-rw-r--r-- 17361 high-ctidh-20210504/mont.c
#include <stdio.h>
#include <string.h>
#include <assert.h>

#include "steps.h"
#include "mont.h"
#include "uintbig.h"
#include "poly.h"
#include "int64mask.h"

// precompute A24.x = A.x+2*A.z, A24.z = 4*A.z
void xA24(proj *A24,const proj *A)
{
  fp_double2(&A24->x, &A->z);
  fp_double2(&A24->z, &A24->x);
  fp_add2(&A24->x, &A->x);
}

void x2(proj *P2, proj const *P)
{
    fp_add3(&P2->x, &P->x, &P->z);
    fp_sq1(&P2->x);
    fp_sub3(&P2->z, &P->x, &P->z);
    fp_sq1(&P2->z);
}

void x2DBL(proj *Q, proj const *P2, const proj *A24, int Aaffine)
{
    fp a, b, c;
    fp_sub3(&c, &P2->x, &P2->z);
    if (Aaffine)
      fp_quadruple2(&b, &P2->z);
    else
      fp_mul3(&b, &A24->z, &P2->z);
    fp_mul3(&Q->x, &P2->x, &b);
    fp_mul3(&a, &c, &A24->x);
    fp_add2(&a, &b);
    fp_mul3(&Q->z, &a, &c);
}

void xDBLADD(proj *R, proj *S, proj const *P, proj const *Q, proj const *PQ, proj const *A24, int Aaffine)
{
    fp tmp0, tmp1, tmp2;

    fp_add3(&tmp0, &P->x, &P->z);
    fp_sub3(&tmp1, &P->x, &P->z);
    fp_sq2(&R->x, &tmp0);
    fp_sub3(&tmp2, &Q->x, &Q->z);
    fp_add3(&S->x, &Q->x, &Q->z);
    fp_mul2(&tmp0, &tmp2);
    fp_sq2(&R->z, &tmp1);
    fp_mul2(&tmp1, &S->x);
    fp_sub3(&tmp2, &R->x, &R->z);
    if (Aaffine)
      fp_quadruple1(&R->z);
    else
      fp_mul2(&R->z, &A24->z);
    fp_mul2(&R->x, &R->z);
    fp_mul3(&S->x, &A24->x, &tmp2);
    fp_sub3(&S->z, &tmp0, &tmp1);
    fp_add2(&R->z, &S->x);
    fp_add3(&S->x, &tmp0, &tmp1);
    fp_mul2(&R->z, &tmp2);
    fp_sq1(&S->z);
    fp_sq1(&S->x);
    fp_mul2(&S->z, &PQ->x);
    fp_mul2(&S->x, &PQ->z);
}

void xDBL(proj *Q, proj const *P, const proj *A24, int Aaffine)
{
    proj P2;
    x2(&P2, P);
    x2DBL(Q, &P2, A24, Aaffine);
}

void xADD(proj *S, proj const *P, proj const *Q, proj const *PQ)
{
    fp a, b, c, d;
    fp_add3(&a, &P->x, &P->z);
    fp_sub3(&b, &P->x, &P->z);
    fp_add3(&c, &Q->x, &Q->z);
    fp_sub3(&d, &Q->x, &Q->z);
    fp_mul2(&a, &d);
    fp_mul2(&b, &c);
    fp_add3(&c, &a, &b);
    fp_sub3(&d, &a, &b);
    fp_sq1(&c);
    fp_sq1(&d);
    fp_mul3(&S->x, &PQ->z, &c);
    fp_mul3(&S->z, &PQ->x, &d);
}

// goal: Q = any point having same order as lP
// where prime l has chain dac,daclen with 0 <= daclen <= maxdaclen
// time depends on maxdaclen, but not dac or daclen
// this implementation does not necessarily return Q=lP
// can simply return P if it notices that P has order <l
// note that Q buffer can overlap P buffer
// XXX: skip copies in cases where dac and daclen are specified to be public
// XXX: skip copies by unrolling
void xMUL_dac(proj *Q, proj const *A24, int Aaffine, proj const *P, long long dac, long long daclen, long long maxdaclen)
{
  proj Pinput = *P; // keeping copy since Q can overlap P
  proj P1 = Pinput;
  proj P2; xDBL(&P2,&P1,A24,Aaffine);
  proj P3; xADD(&P3,&P2,&P1,&P1);
  long long collision = fp_iszero(&Pinput.z);

  for (;;) {
    long long want = 1+int64mask_negative(daclen);
    proj_cmov(Q,&P3,want);
    if (maxdaclen <= 0) break;

