve(Zy, t4); \ } #define do_tangent() { \ /* a = -(3x^2 + cca z^4) */ \ /* b = 2 y z^3 */ \ /* c = -(2 y^2 + x a) */ \ /* a = z^2 a */ \ element_square(a, z2); \ element_mul(a, a, curve_a); \ element_square(b, Zx); \ /* element_mul_si(b, b, 3); */ \ element_double(t0, b); \ element_add(b, b, t0); \ element_add(a, a, b); \ element_neg(a, a); \ \ element_mul(b, z, z2); \ element_mul(b, b, Zy); \ element_mul_si(b, b, 2); \ \ element_mul(c, Zx, a); \ element_mul(a, a, z2); \ element_square(t0, Zy); \ element_mul_si(t0, t0, 2); \ element_add(c, c, t0); \ element_neg(c, c); \ \ d_miller_evalfn(e0, a, b, c, Qx, Qy); \ element_mul(v, v, e0); \ } #define do_line() { \ /* a = -(Py z^3 - Zy) */ \ /* b = Px z^3 - Zx z */ \ /* c = Zx z Py - Zy Px; */ \ \ element_mul(t0, Zx, z); \ element_mul(t1, z2, z); \ \ element_mul(a, Py, t1); \ element_sub(a, Zy, a); \ \ element_mul(b, Px, t1); \ element_sub(b, b, t0); \ \ element_mul(t0, t0, Py); \ element_mul(c, Zy, Px); \ element_sub(c, t0, c); \ \ d_miller_evalfn(e0, a, b, c, Qx, Qy); \ element_mul(v, v, e0); \ } element_init(a, Px->field); element_init(b, a->field); element_init(c, a->field); element_init(t0, a->field); element_init(t1, a->field); element_init(e0, res->field); element_init(z, a->field); element_init(z2, a->field); element_set1(z); element_set1(z2); element_init(v, res->field); element_init(Z, P->field); element_set(Z, P); Zx = curve_x_coord(Z); Zy = curve_x_coord(Z); element_set1(v); m = (mp_bitcnt_t)mpz_sizeinbase(q, 2); m = (m > 2 ? m - 2 : 0); for(;;) { do_tangent(); if (!m) break; proj_double(); if (mpz_tstbit(q, m)) { do_line(); proj_mixin(); } m--; element_square(v, v); } element_set(res, v); element_clear(v); element_clear(Z); element_clear(a); element_clear(b); element_clear(c); element_clear(t0); element_clear(t1); element_clear(e0); element_clear(z); element_clear(z2); #undef proj_double #undef proj_mixin #undef do_tangent #undef do_line }