pq-crypto/sike_r3/sikep434r3_ec_isogeny.c

/********************************************************************************************
* Supersingular Isogeny Key Encapsulation Library
*
* Abstract: elliptic curve and isogeny functions
*********************************************************************************************/

#include "sikep434r3.h"
#include "sikep434r3_fpx.h"
#include "sikep434r3_ec_isogeny.h"

/* Doubling of a Montgomery point in projective coordinates (X:Z).
 * Input: projective Montgomery x-coordinates P = (X1:Z1), where x1=X1/Z1 and Montgomery curve constants A+2C and 4C.
 * Output: projective Montgomery x-coordinates Q = 2*P = (X2:Z2). */
void xDBL(const point_proj_t P, point_proj_t Q, const f2elm_t *A24plus, const f2elm_t *C24)
{
    f2elm_t _t0, _t1;
    f2elm_t *t0=&_t0, *t1=&_t1;

    mp2_sub_p2(&P->X, &P->Z, t0);                     /* t0 = X1-Z1 */
    mp2_add(&P->X, &P->Z, t1);                        /* t1 = X1+Z1 */
    fp2sqr_mont(t0, t0);                              /* t0 = (X1-Z1)^2 */
    fp2sqr_mont(t1, t1);                              /* t1 = (X1+Z1)^2 */
    fp2mul_mont(C24, t0, &Q->Z);                      /* Z2 = C24*(X1-Z1)^2 */
    fp2mul_mont(t1, &Q->Z, &Q->X);                    /* X2 = C24*(X1-Z1)^2*(X1+Z1)^2 */
    mp2_sub_p2(t1, t0, t1);                           /* t1 = (X1+Z1)^2-(X1-Z1)^2 */
    fp2mul_mont(A24plus, t1, t0);                     /* t0 = A24plus*[(X1+Z1)^2-(X1-Z1)^2] */
    mp2_add(&Q->Z, t0, &Q->Z);                        /* Z2 = A24plus*[(X1+Z1)^2-(X1-Z1)^2] + C24*(X1-Z1)^2 */
    fp2mul_mont(&Q->Z, t1, &Q->Z);                    /* Z2 = [A24plus*[(X1+Z1)^2-(X1-Z1)^2] + C24*(X1-Z1)^2]*[(X1+Z1)^2-(X1-Z1)^2] */
}

/* Computes [2^e](X:Z) on Montgomery curve with projective constant via e repeated doublings.
 * Input: projective Montgomery x-coordinates P = (XP:ZP), such that xP=XP/ZP and Montgomery curve constants A+2C and 4C.
 * Output: projective Montgomery x-coordinates Q <- (2^e)*P. */
void xDBLe(const point_proj_t P, point_proj_t Q, const f2elm_t *A24plus, const f2elm_t *C24, const int e)
{
    int i;

    copy_words((const digit_t*)P, (digit_t*)Q, 2*2*S2N_SIKE_P434_R3_NWORDS_FIELD);

    for (i = 0; i < e; i++) {
        xDBL(Q, Q, A24plus, C24);
    }
}

/* Computes the corresponding 4-isogeny of a projective Montgomery point (X4:Z4) of order 4.
 * Input:  projective point of order four P = (X4:Z4).
 * Output: the 4-isogenous Montgomery curve with projective coefficients A+2C/4C and the 3 coefficients
 * that are used to evaluate the isogeny at a point in eval_4_isog(). */
void get_4_isog(const point_proj_t P, f2elm_t *A24plus, f2elm_t *C24, f2elm_t *coeff)
{
    mp2_sub_p2(&P->X, &P->Z, &coeff[1]);               /* coeff[1] = X4-Z4 */
    mp2_add(&P->X, &P->Z, &coeff[2]);                  /* coeff[2] = X4+Z4 */
    fp2sqr_mont(&P->Z, &coeff[0]);                     /* coeff[0] = Z4^2 */
    mp2_add(&coeff[0], &coeff[0], &coeff[0]);          /* coeff[0] = 2*Z4^2 */
    fp2sqr_mont(&coeff[0], C24);                       /* C24 = 4*Z4^4 */
    mp2_add(&coeff[0], &coeff[0], &coeff[0]);          /* coeff[0] = 4*Z4^2 */
    fp2sqr_mont(&P->X, A24plus);                       /* A24plus = X4^2 */
    mp2_add(A24plus, A24plus, A24plus);                /* A24plus = 2*X4^2 */
    fp2sqr_mont(A24plus, A24plus);                     /* A24plus = 4*X4^4 */
}

