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+#include "libsnark/gadgetlib1/gadgets/basic_gadgets.hpp"
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+
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+using namespace libsnark;
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+
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+// Double the EC point (inx,iny) to yield (outx,outy). The input point
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+// must not be the point at infinity.
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+template<typename FieldT>
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+class ec_double_gadget : public gadget<FieldT> {
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+private:
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+ pb_variable<FieldT> lambda, inxsq;
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+public:
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+ const pb_variable<FieldT> outx, outy, inx, iny;
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+
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+ ec_double_gadget(protoboard<FieldT> &pb,
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+ const pb_variable<FieldT> &outx,
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+ const pb_variable<FieldT> &outy,
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+ const pb_variable<FieldT> &inx,
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+ const pb_variable<FieldT> &iny) :
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+ gadget<FieldT>(pb, "ec_double_gadget"), outx(outx), outy(outy),
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+ inx(inx), iny(iny)
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+ {
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+ // Allocate variables to protoboard
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+ // The strings (like "x") are only for debugging purposes
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+
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+ lambda.allocate(this->pb, "lambda");
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+ inxsq.allocate(this->pb, "inxsq");
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+ }
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+
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+ void generate_r1cs_constraints()
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+ {
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+ // inxsq = inx * inx
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+ this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(inx, inx, inxsq));
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+
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+ // 2 * iny * lambda = 3 * inxsq - 3 (a = -3 on our curve)
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+ this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(2 * iny, lambda, 3 * inxsq - 3));
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+
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+ // outx = lambda^2 - 2 * inx
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+ this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(lambda, lambda, outx + 2 * inx));
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+
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+ // outy = lambda * (inx - outx) - iny
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+ this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(lambda, inx - outx, outy + iny));
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+
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+ }
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+
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+ void generate_r1cs_witness()
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+ {
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+ this->pb.val(inxsq) = this->pb.val(inx) * this->pb.val(inx);
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+ this->pb.val(lambda) = (this->pb.val(inxsq) * 3 - 3) * (this->pb.val(iny) * 2).inverse();
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+ this->pb.val(outx) = this->pb.val(lambda).squared() - this->pb.val(inx) * 2;
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+ this->pb.val(outy) = this->pb.val(lambda) * (this->pb.val(inx) - this->pb.val(outx)) - this->pb.val(iny);
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+ }
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+};
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+
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+// Add nothing, G, H, or G+H to the EC point (inx,iny) to yield
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+// (outx,outy). The input point must not be the point at infinity.
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+// The two input bits addG and addH control what is added.
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+template<typename FieldT>
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+class ec_add_GH_gadget : public gadget<FieldT> {
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+private:
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+ pb_variable<FieldT> lambda, sumx, sumy, move;
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+ const FieldT Gx, Gy, Hx, Hy, GHx, GHy;
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+public:
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+ const pb_variable<FieldT> outx, outy, inx, iny, addG, addH;
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+
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+ ec_add_GH_gadget(protoboard<FieldT> &pb,
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+ const pb_variable<FieldT> &outx,
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+ const pb_variable<FieldT> &outy,
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+ const pb_variable<FieldT> &inx,
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+ const pb_variable<FieldT> &iny,
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+ const pb_variable<FieldT> &addG,
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+ const pb_variable<FieldT> &addH) :
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+ gadget<FieldT>(pb, "ec_add_GH_gadget"),
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+ Gx(0),
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+ Gy("11977228949870389393715360594190192321220966033310912010610740966317727761886"),
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+ Hx(1),
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+ Hy("21803877843449984883423225223478944275188924769286999517937427649571474907279"),
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+ GHx("2864090850787705444524344020850508438903451433901276387624248428140647539638"),
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+ GHy("3350168998338968221269367365107720885864670493693161027931048546881356285970"),
