pedersen-libsnark/ecgadget.hpp

#include "libsnark_headers.hpp"

using namespace libsnark;

// Double a public EC point (inx,iny) to yield (outx,outy).  The input
// point must not be the point at infinity.
template<typename FieldT>
static void ec_double_point(FieldT &outx, FieldT &outy,
    const FieldT &inx, const FieldT &iny)
{
    FieldT xsq = inx.squared();
    FieldT lambda = (xsq * 3 - 3) * (iny * 2).inverse();
    outx = lambda.squared() - inx * 2;
    outy = lambda * (inx - outx) - iny;
}

// Add public EC points (inx, iny) and (addx, addy) to yield (outx, outy).
// inx and addx must not be equal.
template<typename FieldT>
static void ec_add_points(FieldT &outx, FieldT &outy,
    const FieldT &inx, const FieldT &iny,
    const FieldT &addx, const FieldT &addy)
{
    FieldT lambda = (addy - iny) * (addx - inx).inverse();
    outx = lambda.squared() - (addx + inx);
    outy = lambda * (inx - outx) - iny;
}

// Double the EC point (inx,iny) to yield (outx,outy).  The input point
// must not be the point at infinity.
template<typename FieldT>
class ec_double_gadget : public gadget<FieldT> {
private:
  pb_variable<FieldT> lambda, inxsq;
public:
  const pb_variable<FieldT> outx, outy, inx, iny;

  ec_double_gadget(protoboard<FieldT> &pb,
              const pb_variable<FieldT> &outx,
              const pb_variable<FieldT> &outy,
              const pb_variable<FieldT> &inx,
              const pb_variable<FieldT> &iny) : 
    gadget<FieldT>(pb, "ec_double_gadget"), outx(outx), outy(outy),
        inx(inx), iny(iny)
  {
    // Allocate variables to protoboard
    // The strings (like "x") are only for debugging purposes
	  
    lambda.allocate(this->pb, "lambda");
    inxsq.allocate(this->pb, "inxsq");
  }

  void generate_r1cs_constraints()
  {
    // inxsq = inx * inx
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(inx, inx, inxsq));

    // 2 * iny * lambda = 3 * inxsq - 3  (a = -3 on our curve)
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(2 * iny, lambda, 3 * inxsq - 3));

    // outx = lambda^2 - 2 * inx
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(lambda, lambda, outx + 2 * inx));

    // outy = lambda * (inx - outx) - iny
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(lambda, inx - outx, outy + iny));

  }

  void generate_r1cs_witness()
  {
    this->pb.val(inxsq) = this->pb.val(inx) * this->pb.val(inx);
    this->pb.val(lambda) = (this->pb.val(inxsq) * 3 - 3) * (this->pb.val(iny) * 2).inverse();
    this->pb.val(outx) = this->pb.val(lambda).squared() - this->pb.val(inx) * 2;
    this->pb.val(outy) = this->pb.val(lambda) * (this->pb.val(inx) - this->pb.val(outx)) - this->pb.val(iny);
  }
};

// Add the EC point (addx,addy) to the EC point (inx,iny) to yield
// (outx,outy).  The input point must not be the point at infinity.
template<typename FieldT>
class ec_add_gadget : public gadget<FieldT> {
private:
  pb_variable<FieldT> lambda;
public:
  const pb_variable<FieldT> outx, outy;
  const pb_linear_combination<FieldT> inx, iny, addx, addy;

  ec_add_gadget(protoboard<FieldT> &pb,
              const pb_variable<FieldT> &outx,
              const pb_variable<FieldT> &outy,
              const pb_linear_combination<FieldT> &inx,
              const pb_linear_combination<FieldT> &iny,
              const pb_linear_combination<FieldT> &addx,
              const pb_linear_combination<FieldT> &addy) :
    gadget<FieldT>(pb, "ec_add_gadget"),
    outx(outx), outy(outy), inx(inx), iny(iny), addx(addx), addy(addy)
  {
    // Allocate variables to protoboard
    // The strings (like "x") are only for debugging purposes
	  
    lambda.allocate(this->pb, "lambda");
  }

  void generate_r1cs_constraints()
  {
    // (addx - inx) * lambda = addy - iny
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(addx - inx, lambda, addy - iny));