/* Evaluates the isogeny at the point (X:Z) in the domain of the isogeny, given a 4-isogeny phi defined
 * by the 3 coefficients in coeff (computed in the function get_4_isog()).
 * Inputs: the coefficients defining the isogeny, and the projective point P = (X:Z).
 * Output: the projective point P = phi(P) = (X:Z) in the codomain.  */
void eval_4_isog(point_proj_t P, f2elm_t *coeff)
{
    f2elm_t _t0, _t1;
    f2elm_t *t0=&_t0, *t1=&_t1;

    mp2_add(&P->X, &P->Z, t0);                        /* t0 = X+Z */
    mp2_sub_p2(&P->X, &P->Z, t1);                     /* t1 = X-Z */
    fp2mul_mont(t0, &coeff[1], &P->X);                /* X = (X+Z)*coeff[1] */
    fp2mul_mont(t1, &coeff[2], &P->Z);                /* Z = (X-Z)*coeff[2] */
    fp2mul_mont(t0, t1, t0);                          /* t0 = (X+Z)*(X-Z) */
    fp2mul_mont(&coeff[0], t0, t0);                   /* t0 = coeff[0]*(X+Z)*(X-Z) */
    mp2_add(&P->X, &P->Z, t1);                        /* t1 = (X-Z)*coeff[2] + (X+Z)*coeff[1] */
    mp2_sub_p2(&P->X, &P->Z, &P->Z);                  /* Z = (X-Z)*coeff[2] - (X+Z)*coeff[1] */
    fp2sqr_mont(t1, t1);                              /* t1 = [(X-Z)*coeff[2] + (X+Z)*coeff[1]]^2 */
    fp2sqr_mont(&P->Z, &P->Z);                        /* Z = [(X-Z)*coeff[2] - (X+Z)*coeff[1]]^2 */
    mp2_add(t1, t0, &P->X);                           /* X = coeff[0]*(X+Z)*(X-Z) + [(X-Z)*coeff[2] + (X+Z)*coeff[1]]^2 */
    mp2_sub_p2(&P->Z, t0, t0);                        /* t0 = [(X-Z)*coeff[2] - (X+Z)*coeff[1]]^2 - coeff[0]*(X+Z)*(X-Z) */
    fp2mul_mont(&P->X, t1, &P->X);                    /* Xfinal */
    fp2mul_mont(&P->Z, t0, &P->Z);                    /* Zfinal */
}

/* Tripling of a Montgomery point in projective coordinates (X:Z).
 * Input: projective Montgomery x-coordinates P = (X:Z), where x=X/Z and Montgomery curve constants A24plus = A+2C and A24minus = A-2C.
 * Output: projective Montgomery x-coordinates Q = 3*P = (X3:Z3).  */
void xTPL(const point_proj_t P, point_proj_t Q, const f2elm_t *A24minus, const f2elm_t *A24plus)
{
    f2elm_t _t0, _t1, _t2, _t3, _t4, _t5, _t6;
    f2elm_t *t0=&_t0, *t1=&_t1, *t2=&_t2, *t3=&_t3, *t4=&_t4, *t5=&_t5, *t6=&_t6;