    // invariant: P1+P2 = P3
    // odd dac: want to replace P1,P2,P3 with P1,P3,P1+P3
    // even dac: want to replace P1,P2,P3 with P2,P3,P2+P3

    proj_cswap(&P1,&P2,1-(dac&1));
    // invariant: P1+P2 = P3
    // all cases: want to replace P1,P2,P3 with P1,P3,P1+P3

    collision |= want&fp_iszero(&P2.z);

    proj next;
    xADD(&next,&P3,&P1,&P2);
    P2 = P3;
    P3 = next;

    maxdaclen -= 1;
    daclen -= 1;
    dac >>= 1;
  }

  proj_cmov(Q,&Pinput,collision);
  // in case of collision, input has the right order
}

/* Montgomery ladder. */
/* P must not be the unique point of order 2. */
/* k must have _at most_ kbits. */
void xMUL(proj *Q, proj const *A, int Aaffine, proj const *P, uintbig const *k, long long kbits)
{
    proj R = *P;
    proj A24;
    const proj Pcopy = *P; /* in case Q = P */

    Q->x = fp_1;
    Q->z = fp_0;

    xA24(&A24,A);

    long long prevbit = 0;
    while (kbits > 0) {
      --kbits;

      long long bit = uintbig_bit(k, kbits);
      long long bitxor = bit^prevbit;

      proj_cswap(Q, &R, bitxor);

      xDBLADD(Q, &R, Q, &R, &Pcopy, &A24, Aaffine);

      prevbit = bit;
    }
    proj_cswap(Q, &R, prevbit);
}

void xMUL_vartime(proj *Q, proj const *A, int Aaffine, proj const *P, uintbig const *k)
{
  long long kbits = BITS;
  while (kbits && !uintbig_bit(k,kbits-1)) --kbits;
  xMUL(Q,A,Aaffine,P,k,kbits);
}

/* replace x with x^k y^8 */
/* constant-time 2-bit window algorithm copied from qisog */
void powpow8mod(fp *x, const fp *y, uint64_t k, uint64_t kupper)
{
  long long bits = 5; /* XXX: also optimize smaller k */
  kupper >>= 5;
  while (kupper) { ++bits; kupper >>= 1; }

  fp x1 = *x;
  fp x2; fp_sq2(&x2,x);
  fp x3; fp_mul3(&x3,&x2,x);

  int xwritten = 0; /* otherwise *x is implicitly 1 */

  for (long long i = (bits - 1) & ~1;i >= 0;i -= 2) {
    if (xwritten) {
      fp_sq1(x);
      if (i == 2) fp_mul2(x,y);
      fp_sq1(x);
    }
    fp ram0 = fp_1;
    fp ram1 = x1;
    long long control1 = 1&(k>>(i+1));
    fp_cmov(&ram0,&x2,control1);
    fp_cmov(&ram1,&x3,control1);
    long long control0 = 1&(k>>i);
    fp_cmov(&ram0,&ram1,control0);

    if (xwritten)
      fp_mul2(x,&ram0);
    else
      *x = ram0;

    xwritten = 1;
  }
  assert(xwritten);
}

void biquad_precompute_point(fp *precomp,const proj *Q)
{
  /* XXX: can do S-M tradeoff */

  fp_sq2(&precomp[0],&Q->x);
  fp_sq2(&precomp[1],&Q->z);
  fp_mul3(&precomp[2],&Q->x,&Q->z);
  fp_add3(&precomp[3],&precomp[2],&precomp[2]);
  fp_sub3(&precomp[4],&precomp[0],&precomp[1]);
  fp_sub3(&precomp[5],&Q->x,&Q->z);
  fp_sq1(&precomp[5]);
}

void biquad_precompute_curve(fp *Aprecomp,const proj *P,const proj *A)
{
  fp Pplus; fp_add3(&Pplus,&P->x,&P->z);
  fp_sq1(&Pplus);
  fp PxPx; fp_sq2(&PxPx,&P->x);
  fp PzPz; fp_sq2(&PzPz,&P->z);
  fp PxPz2; fp_sub3(&PxPz2,&Pplus,&PxPx);
  fp_sub2(&PxPz2,&PzPz);