    // outx = lambda^2 - (addx + inx)
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(lambda, lambda, outx + addx + inx));

    // outy = lambda * (inx - outx) - iny
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(lambda, inx - outx, outy + iny));

  }

  void generate_r1cs_witness()
  {
    this->pb.val(lambda) = (this->pb.lc_val(addy) - this->pb.lc_val(iny)) * (this->pb.lc_val(addx) - this->pb.lc_val(inx)).inverse();
    this->pb.val(outx) = this->pb.val(lambda).squared() - (this->pb.lc_val(addx) + this->pb.lc_val(inx));
    this->pb.val(outy) = this->pb.val(lambda) * (this->pb.lc_val(inx) - this->pb.val(outx)) - this->pb.lc_val(iny);
  }
};

// Add the public EC point P to the EC point (inx,iny) to yield
// (outx,outy).  The input point must not be the point at infinity.
template<typename FieldT>
class ec_public_add_gadget : public gadget<FieldT> {
private:
  pb_variable<FieldT> lambda;
public:
  const pb_variable<FieldT> outx, outy;
  const pb_linear_combination<FieldT> inx, iny;
  const FieldT Px, Py;

  ec_public_add_gadget(protoboard<FieldT> &pb,
              const pb_variable<FieldT> &outx,
              const pb_variable<FieldT> &outy,
              const pb_linear_combination<FieldT> &inx,
              const pb_linear_combination<FieldT> &iny,
              const FieldT &Px, const FieldT &Py) :
    gadget<FieldT>(pb, "ec_public_add_gadget"),
    outx(outx), outy(outy), inx(inx), iny(iny), Px(Px), Py(Py)
  {
    // Allocate variables to protoboard
    // The strings (like "x") are only for debugging purposes
	  
    lambda.allocate(this->pb, "lambda");
  }

  void generate_r1cs_constraints()
  {
    // (Px - inx) * lambda = Py - iny
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(Px - inx, lambda, Py - iny));

    // outx = lambda^2 - (Px + inx)
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(lambda, lambda, outx + Px + inx));

    // outy = lambda * (inx - outx) - iny
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(lambda, inx - outx, outy + iny));

  }

  void generate_r1cs_witness()
  {
    this->pb.val(lambda) = (Py - this->pb.lc_val(iny)) * (Px - this->pb.lc_val(inx)).inverse();
    this->pb.val(outx) = this->pb.val(lambda).squared() - (Px + this->pb.lc_val(inx));
    this->pb.val(outy) = this->pb.val(lambda) * (this->pb.lc_val(inx) - this->pb.val(outx)) - this->pb.lc_val(iny);
  }
};

// Add nothing or the public EC point P to the EC point (inx,iny) to
// yield (outx,outy).  The input point must not be the point at
// infinity.  The input bit do_add controls whether the addition is
// done.
template<typename FieldT>
class ec_conditional_add_gadget : public gadget<FieldT> {
private:
  pb_variable<FieldT> sumx, sumy;
  std::vector<ec_public_add_gadget<FieldT> > adder;
public:
  const pb_variable<FieldT> outx, outy;
  const pb_linear_combination<FieldT> inx, iny;
  const pb_variable<FieldT> do_add;
  const FieldT Px, Py;

  ec_conditional_add_gadget(protoboard<FieldT> &pb,
              const pb_variable<FieldT> &outx,
              const pb_variable<FieldT> &outy,
              const pb_linear_combination<FieldT> &inx,
              const pb_linear_combination<FieldT> &iny,
              const pb_variable<FieldT> &do_add,
              const FieldT &Px, const FieldT &Py) :
    gadget<FieldT>(pb, "ec_conditional_add_gadget"),
    outx(outx), outy(outy), inx(inx), iny(iny), do_add(do_add),
    Px(Px), Py(Py)
  {
    // Allocate variables to protoboard
    // The strings (like "x") are only for debugging purposes
	  
    sumx.allocate(this->pb, "sumx");
    sumy.allocate(this->pb, "sumy");
    adder.emplace_back(this->pb, sumx, sumy, inx, iny, Px, Py);
  }

  void generate_r1cs_constraints()
  {
    // Strategy: we always do the addition, but if do_add = 0, we throw
    // it away later.