    mp2_sub_p2(&P->X, &P->Z, t0);                     /* t0 = X-Z */
    fp2sqr_mont(t0, t2);                              /* t2 = (X-Z)^2 */
    mp2_add(&P->X, &P->Z, t1);                        /* t1 = X+Z */
    fp2sqr_mont(t1, t3);                              /* t3 = (X+Z)^2 */
    mp2_add(&P->X, &P->X, t4);                        /* t4 = 2*X */
    mp2_add(&P->Z, &P->Z, t0);                        /* t0 = 2*Z */
    fp2sqr_mont(t4, t1);                              /* t1 = 4*X^2 */
    mp2_sub_p2(t1, t3, t1);                           /* t1 = 4*X^2 - (X+Z)^2 */
    mp2_sub_p2(t1, t2, t1);                           /* t1 = 4*X^2 - (X+Z)^2 - (X-Z)^2 */
    fp2mul_mont(A24plus, t3, t5);                     /* t5 = A24plus*(X+Z)^2 */
    fp2mul_mont(t3, t5, t3);                          /* t3 = A24plus*(X+Z)^4 */
    fp2mul_mont(A24minus, t2, t6);                    /* t6 = A24minus*(X-Z)^2 */
    fp2mul_mont(t2, t6, t2);                          /* t2 = A24minus*(X-Z)^4 */
    mp2_sub_p2(t2, t3, t3);                           /* t3 = A24minus*(X-Z)^4 - A24plus*(X+Z)^4 */
    mp2_sub_p2(t5, t6, t2);                           /* t2 = A24plus*(X+Z)^2 - A24minus*(X-Z)^2 */
    fp2mul_mont(t1, t2, t1);                          /* t1 = [4*X^2 - (X+Z)^2 - (X-Z)^2]*[A24plus*(X+Z)^2 - A24minus*(X-Z)^2] */
    fp2add(t3, t1, t2);                               /* t2 = [4*X^2 - (X+Z)^2 - (X-Z)^2]*[A24plus*(X+Z)^2 - A24minus*(X-Z)^2] + A24minus*(X-Z)^4 - A24plus*(X+Z)^4 */
    fp2sqr_mont(t2, t2);                              /* t2 = t2^2 */
    fp2mul_mont(t4, t2, &Q->X);                       /* X3 = 2*X*t2 */
    fp2sub(t3, t1, t1);                               /* t1 = A24minus*(X-Z)^4 - A24plus*(X+Z)^4 - [4*X^2 - (X+Z)^2 - (X-Z)^2]*[A24plus*(X+Z)^2 - A24minus*(X-Z)^2] */
    fp2sqr_mont(t1, t1);                              /* t1 = t1^2 */
    fp2mul_mont(t0, t1, &Q->Z);                       /* Z3 = 2*Z*t1 */
}

/* Computes [3^e](X:Z) on Montgomery curve with projective constant via e repeated triplings.
 * Input: projective Montgomery x-coordinates P = (XP:ZP), such that xP=XP/ZP and Montgomery curve constants A24plus = A+2C and A24minus = A-2C.
 * Output: projective Montgomery x-coordinates Q <- (3^e)*P. */
void xTPLe(const point_proj_t P, point_proj_t Q, const f2elm_t *A24minus, const f2elm_t *A24plus, const int e)
{
    int i;

    copy_words((const digit_t*)P, (digit_t*)Q, 2*2*S2N_SIKE_P434_R3_NWORDS_FIELD);

    for (i = 0; i < e; i++) {
        xTPL(Q, Q, A24minus, A24plus);
    }
}

/* Computes the corresponding 3-isogeny of a projective Montgomery point (X3:Z3) of order 3.
 * Input:  projective point of order three P = (X3:Z3).
 * Output: the 3-isogenous Montgomery curve with projective coefficient A/C.  */
void get_3_isog(const point_proj_t P, f2elm_t *A24minus, f2elm_t *A24plus, f2elm_t *coeff)
{
    f2elm_t _t0, _t1, _t2, _t3, _t4;
    f2elm_t *t0=&_t0, *t1=&_t1, *t2=&_t2, *t3=&_t3, *t4=&_t4;