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+ outx(outx), outy(outy), inx(inx), iny(iny), addG(addG), addH(addH)
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+ {
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+ // Allocate variables to protoboard
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+ // The strings (like "x") are only for debugging purposes
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+
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+ lambda.allocate(this->pb, "lambda");
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+ sumx.allocate(this->pb, "sumx");
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+ sumy.allocate(this->pb, "sumy");
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+ move.allocate(this->pb, "move");
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+ }
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+
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+ void generate_r1cs_constraints()
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+ {
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+ // Strategy: if addG = addH = 0, we compute some nonsense but throw
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+ // it away later. Otherwise, the coordinates of addG * G + addH * H
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+ // are a _linear_ function of addG and addH (since G and H are global
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+ // constants)
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+
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+ // In particular:
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+ // G = (0, 11977228949870389393715360594190192321220966033310912010610740966317727761886)
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+ // H = (1, 21803877843449984883423225223478944275188924769286999517937427649571474907279)
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+ // G+H = (2864090850787705444524344020850508438903451433901276387624248428140647539638, 3350168998338968221269367365107720885864670493693161027931048546881356285970)
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+ // so the point to add is ( (GHx - Hx) * addG + (GHx - Gx) * addH + (Gx + Hx - GHx), (GHy - Hy) * addG + (GHy - Gy) * addH + (Gy + Hy - GHy))
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+ // = (2864090850787705444524344020850508438903451433901276387624248428140647539637 * addG + 2864090850787705444524344020850508438903451433901276387624248428140647539638 * addH - 2864090850787705444524344020850508438903451433901276387624248428140647539637, -18453708845111016662153857858371223389324254275593838490006379102690118621309 * addG - 8627059951531421172445993229082471435356295539617750982679692419436371475916 * addH + 30430937794981406055869218452561415710545220308904750500617120069007846383195)
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+
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+ // (addx - inx) * lambda = addy - iny
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+ this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>((GHx - Hx) * addG + (GHx - Gx) * addH + (Gx + Hx - GHx) - inx, lambda, (GHy - Hy) * addG + (GHy - Gy) * addH + (Gy + Hy - GHy) - iny));
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+
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+ // sumx = lambda^2 - (addx + inx)
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+ this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(lambda, lambda, sumx + (GHx - Hx) * addG + (GHx - Gx) * addH + (Gx + Hx - GHx) + inx));
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+
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+ // sumy = lambda * (inx - sumx) - iny
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+ this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(lambda, inx - sumx, sumy + iny));
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+
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+ // Now we want to conditionally move the sum. We want that
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+ // outx = (addG || addH) ? sumx : inx
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+ // outy = (addG || addH) ? sumy : iny
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+
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+ // so we compute move = addG || addH, and then
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+ // outx = inx + (sumx - inx) * move
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+ // outy = iny + (sumy - iny) * move
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+
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+ this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(1 - addG, 1 - addH, 1 - move));
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+ this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(sumx - inx, move, outx - inx));
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+ this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(sumy - iny, move, outy - iny));
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+
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+ }
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+
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+ void generate_r1cs_witness()
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+ {
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+ FieldT addxval = (GHx - Hx) * this->pb.val(addG) + (GHx - Gx) * this->pb.val(addH) + (Gx + Hx - GHx);
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+ FieldT addyval = (GHy - Hy) * this->pb.val(addG) + (GHy - Gy) * this->pb.val(addH) + (Gy + Hy - GHy);
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+ this->pb.val(lambda) = (addyval - this->pb.val(iny)) * (addxval - this->pb.val(inx)).inverse();
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+ this->pb.val(sumx) = this->pb.val(lambda).squared() - (addxval + this->pb.val(inx));
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+ this->pb.val(sumy) = this->pb.val(lambda) * (this->pb.val(inx) - this->pb.val(sumx)) - this->pb.val(iny);
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+
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+ bool aG = this->pb.val(addG) != FieldT(0);
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+ bool aH = this->pb.val(addH) != FieldT(0);
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+ this->pb.val(move) = aG || aH;
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+ this->pb.val(outx) = this->pb.val(aG || aH ? sumx : inx);
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+ this->pb.val(outy) = this->pb.val(aG || aH ? sumy : iny);
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+ }
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+};
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+
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+// Compute a*G + b*H as (outx, outy), given a and b as bit vectors.