    adder[0].generate_r1cs_constraints();

    // Now we want to conditionally move the sum.  We want that
    // outx = do_add ? sumx : inx
    // outy = do_add ? sumy : iny

    // so we compute
    // outx = inx + (sumx - inx) * do_add
    // outy = iny + (sumy - iny) * do_add

    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(sumx - inx, do_add, outx - inx));
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(sumy - iny, do_add, outy - iny));

  }

  void generate_r1cs_witness()
  {
    adder[0].generate_r1cs_witness();

    bool move = this->pb.val(do_add) != FieldT(0);
    this->pb.val(outx) = move ? this->pb.val(sumx) : this->pb.lc_val(inx);
    this->pb.val(outy) = move ? this->pb.val(sumy) : this->pb.lc_val(iny);
  }
};

// Add nothing, or one of the public EC points P1, P2, or P3 to the EC
// point (inx,iny) to yield (outx,outy).  The input point must not be
// the point at infinity.  The two input bits add1 and add2 control what
// is added.  Typically, P3 will equal P1+P2, in which case this gadget
// does two conditional adds simultaneously in just 6 constraints.
template<typename FieldT>
class ec_add_P123_gadget : public gadget<FieldT> {
private:
  pb_variable<FieldT> lambda, sumx, sumy, move;
public:
  const pb_variable<FieldT> outx, outy;
  const pb_linear_combination<FieldT> inx, iny;
  const pb_variable<FieldT> add1, add2;
  const FieldT P1x, P1y, P2x, P2y, P3x, P3y;

  ec_add_P123_gadget(protoboard<FieldT> &pb,
              const pb_variable<FieldT> &outx,
              const pb_variable<FieldT> &outy,
              const pb_linear_combination<FieldT> &inx,
              const pb_linear_combination<FieldT> &iny,
              const pb_variable<FieldT> &add1,
              const pb_variable<FieldT> &add2,
              const FieldT &P1x, const FieldT &P1y,
              const FieldT &P2x, const FieldT &P2y,
              const FieldT &P3x, const FieldT &P3y) : 
    gadget<FieldT>(pb, "ec_add_P123_gadget"),
    outx(outx), outy(outy), inx(inx), iny(iny), add1(add1), add2(add2),
    P1x(P1x), P1y(P1y), P2x(P2x), P2y(P2y), P3x(P3x), P3y(P3y)
  {
    // Allocate variables to protoboard
    // The strings (like "x") are only for debugging purposes
	  
    lambda.allocate(this->pb, "lambda");
    sumx.allocate(this->pb, "sumx");
    sumy.allocate(this->pb, "sumy");
    move.allocate(this->pb, "move");
  }

  void generate_r1cs_constraints()
  {
    // Strategy: if add1 = add2 = 0, we compute some nonsense but throw
    // it away later.  Otherwise, the coordinates of the point to add
    // are a _linear_ function of add1 and add2 (since P1, P2, and P3
    // are public constants)

    // In particular, the point to add is ( (P3x - P2x) * add1 + (P3x -
    // P1x) * add2 + (P1x + P2x - P3x), (P3y - P2y) * add1 + (P3y - P1y) *
    // add2 + (P1y + P2y - P3y))

    // (addx - inx) * lambda = addy - iny
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>((P3x - P2x) * add1 + (P3x - P1x) * add2 + (P1x + P2x - P3x) - inx, lambda, (P3y - P2y) * add1 + (P3y - P1y) * add2 + (P1y + P2y - P3y) - iny));

    // sumx = lambda^2 - (addx + inx)
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(lambda, lambda, sumx + (P3x - P2x) * add1 + (P3x - P1x) * add2 + (P1x + P2x - P3x) + inx));

    // sumy = lambda * (inx - sumx) - iny
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(lambda, inx - sumx, sumy + iny));

    // Now we want to conditionally move the sum.  We want that
    // outx = (add1 || add2) ? sumx : inx
    // outy = (add1 || add2) ? sumy : iny

    // so we compute move = add1 || add2, and then
    // outx = inx + (sumx - inx) * move
    // outy = iny + (sumy - iny) * move

