r1cs_gg_ppzksnark.tcc

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/** @file
*****************************************************************************
 
Implementation of interfaces for a ppzkSNARK for R1CS.
 
See r1cs_gg_ppzksnark.hpp .
 
*****************************************************************************
* @author     This file is part of libsnark, developed by SCIPR Lab
*             and contributors (see AUTHORS).
* @copyright  MIT license (see LICENSE file)
*****************************************************************************/
 
#ifndef R1CS_GG_PPZKSNARK_TCC_
#define R1CS_GG_PPZKSNARK_TCC_
 
#include <algorithm>
#include <cassert>
#include <functional>
#include <iostream>
#include <libff/algebra/scalar_multiplication/multiexp.hpp>
#include <libff/common/profiling.hpp>
#include <libff/common/utils.hpp>
#include <sstream>
 
#ifdef MULTICORE
#include <omp.h>
#endif
 
#include <libsnark/knowledge_commitment/kc_multiexp.hpp>
#include <libsnark/reductions/r1cs_to_qap/r1cs_to_qap.hpp>
 
namespace libsnark
{
 
template<typename ppT>
bool r1cs_gg_ppzksnark_proving_key<ppT>::operator==(
    const r1cs_gg_ppzksnark_proving_key<ppT> &other) const
{
    return (
        this->alpha_g1 == other.alpha_g1 && this->beta_g1 == other.beta_g1 &&
        this->beta_g2 == other.beta_g2 && this->delta_g1 == other.delta_g1 &&
        this->delta_g2 == other.delta_g2 && this->A_query == other.A_query &&
        this->B_query == other.B_query && this->H_query == other.H_query &&
        this->L_query == other.L_query &&
        this->constraint_system == other.constraint_system);
}
 
template<typename ppT>
std::ostream &operator<<(
    std::ostream &out, const r1cs_gg_ppzksnark_proving_key<ppT> &pk)
{
    out << pk.alpha_g1 << OUTPUT_NEWLINE;
    out << pk.beta_g1 << OUTPUT_NEWLINE;
    out << pk.beta_g2 << OUTPUT_NEWLINE;
    out << pk.delta_g1 << OUTPUT_NEWLINE;
    out << pk.delta_g2 << OUTPUT_NEWLINE;
    out << pk.A_query;
    out << pk.B_query;
    out << pk.H_query;
    out << pk.L_query;
    out << pk.constraint_system;
 
    return out;
}
 
template<typename ppT>
std::istream &operator>>(
    std::istream &in, r1cs_gg_ppzksnark_proving_key<ppT> &pk)
{
    in >> pk.alpha_g1;
    libff::consume_OUTPUT_NEWLINE(in);
    in >> pk.beta_g1;
    libff::consume_OUTPUT_NEWLINE(in);
    in >> pk.beta_g2;
    libff::consume_OUTPUT_NEWLINE(in);
    in >> pk.delta_g1;
    libff::consume_OUTPUT_NEWLINE(in);
    in >> pk.delta_g2;
    libff::consume_OUTPUT_NEWLINE(in);
    in >> pk.A_query;
    in >> pk.B_query;
    in >> pk.H_query;
    in >> pk.L_query;
    in >> pk.constraint_system;
 
    return in;
}
 
template<typename ppT>
bool r1cs_gg_ppzksnark_verification_key<ppT>::operator==(
    const r1cs_gg_ppzksnark_verification_key<ppT> &other) const
{
    return (
        this->alpha_g1 == other.alpha_g1 && this->beta_g2 == other.beta_g2 &&
        this->delta_g2 == other.delta_g2 && this->ABC_g1 == other.ABC_g1);
}
 
template<typename ppT>
std::ostream &operator<<(
    std::ostream &out, const r1cs_gg_ppzksnark_verification_key<ppT> &vk)
{
    out << vk.alpha_g1 << OUTPUT_NEWLINE;
    out << vk.beta_g2 << OUTPUT_NEWLINE;
    out << vk.delta_g2 << OUTPUT_NEWLINE;
    out << vk.ABC_g1 << OUTPUT_NEWLINE;
 
    return out;
}
 
template<typename ppT>
std::istream &operator>>(
    std::istream &in, r1cs_gg_ppzksnark_verification_key<ppT> &vk)
{
    in >> vk.alpha_g1;
    libff::consume_OUTPUT_NEWLINE(in);
    in >> vk.beta_g2;
    libff::consume_OUTPUT_NEWLINE(in);
    in >> vk.delta_g2;
    libff::consume_OUTPUT_NEWLINE(in);
    in >> vk.ABC_g1;
    libff::consume_OUTPUT_NEWLINE(in);
 