  fp_mul3(&Aprecomp[0],&A->z,&PxPx);
  fp_mul3(&Aprecomp[1],&A->z,&PzPz);
  fp_mul3(&Aprecomp[2],&A->z,&PxPz2);
  fp_mul3(&Aprecomp[3],&A->x,&PxPz2);

  fp t;
  fp_add3(&t,&Aprecomp[0],&Aprecomp[1]);
  fp_sub3(&Aprecomp[5],&t,&Aprecomp[2]);
  fp_add3(&Aprecomp[6],&t,&Aprecomp[2]);

  fp_add3(&Aprecomp[7],&Aprecomp[3],&Aprecomp[6]);
  fp_neg1(&Aprecomp[7]);

  fp_sub3(&Aprecomp[4],&Aprecomp[0],&Aprecomp[1]);
}

void biquad_postcompute_curve(fp *outplus,fp *outminus,const fp *Aprecomp)
{
  outplus[0] = Aprecomp[5];
  outplus[2] = Aprecomp[5];
  outminus[0] = Aprecomp[6];
  outminus[2] = Aprecomp[6];

  fp_add3(&outplus[1],&Aprecomp[7],&Aprecomp[7]);
  fp_add3(&outminus[1],&Aprecomp[5],&Aprecomp[3]);
  fp_double1(&outminus[1]);
}

void biquad_postcompute_point(fp *out,const fp *precomp,const fp *Aprecomp)
{
  fp v;

  fp_mul3(&out[1],&Aprecomp[7],&precomp[3]);
  fp_mul3(&v,&Aprecomp[2],&precomp[5]);
  fp_sub2(&out[1],&v);

  fp_mul3(&out[2],&Aprecomp[0],&precomp[1]);
  fp_mul3(&v,&Aprecomp[2],&precomp[2]);
  fp_sub2(&out[2],&v);
  fp_mul3(&v,&Aprecomp[1],&precomp[0]);
  fp_add2(&out[2],&v);

  fp_mul3(&out[0],&Aprecomp[4],&precomp[4]);
  fp_add2(&out[0],&out[2]);
}

/* computes the isogeny with kernel point K of order k */
/* returns the new curve coefficient A and the images of P[0],P[1],...,P[Plen-1] */
/* k is odd; klower <= k <= kupper */
/* goal here is for timing to not leak k */
void xISOG_matryoshka(proj *A, proj *P, long long Plen, proj const *K, long long k, long long klower, long long kupper)
{
    assert (3 <= klower);
    assert (klower&1);
    assert (kupper&1);

    long long sqrtvelu = 0;
    long long bs = 0;
    long long gs = 0;

    steps(&bs,&gs,klower);
    if (bs) {
      sqrtvelu = 1;
      assert(bs > 0);
      assert(gs > 0);
      assert(!(bs&1));
    }

    fp tmp0, tmp1, tmp2, tmp3, tmp4;

    proj A24;
    proj Aed; // twisted Edwards curve coefficients
    fp_double2(&Aed.z, &A->z);
    fp_add3(&Aed.x, &A->x, &Aed.z);
    A24.x = Aed.x;
    fp_double2(&A24.z,&Aed.z);
    fp_sub3(&Aed.z, &A->x, &Aed.z);
   
    int Minit[kupper];
    proj M[kupper]; /* XXX: use less space */

    for (long long s = 0;s < kupper;++s) Minit[s] = 0;