    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(1 - add1, 1 - add2, 1 - move));
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(sumx - inx, move, outx - inx));
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(sumy - iny, move, outy - iny));

  }

  void generate_r1cs_witness()
  {
    FieldT addxval = (P3x - P2x) * this->pb.val(add1) + (P3x - P1x) * this->pb.val(add2) + (P1x + P2x - P3x);
    FieldT addyval = (P3y - P2y) * this->pb.val(add1) + (P3y - P1y) * this->pb.val(add2) + (P1y + P2y - P3y);
    this->pb.val(lambda) = (addyval - this->pb.lc_val(iny)) * (addxval - this->pb.lc_val(inx)).inverse();
    this->pb.val(sumx) = this->pb.val(lambda).squared() - (addxval + this->pb.lc_val(inx));
    this->pb.val(sumy) = this->pb.val(lambda) * (this->pb.lc_val(inx) - this->pb.val(sumx)) - this->pb.lc_val(iny);

    bool a1 = this->pb.val(add1) != FieldT(0);
    bool a2 = this->pb.val(add2) != FieldT(0);
    this->pb.val(move) = a1 || a2;
    this->pb.val(outx) = (a1 || a2) ? this->pb.val(sumx) : this->pb.lc_val(inx);
    this->pb.val(outy) = (a1 || a2) ? this->pb.val(sumy) : this->pb.lc_val(iny);
  }
};

// Compute a*P as (outx, outy) for a given public point P, given a
// as a bit vector.  The _caller_ is responsible for proving that the
// elements of avec are bits.
template<typename FieldT>
class ec_scalarmul_vec_gadget : public gadget<FieldT> {
private:
  FieldT Cx, Cy, CPx, CPy;
  pb_variable_array<FieldT> accumx, accumy;
  std::vector<ec_add_P123_gadget<FieldT> > conddoubleadders;
  std::vector<ec_conditional_add_gadget<FieldT> > condsingleadders;
  std::vector<ec_public_add_gadget<FieldT> > singleadders;
public:
  const pb_variable<FieldT> outx, outy;
  const pb_variable_array<FieldT> avec;
  const FieldT Px, Py;

  // Strategy: We compute (in public) (powers of 2) times P, and then
  // conditionally add them into an accumulator.  Because our adder cannot
  // handle the point at infinity O, we start the accumulator with a value
  // of C, whose discrete log with respect to P should be unknown, so that
  // we won't encounter O along the way.  (Also, a should not be 0 or
  // the group order.)  We actually start the accumulator with either
  // C or C+P depending on avec[0], so we get the first conditional add
  // "for free".  Then we use the ec_add_P123_gadget to do the conditional
  // adds two at a time (at a cost of 6 constraints per pair, as opposed to
  // 5 for a single conditional add).  If the length of avec is even, then
  // there will be one left over, and we do a single conditional add for
  // that one.  Finally, we add the public point -C.

  ec_scalarmul_vec_gadget(protoboard<FieldT> &pb,
              const pb_variable<FieldT> &outx,
              const pb_variable<FieldT> &outy,
              const pb_variable_array<FieldT> &avec,
              const FieldT &Px, const FieldT &Py) :
    gadget<FieldT>(pb, "ec_pedersen_vec_gadget"),
    // Precomputed coordinates of C
    Cx(2),
    Cy("4950745124018817972378217179409499695353526031437053848725554590521829916331"),
    outx(outx), outy(outy), avec(avec), Px(Px), Py(Py)
  {
    size_t numbits = avec.size();
    accumx.allocate(this->pb, numbits/2+1, "accumx");
    accumy.allocate(this->pb, numbits/2+1, "accumy");

    ec_add_points(CPx, CPy, Cx, Cy, Px, Py);