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+// a and b must be of the same size. The _caller_ is responsible for
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+// proving that the elements of avec and bvec are bits.
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+template<typename FieldT>
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+class ec_pedersen_vec_gadget : public gadget<FieldT> {
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+private:
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+ pb_variable_array<FieldT> accumx, accumy, daccumx, daccumy;
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+ pb_variable<FieldT> lambda;
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+ std::vector<ec_double_gadget<FieldT> > doublers;
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+ std::vector<ec_add_GH_gadget<FieldT> > adders;
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+ const FieldT Cx, Cy, CGx, CGy, CHx, CHy, CGHx, CGHy;
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+ FieldT m2nCx, m2nCy;
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+
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+ // Compute m2nC = -2^n * C. We can precomute the answers for values
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+ // of n we expect to get.
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+ void compute_m2nC(FieldT &m2nCx, FieldT &m2nCy, size_t n)
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+ {
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+ if (n == 253) {
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+ m2nCx = FieldT("2630025903576807331238993847875694711243784786568881628418508626984487096258");
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+ m2nCy = FieldT("17628834417659968531880949658739649785248429713924280788649629869316127047701");
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+ } else {
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+ // Invariant: m2iC = -2^i * C
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+ FieldT m2iCx = Cx;
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+ FieldT m2iCy = -Cy;
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+ size_t i = 0;
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+ while (i < n) {
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+ FieldT xsq = m2iCx.squared();
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+ FieldT lambda = (xsq * 3 - 3) * (m2iCy * 2).inverse();
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+ FieldT m2iCxo = lambda.squared() - m2iCx * 2;
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+ FieldT m2iCyo = lambda * (m2iCx - m2iCxo) - m2iCy;
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+ m2iCx = m2iCxo;
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+ m2iCy = m2iCyo;
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+ ++i;
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+ }
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+ m2nCx = m2iCx;
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+ m2nCy = m2iCy;
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+ }
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+ }
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+
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+public:
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+ const pb_variable<FieldT> outx, outy;
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+ const pb_variable_array<FieldT> avec, bvec;
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+
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+ ec_pedersen_vec_gadget(protoboard<FieldT> &pb,
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+ const pb_variable<FieldT> &outx,
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+ const pb_variable<FieldT> &outy,
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+ const pb_variable_array<FieldT> &avec,
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+ const pb_variable_array<FieldT> &bvec) :
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+ gadget<FieldT>(pb, "ec_pedersen_vec_gadget"),
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+ Cx(2),
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+ Cy("4950745124018817972378217179409499695353526031437053848725554590521829916331"),
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+ CGx("4998993376791159436553350546778310121346937620672073819457843493128326049156"),
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+ CGy("11119675827304465476900978353730540420130346377889406728458325551400357147144"),
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+ CHx("19614539896004018833724771305328960655474424364705508053472946746883341111010"),
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+ CHy("9853241351900213537247225242092949438866383394579783148395572971112906592855"),
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+ CGHx("10755582242294898568680134375159803731902153202607833320871336755950640390928"),
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+ CGHy("3110667473759844579409644567672992116704859238881299917617768683686288881761"),
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+ outx(outx), outy(outy), avec(avec), bvec(bvec)
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+ {
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+ // Allocate variables to protoboard
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+ // The strings (like "x") are only for debugging purposes
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+
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+ size_t numbits = avec.size();
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+ accumx.allocate(this->pb, numbits, "accumx");
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+ accumy.allocate(this->pb, numbits, "accumy");
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+ daccumx.allocate(this->pb, numbits-1, "daccumx");
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+ daccumy.allocate(this->pb, numbits-1, "daccumy");
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+ lambda.allocate(this->pb, "lambda");
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+
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+ for (size_t i = 0; i < numbits-1; ++i) {
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+ doublers.emplace_back(this->pb, daccumx[i], daccumy[i],
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+ accumx[i], accumy[i]);
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+ adders.emplace_back(this->pb, accumx[i+1], accumy[i+1],
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+ daccumx[i], daccumy[i], avec[numbits-2-i], bvec[numbits-2-i]);
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+ }
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+
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+ compute_m2nC(m2nCx, m2nCy, numbits-1);
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+ }
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+
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+ void generate_r1cs_constraints()
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+ {
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+ // Strategy: We do a basic double-and-add, using the variant of a
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+ // single double and an add of one of (O, G, H, G+H) at each step.