    M[1] = *K; Minit[1] = 1;
    xDBL(&M[2],K,&A24,0); Minit[2] = 1;

    if (sqrtvelu) {
      for (long long s = 3;s < kupper;++s) {
        if (s & 1) {
          long long i = s/2;
          assert(s == 2*i+1);
          if (i < bs) {
            if (s == 3) {
              assert(Minit[1]);
              assert(Minit[2]);
              xADD(&M[s],&M[2],&M[1],&M[1]);
              Minit[s] = 1;
              continue;
            }
            assert(Minit[s-2]);
            assert(Minit[s-4]);
            assert(Minit[2]);
            xADD(&M[s],&M[s-2],&M[2],&M[s-4]);
            Minit[s] = 1;
            continue;
          }
        } else {
          long long i = s/2-1;
          assert(s == 2*i+2);
          if (i < (kupper-1)/2-2*bs*gs) {
            if (s == 4) {
              assert(Minit[2]);
              xDBL(&M[s],&M[2],&A24,0);
              Minit[s] = 1;
              continue;
            }
            assert(Minit[s-2]);
            assert(Minit[s-4]);
            assert(Minit[2]);
            xADD(&M[s],&M[s-2],&M[2],&M[s-4]);
            Minit[s] = 1;
            continue;
          }
        }
  
        if (bs > 0) {
          if (s == 2*bs) {
            assert(Minit[bs-1]);
            assert(Minit[bs+1]);
            assert(Minit[2]);
            xADD(&M[s],&M[bs+1],&M[bs-1],&M[2]);
            Minit[s] = 1;
            continue;
          } else if (s == 4*bs) {
            assert(Minit[2*bs]);
            xDBL(&M[s],&M[2*bs],&A24,0);
            Minit[s] = 1;
            continue;
          } else if (s == 6*bs) {
            assert(Minit[2*bs]);
            assert(Minit[4*bs]);
            xADD(&M[s],&M[4*bs],&M[2*bs],&M[2*bs]);
            Minit[s] = 1;
            continue;
          } else if (s%(4*bs) == 2*bs) {
            long long j = s/(4*bs);
            assert(s == 2*bs*(2*j+1));
            if (j < gs) {
              assert(Minit[s-4*bs]);
              assert(Minit[s-8*bs]);
              assert(Minit[4*bs]);
              xADD(&M[s],&M[s-4*bs],&M[4*bs],&M[s-8*bs]);
              Minit[s] = 1;
              continue;
            }
          }
        }
      }
    } else {
      for (long long i = 3;i <= (kupper-1)/2;++i) {
	Minit[i] = 1;
        xADD(&M[i],&M[i-1],K,&M[i-2]);
      }
    }

    proj Abatch;
    Abatch.x = fp_1;
    Abatch.z = fp_1;
    proj Qbatch[Plen];
    for (long long h = 0;h < Plen;++h) {
      Qbatch[h].x = fp_1;
      Qbatch[h].z = fp_1;
    }
    fp Psum[Plen], Pdif[Plen];
    for (long long h = 0;h < Plen;++h) {
      fp_add3(&Psum[h], &P[h].x, &P[h].z);   //precomputations
      fp_sub3(&Pdif[h], &P[h].x, &P[h].z);
    }

    if (sqrtvelu) {
      long long TIlen = 2*bs+poly_tree1size(bs);
      fp TI[TIlen];
  
      for (long long i = 0;i < bs;++i) {
        assert(Minit[2*i+1]);
        fp_neg2(&TI[2*i],&M[2*i+1].x);
        TI[2*i+1] = M[2*i+1].z;
      }
  
      poly_tree1(TI+2*bs,TI,bs);
  
      fp Aprecomp[gs][8];
      fp T1[3*gs];
      fp Tminus1[3*gs];

      for (long long j = 0;j < gs;++j) {
        assert(Minit[2*bs*(2*j+1)]);
        biquad_precompute_curve(Aprecomp[j],&M[2*bs*(2*j+1)],A);
        biquad_postcompute_curve(T1+3*j,Tminus1+3*j,Aprecomp[j]);
      }
      poly_multiprod2_selfreciprocal(T1,gs);
      poly_multiprod2_selfreciprocal(Tminus1,gs);

      long long precompsize = poly_multieval_precomputesize(bs,2*gs+1);
      fp precomp[precompsize];
      poly_multieval_precompute(precomp,bs,2*gs+1,TI,TI+2*bs);

      fp v[bs];

      poly_multieval_postcompute(v,bs,T1,2*gs+1,TI,TI+2*bs,precomp);
      Abatch.x = v[0];
      for (long long i = 1;i < bs;++i) fp_mul2(&Abatch.x,&v[i]);

      poly_multieval_postcompute(v,bs,Tminus1,2*gs+1,TI,TI+2*bs,precomp);
      Abatch.z = v[0];
      for (long long i = 1;i < bs;++i) fp_mul2(&Abatch.z,&v[i]);