    mp2_sub_p2(&P->X, &P->Z, &coeff[0]);               /* coeff0 = X-Z */
    fp2sqr_mont(&coeff[0], t0);                        /* t0 = (X-Z)^2 */
    mp2_add(&P->X, &P->Z, &coeff[1]);                  /* coeff1 = X+Z */
    fp2sqr_mont(&coeff[1], t1);                        /* t1 = (X+Z)^2 */
    mp2_add(&P->X, &P->X, t3);                         /* t3 = 2*X */
    fp2sqr_mont(t3, t3);                               /* t3 = 4*X^2 */
    fp2sub(t3, t0, t2);                                /* t2 = 4*X^2 - (X-Z)^2 */
    fp2sub(t3, t1, t3);                                /* t3 = 4*X^2 - (X+Z)^2 */
    mp2_add(t0, t3, t4);                               /* t4 = 4*X^2 - (X+Z)^2 + (X-Z)^2 */
    mp2_add(t4, t4, t4);                               /* t4 = 2(4*X^2 - (X+Z)^2 + (X-Z)^2) */
    mp2_add(t1, t4, t4);                               /* t4 = 8*X^2 - (X+Z)^2 + 2*(X-Z)^2 */
    fp2mul_mont(t2, t4, A24minus);                     /* A24minus = [4*X^2 - (X-Z)^2]*[8*X^2 - (X+Z)^2 + 2*(X-Z)^2] */
    mp2_add(t1, t2, t4);                               /* t4 = 4*X^2 + (X+Z)^2 - (X-Z)^2 */
    mp2_add(t4, t4, t4);                               /* t4 = 2(4*X^2 + (X+Z)^2 - (X-Z)^2) */
    mp2_add(t0, t4, t4);                               /* t4 = 8*X^2 + 2*(X+Z)^2 - (X-Z)^2 */
    fp2mul_mont(t3, t4, A24plus);                      /* A24plus = [4*X^2 - (X+Z)^2]*[8*X^2 + 2*(X+Z)^2 - (X-Z)^2] */
}

/* Computes the 3-isogeny R=phi(X:Z), given projective point (X3:Z3) of order 3 on a Montgomery curve and
 * a point P with 2 coefficients in coeff (computed in the function get_3_isog()).
 * Inputs: projective points P = (X3:Z3) and Q = (X:Z).
 * Output: the projective point Q <- phi(Q) = (X3:Z3).  */
void eval_3_isog(point_proj_t Q, const f2elm_t *coeff)
{
    f2elm_t _t0, _t1, _t2;
    f2elm_t *t0=&_t0, *t1=&_t1, *t2=&_t2;

    mp2_add(&Q->X, &Q->Z, t0);                      /* t0 = X+Z */
    mp2_sub_p2(&Q->X, &Q->Z, t1);                   /* t1 = X-Z */
    fp2mul_mont(&coeff[0], t0, t0);                 /* t0 = coeff0*(X+Z) */
    fp2mul_mont(&coeff[1], t1, t1);                 /* t1 = coeff1*(X-Z) */
    mp2_add(t0, t1, t2);                            /* t2 = coeff0*(X+Z) + coeff1*(X-Z) */
    mp2_sub_p2(t1, t0, t0);                         /* t0 = coeff1*(X-Z) - coeff0*(X+Z) */
    fp2sqr_mont(t2, t2);                            /* t2 = [coeff0*(X+Z) + coeff1*(X-Z)]^2 */
    fp2sqr_mont(t0, t0);                            /* t0 = [coeff1*(X-Z) - coeff0*(X+Z)]^2 */
    fp2mul_mont(&Q->X, t2, &Q->X);                  /* X3final = X*[coeff0*(X+Z) + coeff1*(X-Z)]^2 */
    fp2mul_mont(&Q->Z, t0, &Q->Z);                  /* Z3final = Z*[coeff1*(X-Z) - coeff0*(X+Z)]^2 */
}