    return in;
}
 
template<typename ppT>
bool r1cs_gg_ppzksnark_processed_verification_key<ppT>::operator==(
    const r1cs_gg_ppzksnark_processed_verification_key<ppT> &other) const
{
    return (
        this->vk_alpha_g1_precomp == other.vk_alpha_g1_precomp &&
        this->vk_beta_g2_precomp == other.vk_beta_g2_precomp &&
        this->vk_generator_g2_precomp == other.vk_generator_g2_precomp &&
        this->vk_delta_g2_precomp == other.vk_delta_g2_precomp &&
        this->ABC_g1 == other.ABC_g1);
}
 
template<typename ppT>
std::ostream &operator<<(
    std::ostream &out,
    const r1cs_gg_ppzksnark_processed_verification_key<ppT> &pvk)
{
    out << pvk.vk_alpha_g1_precomp << OUTPUT_NEWLINE;
    out << pvk.vk_beta_g2_precomp << OUTPUT_NEWLINE;
    out << pvk.vk_generator_g2_precomp << OUTPUT_NEWLINE;
    out << pvk.vk_delta_g2_precomp << OUTPUT_NEWLINE;
    out << pvk.ABC_g1 << OUTPUT_NEWLINE;
 
    return out;
}
 
template<typename ppT>
std::istream &operator>>(
    std::istream &in, r1cs_gg_ppzksnark_processed_verification_key<ppT> &pvk)
{
    in >> pvk.vk_alpha_g1_precomp;
    libff::consume_OUTPUT_NEWLINE(in);
    in >> pvk.vk_beta_g2_precomp;
    libff::consume_OUTPUT_NEWLINE(in);
    in >> pvk.vk_generator_g2_precomp;
    libff::consume_OUTPUT_NEWLINE(in);
    in >> pvk.vk_delta_g2_precomp;
    libff::consume_OUTPUT_NEWLINE(in);
    in >> pvk.ABC_g1;
    libff::consume_OUTPUT_NEWLINE(in);
 
    return in;
}
 
template<typename ppT>
bool r1cs_gg_ppzksnark_proof<ppT>::operator==(
    const r1cs_gg_ppzksnark_proof<ppT> &other) const
{
    return (
        this->g_A == other.g_A && this->g_B == other.g_B &&
        this->g_C == other.g_C);
}
 
template<typename ppT>
std::ostream &operator<<(
    std::ostream &out, const r1cs_gg_ppzksnark_proof<ppT> &proof)
{
    out << proof.g_A << OUTPUT_NEWLINE;
    out << proof.g_B << OUTPUT_NEWLINE;
    out << proof.g_C << OUTPUT_NEWLINE;
 
    return out;
}
 
template<typename ppT>
std::istream &operator>>(std::istream &in, r1cs_gg_ppzksnark_proof<ppT> &proof)
{
    in >> proof.g_A;
    libff::consume_OUTPUT_NEWLINE(in);
    in >> proof.g_B;
    libff::consume_OUTPUT_NEWLINE(in);
    in >> proof.g_C;
    libff::consume_OUTPUT_NEWLINE(in);
 
    return in;
}
 
template<typename ppT>
r1cs_gg_ppzksnark_verification_key<ppT> r1cs_gg_ppzksnark_verification_key<
    ppT>::dummy_verification_key(const size_t input_size)
{
    r1cs_gg_ppzksnark_verification_key<ppT> result;
    result.alpha_g1 = libff::G1<ppT>::random_element();
    result.beta_g2 = libff::G2<ppT>::random_element();
    result.delta_g2 = libff::G2<ppT>::random_element();
 
    libff::G1<ppT> base = libff::G1<ppT>::random_element();
    libff::G1_vector<ppT> v;
    for (size_t i = 0; i < input_size; ++i) {
        v.emplace_back(libff::G1<ppT>::random_element());
    }
 
    result.ABC_g1 =
        accumulation_vector<libff::G1<ppT>>(std::move(base), std::move(v));
 
    return result;
}
 
template<typename ppT, libff::multi_exp_base_form BaseForm>
r1cs_gg_ppzksnark_keypair<ppT> r1cs_gg_ppzksnark_generator_from_secrets(
    const r1cs_gg_ppzksnark_constraint_system<ppT> &r1cs,
    const libff::Fr<ppT> &t,
    const libff::Fr<ppT> &alpha,
    const libff::Fr<ppT> &beta,
    const libff::Fr<ppT> &delta,
    const libff::G1<ppT> &g1_generator,
    const libff::G2<ppT> &g2_generator,
    bool force_pow_2_domain)
{
    libff::enter_block("Call to r1cs_gg_ppzksnark_generator_from_secrets");
 