      for (long long h = 0;h < Plen;++h) {
        fp Pprecomp[6];
        biquad_precompute_point(Pprecomp,&P[h]);
  
        fp TP[3*gs];
        for (long long j = 0;j < gs;++j)
          biquad_postcompute_point(TP+3*j,Pprecomp,Aprecomp[j]);
        poly_multiprod2(TP,gs);
  
        fp TPinv[2*gs+1];
        for (long long j = 0;j < 2*gs+1;++j)
          TPinv[j] = TP[2*gs-j];
  
        poly_multieval_postcompute(v,bs,TP,2*gs+1,TI,TI+2*bs,precomp);
        Qbatch[h].z = v[0];
        for (long long i = 1;i < bs;++i) fp_mul2(&Qbatch[h].z,&v[i]);
  
        poly_multieval_postcompute(v,bs,TPinv,2*gs+1,TI,TI+2*bs,precomp);
        Qbatch[h].x = v[0];
        for (long long i = 1;i < bs;++i) fp_mul2(&Qbatch[h].x,&v[i]);
      }

      long long ignore = (k-1)/2-2*bs*gs; /* skip i >= ignore */

      for (long long i = 0;i < (kupper-1)/2-2*bs*gs;++i) {
        long long want = -((i-ignore)>>61); /* 61 to reduce risk of compiler doing something silly */
        assert(Minit[2*i+2]);
        fp_sub3(&tmp4, &M[2*i+2].x, &M[2*i+2].z);
        fp_add3(&tmp3, &M[2*i+2].x, &M[2*i+2].z);
        fp_mul3(&tmp2,&Abatch.x,&tmp4);
        fp_cmov(&Abatch.x,&tmp2,want);
        fp_mul3(&tmp2,&Abatch.z,&tmp3);
        fp_cmov(&Abatch.z,&tmp2,want);
        for (long long h = 0;h < Plen;++h) {
          fp_mul3(&tmp1, &tmp4, &Psum[h]);
          fp_mul3(&tmp0, &tmp3, &Pdif[h]);
          fp_add3(&tmp2, &tmp0, &tmp1);
          fp_mul2(&tmp2,&Qbatch[h].x);
          fp_cmov(&Qbatch[h].x,&tmp2,want);
          fp_sub3(&tmp2, &tmp0, &tmp1);
          fp_mul2(&tmp2,&Qbatch[h].z);
          fp_cmov(&Qbatch[h].z,&tmp2,want);
        }
      }
    } else {
      long long ignore = (k+1)/2; /* skip i >= ignore */

      assert(Minit[1]);
      fp_sub3(&tmp4, &M[1].x, &M[1].z);
      fp_add3(&tmp3, &M[1].x, &M[1].z);
      Abatch.x = tmp4;
      Abatch.z = tmp3;
      for (long long h = 0;h < Plen;++h) {
        fp_mul3(&tmp1, &tmp4, &Psum[h]);
        fp_mul3(&tmp0, &tmp3, &Pdif[h]);
        fp_add3(&Qbatch[h].x, &tmp0, &tmp1);
        fp_sub3(&Qbatch[h].z, &tmp0, &tmp1);
      }

      for (long long i = 2;i <= (kupper-1)/2;++i) {
        long long want = -((i-ignore)>>61);

        assert(Minit[i]);
        fp_sub3(&tmp4, &M[i].x, &M[i].z);
        fp_add3(&tmp3, &M[i].x, &M[i].z);
        fp_mul3(&tmp2,&Abatch.x,&tmp4);
        fp_cmov(&Abatch.x,&tmp2,want);
        fp_mul3(&tmp2,&Abatch.z,&tmp3);
        fp_cmov(&Abatch.z,&tmp2,want);
        for (long long h = 0;h < Plen;++h) {
          fp_mul3(&tmp1, &tmp4, &Psum[h]);
          fp_mul3(&tmp0, &tmp3, &Pdif[h]);
          fp_add3(&tmp2, &tmp0, &tmp1);
          fp_mul2(&tmp2,&Qbatch[h].x);
          fp_cmov(&Qbatch[h].x,&tmp2,want);
          fp_sub3(&tmp2, &tmp0, &tmp1);
          fp_mul2(&tmp2,&Qbatch[h].z);
          fp_cmov(&Qbatch[h].z,&tmp2,want);
        }
      }
    }


    for (long long h = 0;h < Plen;++h) {
      // point evaluation
      fp_sq1(&Qbatch[h].x);
      fp_sq1(&Qbatch[h].z);
      fp_mul2(&P[h].x, &Qbatch[h].x);
      fp_mul2(&P[h].z, &Qbatch[h].z);
    }

    powpow8mod(&Aed.x,&Abatch.z,k,kupper);
    powpow8mod(&Aed.z,&Abatch.x,k,kupper);