/* 3-way simultaneous inversion
 * Input:  z1,z2,z3
 * Output: 1/z1,1/z2,1/z3 (override inputs). */
void inv_3_way(f2elm_t *z1, f2elm_t *z2, f2elm_t *z3)
{
    f2elm_t _t0, _t1, _t2, _t3;
    f2elm_t *t0=&_t0, *t1=&_t1, *t2=&_t2, *t3=&_t3;

    fp2mul_mont(z1, z2, t0);                      /* t0 = z1*z2 */
    fp2mul_mont(z3, t0, t1);                      /* t1 = z1*z2*z3 */
    fp2inv_mont(t1);                              /* t1 = 1/(z1*z2*z3) */
    fp2mul_mont(z3, t1, t2);                      /* t2 = 1/(z1*z2) */
    fp2mul_mont(t2, z2, t3);                      /* t3 = 1/z1 */
    fp2mul_mont(t2, z1, z2);                      /* z2 = 1/z2 */
    fp2mul_mont(t0, t1, z3);                      /* z3 = 1/z3 */
    fp2copy(t3, z1);                              /* z1 = 1/z1 */
}

/* Given the x-coordinates of P, Q, and R, returns the value A corresponding to the
 *     Montgomery curve E_A: y^2=x^3+A*x^2+x such that R=Q-P on E_A.
 * Input:  the x-coordinates xP, xQ, and xR of the points P, Q and R.
 * Output: the coefficient A corresponding to the curve E_A: y^2=x^3+A*x^2+x. */
void get_A(const f2elm_t *xP, const f2elm_t *xQ, const f2elm_t *xR, f2elm_t *A)
{
    f2elm_t _t0, _t1, one = {0};
    f2elm_t *t0=&_t0, *t1=&_t1;


    fpcopy((const digit_t*)&Montgomery_one,one.e[0]);
    fp2add(xP, xQ, t1);                           /* t1 = xP+xQ */
    fp2mul_mont(xP, xQ, t0);                      /* t0 = xP*xQ */
    fp2mul_mont(xR, t1, A);                       /* A = xR*t1 */
    fp2add(t0, A, A);                             /* A = A+t0 */
    fp2mul_mont(t0, xR, t0);                      /* t0 = t0*xR */
    fp2sub(A, &one, A);                           /* A = A-1 */
    fp2add(t0, t0, t0);                           /* t0 = t0+t0 */
    fp2add(t1, xR, t1);                           /* t1 = t1+xR */
    fp2add(t0, t0, t0);                           /* t0 = t0+t0 */
    fp2sqr_mont(A, A);                            /* A = A^2 */
    fp2inv_mont(t0);                              /* t0 = 1/t0 */
    fp2mul_mont(A, t0, A);                        /* A = A*t0 */
    fp2sub(A, t1, A);                             /* Afinal = A-t1 */
}

/* Computes the j-invariant of a Montgomery curve with projective constant.
 * Input: A,C in GF(p^2).
 * Output: j=256*(A^2-3*C^2)^3/(C^4*(A^2-4*C^2)), which is the j-invariant of the Montgomery curve
 *     B*y^2=x^3+(A/C)*x^2+x or (equivalently) j-invariant of B'*y^2=C*x^3+A*x^2+C*x. */
void j_inv(const f2elm_t *A, const f2elm_t *C, f2elm_t *jinv)
{
    f2elm_t _t0, _t1;
    f2elm_t *t0=&_t0, *t1=&_t1;