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+ // *However*, there's a twist. Our doubling and adding routines
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+ // don't handle the point at infinity O as the point to add to or to
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+ // double. (Adding O as one of the four options above is fine.)
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+ // Normally, the double-and-add algorithm starts with an accumulator
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+ // of O, and that won't work for us. So instead, we start the
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+ // accumulator at a different base point C, whose discrete log
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+ // with respect to the (G,H) basis is unknown. Then we'll end up
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+ // with an extra 2^n * C in the accumulator (where n is the number
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+ // of doublings we do), so at the end, we'll add the (constant!)
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+ // point -2^n * C to get the final result. That the discrete log of
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+ // C is unknown means we won't encounter O along the way, either (if
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+ // we did, we could compute the DL of C in the (G,H) basis).
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+
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+ // For the first bit, we just precompute C, C+G, C+H, C+G+H and use
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+ // the top bit of a and b to choose which one to start with.
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+
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+ // accumx[0] = Cx + (CGx - Cx) * avec[numbits-1] + (CHx - Cx) *
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+ // bvec[numbits-1] + (CGHx - CGx - CHx + Cx) *
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+ // avec[numbits-1]*bvec[numbits-1] (and similarly for y)
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+
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+ // We could possibly optimize this later by computing the a*b
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+ // product once, but then we'd have to pass a large linear
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+ // combination to a gadget, which it probably doesn't like? It
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+ // would save just one constraint, so probably not so important?
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+ size_t numbits = avec.size();
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+ 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])));
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+ 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])));
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+
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+ // After that, for each remaining bit of a and b, we use the
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+ // ec_double_gadget and the ec_add_GH_gadget to accumulate
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+ // the answer.
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+ for (size_t i = 0; i < numbits-1; ++i) {
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+ doublers[i].generate_r1cs_constraints();
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+ adders[i].generate_r1cs_constraints();
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+ }
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+
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+ // Finally, we add the constant point -2^n * C to the result, where
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+ // n = numbits-1
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+ // (m2nCx - accumx[numbits-1]) * lambda = m2nCy - accumy[numbits-1]
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+ this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(m2nCx - accumx[numbits-1], lambda, m2nCy - accumy[numbits-1]));
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+ // outx = lambda^2 - (m2nCx + accumx[numbits-1])
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+ this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(lambda, lambda, outx + m2nCx + accumx[numbits-1]));
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+ // outy = lambda * (accumx[numbits-1] - outx) - accumy[numbits-1]
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+ this->pb.add_r1cs_constraint(r1cs_constraint<FieldT>(lambda, accumx[numbits-1] - outx, outy + accumy[numbits-1]));
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+
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+ }
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+
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+ void generate_r1cs_witness()
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+ {
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+ size_t numbits = avec.size();
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+ 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]);
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+ 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_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_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)
|
|
|
+ {
|
|
|
+ // Allocate variables to protoboard
|
|
|
+ // The strings (like "x") are only for debugging purposes
|
|
|
+
|
|
|
+ FieldT minus1(-1);
|
|
|
+ size_t numbits = FieldT::num_bits;
|
|
|
+ avec.allocate(this->pb, numbits, "a");
|
|
|
+ bvec.allocate(this->pb, numbits, "b");
|
|
|
+ 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();
|
|
|
+ }
|
|
|
+};
|