    /* Make the B_query "lighter" if possible */
    r1cs_gg_ppzksnark_constraint_system<ppT> r1cs_copy(r1cs);
    r1cs_copy.swap_AB_if_beneficial();
 
    const libff::Fr<ppT> delta_inverse = delta.inverse();
 
    /* A quadratic arithmetic program evaluated at t. */
    qap_instance_evaluation<libff::Fr<ppT>> qap =
        r1cs_to_qap_instance_map_with_evaluation(
            r1cs_copy, t, force_pow_2_domain);
 
    libff::print_indent();
    printf("* QAP number of variables: %zu\n", qap.num_variables());
    libff::print_indent();
    printf("* QAP pre degree: %zu\n", r1cs_copy.constraints.size());
    libff::print_indent();
    printf("* QAP degree: %zu\n", qap.degree());
    libff::print_indent();
    printf("* QAP number of input variables: %zu\n", qap.num_inputs());
 
    libff::enter_block("Compute query densities");
    size_t non_zero_At = 0;
    size_t non_zero_Bt = 0;
    for (size_t i = 0; i < qap.num_variables() + 1; ++i) {
        if (!qap.At[i].is_zero()) {
            ++non_zero_At;
        }
        if (!qap.Bt[i].is_zero()) {
            ++non_zero_Bt;
        }
    }
    libff::leave_block("Compute query densities");
 
    /* qap.{At,Bt,Ct,Ht} are now in unspecified state, but we do not use them
     * later */
    libff::Fr_vector<ppT> At = std::move(qap.At);
    libff::Fr_vector<ppT> Bt = std::move(qap.Bt);
    libff::Fr_vector<ppT> Ct = std::move(qap.Ct);
    libff::Fr_vector<ppT> Ht = std::move(qap.Ht);
 
    /* The product component: (beta*A_i(t) + alpha*B_i(t) + C_i(t)). */
    libff::enter_block("Compute ABC for R1CS verification key");
    libff::Fr_vector<ppT> ABC;
    ABC.reserve(qap.num_inputs());
 
    const libff::Fr<ppT> ABC_0 = beta * At[0] + alpha * Bt[0] + Ct[0];
    for (size_t i = 1; i < qap.num_inputs() + 1; ++i) {
        ABC.emplace_back(beta * At[i] + alpha * Bt[i] + Ct[i]);
    }
    libff::leave_block("Compute ABC for R1CS verification key");
 
    /* The delta inverse product component: (beta*A_i(t) + alpha*B_i(t) +
     * C_i(t)) * delta^{-1}. */
    libff::enter_block("Compute L query for R1CS proving key");
    libff::Fr_vector<ppT> Lt;
    Lt.reserve(qap.num_variables() - qap.num_inputs());
 
    const size_t Lt_offset = qap.num_inputs() + 1;
    for (size_t i = Lt_offset; i < qap.num_variables() + 1; ++i) {
        Lt.emplace_back((beta * At[i] + alpha * Bt[i] + Ct[i]) * delta_inverse);
    }
    libff::leave_block("Compute L query for R1CS proving key");
 
    /**
     * Note that H for Groth's proof system is degree d-2, but the QAP
     * reduction returns coefficients for degree d polynomial H (in
     * style of PGHR-type proof systems)
     */
    Ht.resize(Ht.size() - 2);
 