    fp2sqr_mont(A, jinv);                           /* jinv = A^2 */
    fp2sqr_mont(C, t1);                             /* t1 = C^2 */
    fp2add(t1, t1, t0);                             /* t0 = t1+t1 */
    fp2sub(jinv, t0, t0);                           /* t0 = jinv-t0 */
    fp2sub(t0, t1, t0);                             /* t0 = t0-t1 */
    fp2sub(t0, t1, jinv);                           /* jinv = t0-t1 */
    fp2sqr_mont(t1, t1);                            /* t1 = t1^2 */
    fp2mul_mont(jinv, t1, jinv);                    /* jinv = jinv*t1 */
    fp2add(t0, t0, t0);                             /* t0 = t0+t0 */
    fp2add(t0, t0, t0);                             /* t0 = t0+t0 */
    fp2sqr_mont(t0, t1);                            /* t1 = t0^2 */
    fp2mul_mont(t0, t1, t0);                        /* t0 = t0*t1 */
    fp2add(t0, t0, t0);                             /* t0 = t0+t0 */
    fp2add(t0, t0, t0);                             /* t0 = t0+t0 */
    fp2inv_mont(jinv);                              /* jinv = 1/jinv */
    fp2mul_mont(jinv, t0, jinv);                    /* jinv = t0*jinv */
}

/* Simultaneous doubling and differential addition.
 * Input: projective Montgomery points P=(XP:ZP) and Q=(XQ:ZQ) such that xP=XP/ZP and xQ=XQ/ZQ,
 *     affine difference xPQ=x(P-Q) and Montgomery curve constant A24=(A+2)/4.
 * Output: projective Montgomery points P <- 2*P = (X2P:Z2P) such that x(2P)=X2P/Z2P,
 *     and Q <- P+Q = (XQP:ZQP) such that = x(Q+P)=XQP/ZQP.  */
static void xDBLADD(point_proj_t P, point_proj_t Q, const f2elm_t *xPQ, const f2elm_t *A24)
{
    f2elm_t _t0, _t1, _t2;
    f2elm_t *t0=&_t0, *t1=&_t1, *t2=&_t2;

    mp2_add(&P->X, &P->Z, t0);                        /* t0 = XP+ZP */
    mp2_sub_p2(&P->X, &P->Z, t1);                     /* t1 = XP-ZP */
    fp2sqr_mont(t0, &P->X);                           /* XP = (XP+ZP)^2 */
    mp2_sub_p2(&Q->X, &Q->Z, t2);                     /* t2 = XQ-ZQ */
    mp2_add(&Q->X, &Q->Z, &Q->X);                     /* XQ = XQ+ZQ */
    fp2mul_mont(t0, t2, t0);                          /* t0 = (XP+ZP)*(XQ-ZQ) */
    fp2sqr_mont(t1, &P->Z);                           /* ZP = (XP-ZP)^2 */
    fp2mul_mont(t1, &Q->X, t1);                       /* t1 = (XP-ZP)*(XQ+ZQ) */
    mp2_sub_p2(&P->X, &P->Z, t2);                     /* t2 = (XP+ZP)^2-(XP-ZP)^2 */
    fp2mul_mont(&P->X, &P->Z, &P->X);                 /* XP = (XP+ZP)^2*(XP-ZP)^2 */
    fp2mul_mont(A24, t2, &Q->X);                      /* XQ = A24*[(XP+ZP)^2-(XP-ZP)^2] */
    mp2_sub_p2(t0, t1, &Q->Z);                        /* ZQ = (XP+ZP)*(XQ-ZQ)-(XP-ZP)*(XQ+ZQ) */
    mp2_add(&Q->X, &P->Z, &P->Z);                     /* ZP = A24*[(XP+ZP)^2-(XP-ZP)^2]+(XP-ZP)^2 */
    mp2_add(t0, t1, &Q->X);                           /* XQ = (XP+ZP)*(XQ-ZQ)+(XP-ZP)*(XQ+ZQ) */
    fp2mul_mont(&P->Z, t2, &P->Z);                    /* ZP = [A24*[(XP+ZP)^2-(XP-ZP)^2]+(XP-ZP)^2]*[(XP+ZP)^2-(XP-ZP)^2] */
    fp2sqr_mont(&Q->Z, &Q->Z);                        /* ZQ = [(XP+ZP)*(XQ-ZQ)-(XP-ZP)*(XQ+ZQ)]^2 */
    fp2sqr_mont(&Q->X, &Q->X);                        /* XQ = [(XP+ZP)*(XQ-ZQ)+(XP-ZP)*(XQ+ZQ)]^2 */
    fp2mul_mont(&Q->Z, xPQ, &Q->Z);                   /* ZQ = xPQ*[(XP+ZP)*(XQ-ZQ)-(XP-ZP)*(XQ+ZQ)]^2 */
}