    //compute Montgomery params
    fp_add3(&A->x, &Aed.x, &Aed.z);
    fp_sub3(&A->z, &Aed.x, &Aed.z);
    fp_double1(&A->x);
}

void xISOG(proj *A, proj *P, long long Plen, proj const *K, long long k)
{
  xISOG_matryoshka(A,P,Plen,K,k,k,k);
}

// replace x with x^k
void exp_by_squaring_(fp* x, uint64_t k)
{
        fp result1 = fp_1;

    while (k)
    {
        if (k & 1){
          fp_mul2(&result1, x);
        }
        fp_sq1(x);
        k >>= 1;
    }

    fp_cswap(&result1, x, 1);

}

void xISOG_old(proj *A, proj *P, proj const *K, long long k)
{
    assert (k >= 3);
    assert (k % 2 == 1);

    fp tmp0, tmp1, tmp2, Psum, Pdif;
    proj Q, Aed, prod;

    fp_add3(&Aed.z, &A->z, &A->z);  //compute twisted Edwards curve coefficients
    fp_add3(&Aed.x, &A->x, &Aed.z);
    fp_sub3(&Aed.z, &A->x, &Aed.z);
   
    fp_add3(&Psum, &P->x, &P->z);   //precomputations
    fp_sub3(&Pdif, &P->x, &P->z);

    fp_sub3(&prod.x, &K->x, &K->z);
    fp_add3(&prod.z, &K->x, &K->z);
    
    fp_mul3(&tmp1, &prod.x, &Psum);
    fp_mul3(&tmp0, &prod.z, &Pdif);
    fp_add3(&Q.x, &tmp0, &tmp1);
    fp_sub3(&Q.z, &tmp0, &tmp1);

    proj M[k];
    proj A24;
    xA24(&A24,A);

    M[0] = *K;
    xDBL(&M[1],K,&A24,0);
    for (long long i = 2; i < k / 2; ++i) {
        xADD(&M[i], &M[(i - 1)], K, &M[(i - 2)]);
    }

    for (long long i = 1; i < k / 2; ++i) {
        fp_sub3(&tmp1, &M[i].x, &M[i].z);
            fp_add3(&tmp0, &M[i].x, &M[i].z);
        fp_mul2(&prod.x, &tmp1);
        fp_mul2(&prod.z, &tmp0);
            fp_mul2(&tmp1, &Psum);
            fp_mul2(&tmp0, &Pdif);
            fp_add3(&tmp2, &tmp0, &tmp1);
        fp_mul2(&Q.x, &tmp2);
            fp_sub3(&tmp2, &tmp0, &tmp1);
        fp_mul2(&Q.z, &tmp2);

    }


    // point evaluation
    fp_sq1(&Q.x);
    fp_sq1(&Q.z);
    fp_mul2(&P->x, &Q.x);
    fp_mul2(&P->z, &Q.z);

    //compute Aed.x^k, Aed.z^k
    exp_by_squaring_(&Aed.x, k);
    exp_by_squaring_(&Aed.z, k);

    //compute prod.x^8, prod.z^8
    fp_sq1(&prod.x);
    fp_sq1(&prod.x);
    fp_sq1(&prod.x);
    fp_sq1(&prod.z);
    fp_sq1(&prod.z);
    fp_sq1(&prod.z);

    //compute image curve parameters
    fp_mul2(&Aed.z, &prod.x);
    fp_mul2(&Aed.x, &prod.z);

    //compute Montgomery params
    fp_add3(&A->x, &Aed.x, &Aed.z);
    fp_sub3(&A->z, &Aed.x, &Aed.z);
    fp_add2(&A->x, &A->x);
}