#ifdef MULTICORE
    const size_t chunks =
        omp_get_max_threads(); // to override, set OMP_NUM_THREADS env var or
                               // call omp_set_num_threads()
#else
    const size_t chunks = 1;
#endif
 
    libff::enter_block("Generating G1 MSM window table");
    const size_t g1_scalar_count =
        non_zero_At + non_zero_Bt + qap.num_variables();
    const size_t g1_scalar_size = libff::Fr<ppT>::size_in_bits();
    const size_t g1_window_size =
        libff::get_exp_window_size<libff::G1<ppT>>(g1_scalar_count);
 
    libff::print_indent();
    printf("* G1 window: %zu\n", g1_window_size);
    libff::window_table<libff::G1<ppT>> g1_table =
        libff::get_window_table(g1_scalar_size, g1_window_size, g1_generator);
    libff::leave_block("Generating G1 MSM window table");
 
    libff::enter_block("Generating G2 MSM window table");
    const size_t g2_scalar_count = non_zero_Bt;
    const size_t g2_scalar_size = libff::Fr<ppT>::size_in_bits();
    size_t g2_window_size =
        libff::get_exp_window_size<libff::G2<ppT>>(g2_scalar_count);
 
    libff::print_indent();
    printf("* G2 window: %zu\n", g2_window_size);
    libff::window_table<libff::G2<ppT>> g2_table =
        libff::get_window_table(g2_scalar_size, g2_window_size, g2_generator);
    libff::leave_block("Generating G2 MSM window table");
 
    libff::enter_block("Generate R1CS proving key");
    libff::G1<ppT> alpha_g1 = alpha * g1_generator;
    libff::G1<ppT> beta_g1 = beta * g1_generator;
    libff::G2<ppT> beta_g2 = beta * g2_generator;
    libff::G1<ppT> delta_g1 = delta * g1_generator;
    libff::G2<ppT> delta_g2 = delta * g2_generator;
 
    libff::enter_block("Generate queries");
    libff::enter_block("Compute the A-query", false);
    libff::G1_vector<ppT> A_query =
        batch_exp(g1_scalar_size, g1_window_size, g1_table, At);
    if (BaseForm == libff::multi_exp_base_form_special) {
        libff::batch_to_special<libff::G1<ppT>>(A_query);
    }
    libff::leave_block("Compute the A-query", false);
 
    libff::enter_block("Compute the B-query", false);
    // Force kc_batch_exp to convert to special form, if BaseForm ==
    // libff::multi_exp_base_form_special.
    knowledge_commitment_vector<libff::G2<ppT>, libff::G1<ppT>> B_query =
        kc_batch_exp(
            libff::Fr<ppT>::size_in_bits(),
            g2_window_size,
            g1_window_size,
            g2_table,
            g1_table,
            libff::Fr<ppT>::one(),
            libff::Fr<ppT>::one(),
            Bt,
            chunks,
            BaseForm == libff::multi_exp_base_form_special);
    libff::leave_block("Compute the B-query", false);
 
    libff::enter_block("Compute the H-query", false);
    libff::G1_vector<ppT> H_query = batch_exp_with_coeff(
        g1_scalar_size, g1_window_size, g1_table, qap.Zt * delta_inverse, Ht);
    if (BaseForm == libff::multi_exp_base_form_special) {
        libff::batch_to_special<libff::G1<ppT>>(H_query);
    }
    libff::leave_block("Compute the H-query", false);
 
    libff::enter_block("Compute the L-query", false);
    libff::G1_vector<ppT> L_query =
        batch_exp(g1_scalar_size, g1_window_size, g1_table, Lt);
    if (BaseForm == libff::multi_exp_base_form_special) {
        libff::batch_to_special<libff::G1<ppT>>(L_query);
    }
    libff::leave_block("Compute the L-query", false);
    libff::leave_block("Generate queries");
 
    libff::leave_block("Generate R1CS proving key");
 
    libff::enter_block("Generate R1CS verification key");
 
    libff::enter_block("Encode ABC for R1CS verification key");
    libff::G1<ppT> ABC_g1_0 = ABC_0 * g1_generator;
    libff::G1_vector<ppT> ABC_g1_values =
        batch_exp(g1_scalar_size, g1_window_size, g1_table, ABC);
    libff::leave_block("Encode ABC for R1CS verification key");
    libff::leave_block("Generate R1CS verification key");
 
    libff::leave_block("Call to r1cs_gg_ppzksnark_generator_from_secrets");
 
    accumulation_vector<libff::G1<ppT>> ABC_g1(
        std::move(ABC_g1_0), std::move(ABC_g1_values));
 
    r1cs_gg_ppzksnark_verification_key<ppT> vk =
        r1cs_gg_ppzksnark_verification_key<ppT>(
            alpha_g1, beta_g2, delta_g2, ABC_g1);
 