/* Swap points.
 * If option = 0 then P <- P and Q <- Q, else if option = 0xFF...FF then P <- Q and Q <- P */
static void swap_points(point_proj_t P, point_proj_t Q, const digit_t option)
{
    unsigned int i;

    for (i = 0; i < S2N_SIKE_P434_R3_NWORDS_FIELD; i++) {
        digit_t temp = option & (P->X.e[0][i] ^ Q->X.e[0][i]);
        P->X.e[0][i] = temp ^ P->X.e[0][i];
        Q->X.e[0][i] = temp ^ Q->X.e[0][i];
        temp = option & (P->X.e[1][i] ^ Q->X.e[1][i]);
        P->X.e[1][i] = temp ^ P->X.e[1][i];
        Q->X.e[1][i] = temp ^ Q->X.e[1][i];
        temp = option & (P->Z.e[0][i] ^ Q->Z.e[0][i]);
        P->Z.e[0][i] = temp ^ P->Z.e[0][i];
        Q->Z.e[0][i] = temp ^ Q->Z.e[0][i];
        temp = option & (P->Z.e[1][i] ^ Q->Z.e[1][i]);
        P->Z.e[1][i] = temp ^ P->Z.e[1][i];
        Q->Z.e[1][i] = temp ^ Q->Z.e[1][i];
    }
}

void LADDER3PT(const f2elm_t *xP, const f2elm_t *xQ, const f2elm_t *xPQ, const digit_t* m,
        const unsigned int AliceOrBob, point_proj_t R, const f2elm_t *A)
{
    point_proj_t R0 = {0}, R2 = {0};
    f2elm_t _A24 = {0};
    f2elm_t *A24 = &_A24;
    digit_t mask;
    int i, nbits, swap, prevbit = 0;

    if (AliceOrBob == S2N_SIKE_P434_R3_ALICE) {
        nbits = S2N_SIKE_P434_R3_OALICE_BITS;
    } else {
        nbits = S2N_SIKE_P434_R3_OBOB_BITS - 1;
    }

    /* Initializing constant */
    fpcopy((const digit_t*)&Montgomery_one, A24->e[0]);
    mp2_add(A24, A24, A24);
    mp2_add(A, A24, A24);
    fp2div2(A24, A24);
    fp2div2(A24, A24);  /* A24 = (A+2)/4 */

    /* Initializing points */
    fp2copy(xQ, &R0->X);
    fpcopy((const digit_t*)&Montgomery_one, (digit_t*)&R0->Z);
    fp2copy(xPQ, &R2->X);
    fpcopy((const digit_t*)&Montgomery_one, (digit_t*)&R2->Z);
    fp2copy(xP, &R->X);
    fpcopy((const digit_t*)&Montgomery_one, (digit_t*)&R->Z);
    fpzero((digit_t*)(R->Z.e)[1]);

    /* Main loop */
    for (i = 0; i < nbits; i++) {
        int bit = (m[i >> S2N_SIKE_P434_R3_LOG2RADIX] >> (i & (S2N_SIKE_P434_R3_RADIX-1))) & 1;
        swap = bit ^ prevbit;
        prevbit = bit;
        mask = 0 - (digit_t)swap;

        swap_points(R, R2, mask);
        xDBLADD(R0, R2, &R->X, A24);
        fp2mul_mont(&R2->X, &R->Z, &R2->X);
    }
    swap = 0 ^ prevbit;
    mask = 0 - (digit_t)swap;
    swap_points(R, R2, mask);
}