    FieldT twoiPx, twoiPy, twoi1Px, twoi1Py, twoi3Px, twoi3Py;
    size_t i = 1;

    ec_double_point(twoiPx, twoiPy, Px, Py);

    while(i < numbits) {
        // Invariants: i is odd, and twoiP = 2^i * P
        // Compute twoi1P = 2^{i+1} * P = 2 * twoiP and
        //         twoi3P = 2^i * 3 * P = 3 * twoiP
        ec_double_point(twoi1Px, twoi1Py, twoiPx, twoiPy);
        ec_add_points(twoi3Px, twoi3Py, twoi1Px, twoi1Py, twoiPx, twoiPy);

        if (i == numbits-1) {
            // There's only one bit of avec left; use a single conditional
            // add.
            condsingleadders.emplace_back(this->pb,
                accumx[(i+1)/2], accumy[(i+1)/2],
                accumx[(i-1)/2], accumy[(i-1)/2],
                avec[i],
                twoiPx, twoiPy);
        } else {
            conddoubleadders.emplace_back(this->pb,
                accumx[(i+1)/2], accumy[(i+1)/2],
                accumx[(i-1)/2], accumy[(i-1)/2],
                avec[i], avec[i+1],
                twoiPx, twoiPy, twoi1Px, twoi1Py, twoi3Px, twoi3Py);
        }

        ec_double_point(twoiPx, twoiPy, twoi1Px, twoi1Py);
        i += 2;
    }

    // If numbits is even, the output so far is in accum[(numbits)/2].
    // If numbits is odd, it is in accum[(numbits-1)/2].  So in either
    // case, it is in accum[numbits/2].
    singleadders.emplace_back(this->pb,
        outx, outy, accumx[numbits/2], accumy[numbits/2],
        Cx, -Cy);
  }

  void generate_r1cs_constraints()
  {
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(Cx + (CPx-Cx) * avec[0], 1, accumx[0]));
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(Cy + (CPy-Cy) * avec[0], 1, accumy[0]));

    for (auto&& gadget : conddoubleadders) {
        gadget.generate_r1cs_constraints();
    }
    for (auto&& gadget : condsingleadders) {
        gadget.generate_r1cs_constraints();
    }
    for (auto&& gadget : singleadders) {
        gadget.generate_r1cs_constraints();
    }
  }

  void generate_r1cs_witness()
  {
    this->pb.val(accumx[0]) = Cx + (CPx-Cx) * this->pb.val(avec[0]);
    this->pb.val(accumy[0]) = Cy + (CPy-Cy) * this->pb.val(avec[0]);

    for (auto&& gadget : conddoubleadders) {
        gadget.generate_r1cs_witness();
    }
    for (auto&& gadget : condsingleadders) {
        gadget.generate_r1cs_witness();
    }
    for (auto&& gadget : singleadders) {
        gadget.generate_r1cs_witness();
    }
  }
};

// Compute a*P as (outx, outy) for a given public point P, given a
// as a field element.
template<typename FieldT>
class ec_scalarmul_gadget : public gadget<FieldT> {
private:
  pb_variable_array<FieldT> avec;
  std::vector<packing_gadget<FieldT> > packers;
  std::vector<ec_scalarmul_vec_gadget<FieldT> > vecgadget;

public:
  const pb_variable<FieldT> outx, outy, a;
  const FieldT Px, Py;

  ec_scalarmul_gadget(protoboard<FieldT> &pb,
              const pb_variable<FieldT> &outx,
              const pb_variable<FieldT> &outy,
              const pb_variable<FieldT> &a,
              const FieldT &Px, const FieldT &Py) :
    gadget<FieldT>(pb, "ec_scalarmul_gadget"),
    outx(outx), outy(outy), a(a), Px(Px), Py(Py)
  {
    // Allocate variables to protoboard
    // The strings (like "x") are only for debugging purposes
	  
    size_t numbits = FieldT::num_bits;
    avec.allocate(this->pb, numbits, "avec");
    packers.emplace_back(this->pb, avec, a);
    vecgadget.emplace_back(this->pb, outx, outy, avec, Px, Py);
  }

  void generate_r1cs_constraints()
  {
    packers[0].generate_r1cs_constraints(true);
    vecgadget[0].generate_r1cs_constraints();
  }

  void generate_r1cs_witness()
  {
    packers[0].generate_r1cs_witness_from_packed();
    vecgadget[0].generate_r1cs_witness();
  }
};

// Compute a*G + b*H as (outx, outy), given a and b as bit vectors.
// a and b must be of the same size.  The _caller_ is responsible for
// proving that the elements of avec and bvec are bits.
template<typename FieldT>
class ec_pedersen_vec_gadget : public gadget<FieldT> {
private:
  pb_variable_array<FieldT> accumx, accumy, daccumx, daccumy;
  pb_variable<FieldT> lambda;
  std::vector<ec_double_gadget<FieldT> > doublers;
  std::vector<ec_add_P123_gadget<FieldT> > adders;
  const FieldT Cx, Cy, CGx, CGy, CHx, CHy, CGHx, CGHy;
  FieldT m2nCx, m2nCy;

  // Compute m2nC = -2^n * C.  We can precomute the answers for values
  // of n we expect to get.
  void compute_m2nC(FieldT &m2nCx, FieldT &m2nCy, size_t n)
  {
    if (n == 253) {
        // Precomputed coordinates of -2^253*C
        m2nCx = FieldT("2630025903576807331238993847875694711243784786568881628418508626984487096258");
        m2nCy = FieldT("17628834417659968531880949658739649785248429713924280788649629869316127047701");
    } else {
        // Invariant: m2iC = -2^i * C
        FieldT m2iCx = Cx;
        FieldT m2iCy = -Cy;
        size_t i = 0;
        while (i < n) {
            FieldT m2iCxo, m2iCyo;
            ec_double_point(m2iCxo, m2iCyo, m2iCx, m2iCy);
            m2iCx = m2iCxo;
            m2iCy = m2iCyo;
            ++i;
        }
        m2nCx = m2iCx;
        m2nCy = m2iCy;
    }
  }

public:
  const pb_variable<FieldT> outx, outy;
  const pb_variable_array<FieldT> avec, bvec;

  ec_pedersen_vec_gadget(protoboard<FieldT> &pb,
              const pb_variable<FieldT> &outx,
              const pb_variable<FieldT> &outy,
              const pb_variable_array<FieldT> &avec,
              const pb_variable_array<FieldT> &bvec) :
    gadget<FieldT>(pb, "ec_pedersen_vec_gadget"),
    // Precomputed coordinates of C, C+G, C+H, and C+G+H
    Cx(2),
    Cy("4950745124018817972378217179409499695353526031437053848725554590521829916331"),
    CGx("4998993376791159436553350546778310121346937620672073819457843493128326049156"),
    CGy("11119675827304465476900978353730540420130346377889406728458325551400357147144"),
    CHx("19614539896004018833724771305328960655474424364705508053472946746883341111010"),
    CHy("9853241351900213537247225242092949438866383394579783148395572971112906592855"),
    CGHx("10755582242294898568680134375159803731902153202607833320871336755950640390928"),
    CGHy("3110667473759844579409644567672992116704859238881299917617768683686288881761"),
    outx(outx), outy(outy), avec(avec), bvec(bvec)
  {
    // Allocate variables to protoboard
    // The strings (like "x") are only for debugging purposes
	  
    size_t numbits = avec.size();
    accumx.allocate(this->pb, numbits, "accumx");
    accumy.allocate(this->pb, numbits, "accumy");
    daccumx.allocate(this->pb, numbits-1, "daccumx");
    daccumy.allocate(this->pb, numbits-1, "daccumy");
    lambda.allocate(this->pb, "lambda");

    for (size_t i = 0; i < numbits-1; ++i) {
        doublers.emplace_back(this->pb, daccumx[i], daccumy[i],
            accumx[i], accumy[i]);
        adders.emplace_back(this->pb, accumx[i+1], accumy[i+1],
            daccumx[i], daccumy[i], avec[numbits-2-i], bvec[numbits-2-i],
            // The coordinates of G, H, and G+H
            FieldT(0), FieldT("11977228949870389393715360594190192321220966033310912010610740966317727761886"),
            FieldT(1), FieldT("21803877843449984883423225223478944275188924769286999517937427649571474907279"),
            FieldT("2864090850787705444524344020850508438903451433901276387624248428140647539638"),
            FieldT("3350168998338968221269367365107720885864670493693161027931048546881356285970"));
    }

    compute_m2nC(m2nCx, m2nCy, numbits-1);
  }

  void generate_r1cs_constraints()
  {
    // Strategy: We do a basic double-and-add, using the variant of a
    // single double and an add of one of (O, G, H, G+H) at each step.
    // *However*, there's a twist.  Our doubling and adding routines
    // don't handle the point at infinity O as the point to add to or to
    // double.  (Adding O as one of the four options above is fine.)
    // Normally, the double-and-add algorithm starts with an accumulator
    // of O, and that won't work for us.  So instead, we start the
    // accumulator at a different base point C, whose discrete log
    // with respect to the (G,H) basis is unknown.  Then we'll end up
    // with an extra 2^n * C in the accumulator (where n is the number
    // of doublings we do), so at the end, we'll add the (constant!)
    // point -2^n * C to get the final result.  That the discrete log of
    // C is unknown means we won't encounter O along the way, either (if
    // we did, we could compute the DL of C in the (G,H) basis).

    // For the first bit, we just precompute C, C+G, C+H, C+G+H (the
    // values are above) and use the top bit of a and b to choose which
    // one to start with.

    // accumx[0] = Cx + (CGx - Cx) * avec[numbits-1] + (CHx - Cx) *
    // bvec[numbits-1] + (CGHx - CGx - CHx + Cx) *
    // avec[numbits-1]*bvec[numbits-1] (and similarly for y)

    // We could possibly optimize this later by computing the a*b
    // product once, but then we'd have to pass a large linear
    // combination to a gadget, which it probably doesn't like?  It
    // would save just one constraint, so probably not so important?
    size_t numbits = avec.size();
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>((CGHx - CGx - CHx + Cx) * avec[numbits-1], bvec[numbits-1], accumx[0] - (Cx + (CGx - Cx) * avec[numbits-1] + (CHx - Cx) * bvec[numbits-1])));
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>((CGHy - CGy - CHy + Cy) * avec[numbits-1], bvec[numbits-1], accumy[0] - (Cy + (CGy - Cy) * avec[numbits-1] + (CHy - Cy) * bvec[numbits-1])));

    // After that, for each remaining bit of a and b, we use the
    // ec_double_gadget and the ec_add_GH_gadget to accumulate
    // the answer.
    for (size_t i = 0; i < numbits-1; ++i) {
        doublers[i].generate_r1cs_constraints();
        adders[i].generate_r1cs_constraints();
    }

    // Finally, we add the constant point -2^n * C to the result, where
    // n = numbits-1
    // (m2nCx - accumx[numbits-1]) * lambda = m2nCy - accumy[numbits-1]
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(m2nCx - accumx[numbits-1], lambda, m2nCy - accumy[numbits-1]));
    // outx = lambda^2 - (m2nCx + accumx[numbits-1])
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(lambda, lambda, outx + m2nCx + accumx[numbits-1]));
    // outy = lambda * (accumx[numbits-1] - outx) - accumy[numbits-1]
    this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(lambda, accumx[numbits-1] - outx, outy + accumy[numbits-1]));

  }

  void generate_r1cs_witness()
  {
    size_t numbits = avec.size();
    this->pb.val(accumx[0]) = Cx + (CGx - Cx) * this->pb.val(avec[numbits-1]) + (CHx - Cx) * this->pb.val(bvec[numbits-1]) + (CGHx - CGx - CHx + Cx) * this->pb.val(avec[numbits-1])*this->pb.val(bvec[numbits-1]);
    this->pb.val(accumy[0]) = Cy + (CGy - Cy) * this->pb.val(avec[numbits-1]) + (CHy - Cy) * this->pb.val(bvec[numbits-1]) + (CGHy - CGy - CHy + Cy) * this->pb.val(avec[numbits-1])*this->pb.val(bvec[numbits-1]);

    for (size_t i = 0; i < numbits-1; ++i) {
        doublers[i].generate_r1cs_witness();
        adders[i].generate_r1cs_witness();
    }

    this->pb.val(lambda) = (m2nCy - this->pb.val(accumy[numbits-1])) *
                           (m2nCx - this->pb.val(accumx[numbits-1])).inverse();
    this->pb.val(outx) = this->pb.val(lambda).squared() -
                         (m2nCx + this->pb.val(accumx[numbits-1]));
    this->pb.val(outy) = this->pb.val(lambda) *
                         (this->pb.val(accumx[numbits-1]) - this->pb.val(outx))
                         - this->pb.val(accumy[numbits-1]);
    
  }
};

// Compute a*G + b*H as (outx, outy), given a and b as field elements.
template<typename FieldT>
class ec_old_pedersen_gadget : public gadget<FieldT> {
private:
  pb_variable_array<FieldT> avec, bvec;
  std::vector<packing_gadget<FieldT> > packers;
  std::vector<ec_pedersen_vec_gadget<FieldT> > vecgadget;

public:
  const pb_variable<FieldT> outx, outy, a, b;

  ec_old_pedersen_gadget(protoboard<FieldT> &pb,
              const pb_variable<FieldT> &outx,
              const pb_variable<FieldT> &outy,
              const pb_variable<FieldT> &a,
              const pb_variable<FieldT> &b) :
    gadget<FieldT>(pb, "ec_old_pedersen_gadget"),
    outx(outx), outy(outy), a(a), b(b)
  {
    // Allocate variables to protoboard
    // The strings (like "x") are only for debugging purposes
	  
    size_t numbits = FieldT::num_bits;
    avec.allocate(this->pb, numbits, "avec");
    bvec.allocate(this->pb, numbits, "bvec");
    packers.emplace_back(this->pb, avec, a);
    packers.emplace_back(this->pb, bvec, b);
    vecgadget.emplace_back(this->pb, outx, outy, avec, bvec);
  }

  void generate_r1cs_constraints()
  {
    packers[0].generate_r1cs_constraints(true);
    packers[1].generate_r1cs_constraints(true);
    vecgadget[0].generate_r1cs_constraints();
  }

  void generate_r1cs_witness()
  {
    packers[0].generate_r1cs_witness_from_packed();
    packers[1].generate_r1cs_witness_from_packed();
    vecgadget[0].generate_r1cs_witness();
  }
};

// Compute a*G + b*H as (outx, outy), given a and b as field elements.
template<typename FieldT>
class ec_pedersen_gadget : public gadget<FieldT> {
private:
  pb_variable<FieldT> aoutx, aouty, boutx, bouty;
  std::vector<ec_scalarmul_gadget<FieldT> > mulgadgets;
  std::vector<ec_add_gadget<FieldT> > addgadget;
  const FieldT Gx, Gy, Hx, Hy;

public:
  const pb_variable<FieldT> outx, outy, a, b;

  ec_pedersen_gadget(protoboard<FieldT> &pb,
              const pb_variable<FieldT> &outx,
              const pb_variable<FieldT> &outy,
              const pb_variable<FieldT> &a,
              const pb_variable<FieldT> &b) :
    gadget<FieldT>(pb, "ec_pedersen_gadget"),
    outx(outx), outy(outy), a(a), b(b),
  // Precomputed coordinates of G and H
  Gx(0),
  Gy("11977228949870389393715360594190192321220966033310912010610740966317727761886"),
  Hx(1),
  Hy("21803877843449984883423225223478944275188924769286999517937427649571474907279")
  {
    // Allocate variables to protoboard
    // The strings (like "x") are only for debugging purposes
	  
    aoutx.allocate(this->pb, "aoutx");
    aouty.allocate(this->pb, "aouty");
    boutx.allocate(this->pb, "boutx");
    bouty.allocate(this->pb, "bouty");
    mulgadgets.emplace_back(this->pb, aoutx, aouty, a, Gx, Gy);
    mulgadgets.emplace_back(this->pb, boutx, bouty, b, Hx, Hy);
    addgadget.emplace_back(this->pb, outx, outy, aoutx, aouty, boutx, bouty);
  }

  void generate_r1cs_constraints()
  {
    mulgadgets[0].generate_r1cs_constraints();
    mulgadgets[1].generate_r1cs_constraints();
    addgadget[0].generate_r1cs_constraints();
  }

  void generate_r1cs_witness()
  {
    mulgadgets[0].generate_r1cs_witness();
    mulgadgets[1].generate_r1cs_witness();
    addgadget[0].generate_r1cs_witness();
  }
};