    r1cs_gg_ppzksnark_proving_key<ppT> pk = r1cs_gg_ppzksnark_proving_key<ppT>(
        std::move(alpha_g1),
        std::move(beta_g1),
        std::move(beta_g2),
        std::move(delta_g1),
        std::move(delta_g2),
        std::move(A_query),
        std::move(B_query),
        std::move(H_query),
        std::move(L_query),
        std::move(r1cs_copy));
 
    pk.print_size();
    vk.print_size();
 
    return r1cs_gg_ppzksnark_keypair<ppT>(std::move(pk), std::move(vk));
}
 
template<typename ppT, libff::multi_exp_base_form BaseForm>
r1cs_gg_ppzksnark_keypair<ppT> r1cs_gg_ppzksnark_generator(
    const r1cs_gg_ppzksnark_constraint_system<ppT> &r1cs,
    bool force_pow_2_domain)
{
    libff::enter_block("Call to r1cs_gg_ppzksnark_generator");
 
    /* Generate secret randomness */
    const libff::Fr<ppT> t = libff::Fr<ppT>::random_element();
    const libff::Fr<ppT> alpha = libff::Fr<ppT>::random_element();
    const libff::Fr<ppT> beta = libff::Fr<ppT>::random_element();
    const libff::Fr<ppT> delta = libff::Fr<ppT>::random_element();
    const libff::G1<ppT> g1_generator = libff::G1<ppT>::one();
    const libff::G2<ppT> g2_generator = libff::G2<ppT>::one();
 
    r1cs_gg_ppzksnark_keypair<ppT> key_pair =
        r1cs_gg_ppzksnark_generator_from_secrets<ppT, BaseForm>(
            r1cs,
            t,
            alpha,
            beta,
            delta,
            g1_generator,
            g2_generator,
            force_pow_2_domain);
 
    libff::leave_block("Call to r1cs_gg_ppzksnark_generator");
 
    return key_pair;
}
 
template<
    typename ppT,
    libff::multi_exp_method Method,
    libff::multi_exp_base_form BaseForm>
r1cs_gg_ppzksnark_proof<ppT> r1cs_gg_ppzksnark_prover(
    const r1cs_gg_ppzksnark_proving_key<ppT> &pk,
    const r1cs_gg_ppzksnark_primary_input<ppT> &primary_input,
    const r1cs_gg_ppzksnark_auxiliary_input<ppT> &auxiliary_input,
    bool force_pow_2_domain)
{
    libff::enter_block("Call to r1cs_gg_ppzksnark_prover");
 
#ifdef DEBUG
    assert(pk.constraint_system.is_satisfied(primary_input, auxiliary_input));
#endif
 
    libff::enter_block("Compute the polynomial H");
    const qap_witness<libff::Fr<ppT>> qap_wit = r1cs_to_qap_witness_map(
        pk.constraint_system,
        primary_input,
        auxiliary_input,
        libff::Fr<ppT>::zero(),
        libff::Fr<ppT>::zero(),
        libff::Fr<ppT>::zero(),
        force_pow_2_domain);
 
    /* We are dividing degree 2(d-1) polynomial by degree d polynomial
       and not adding a PGHR-style ZK-patch, so our H is degree d-2 */
    assert(!qap_wit.coefficients_for_H[qap_wit.degree() - 2].is_zero());
    assert(qap_wit.coefficients_for_H[qap_wit.degree() - 1].is_zero());
    assert(qap_wit.coefficients_for_H[qap_wit.degree()].is_zero());
    libff::leave_block("Compute the polynomial H");
 
#ifdef DEBUG
    const libff::Fr<ppT> t = libff::Fr<ppT>::random_element();
    qap_instance_evaluation<libff::Fr<ppT>> qap_inst =
        r1cs_to_qap_instance_map_with_evaluation(
            pk.constraint_system, t, force_pow_2_domain);
    assert(qap_inst.is_satisfied(qap_wit));
#endif
 
    /* Choose two random field elements for prover zero-knowledge. */
    const libff::Fr<ppT> r = libff::Fr<ppT>::random_element();
    const libff::Fr<ppT> s = libff::Fr<ppT>::random_element();
 
#ifdef DEBUG
    assert(qap_wit.coefficients_for_ABCs.size() == qap_wit.num_variables());
    assert(pk.A_query.size() == qap_wit.num_variables() + 1);
    assert(pk.B_query.domain_size() == qap_wit.num_variables() + 1);
    assert(pk.H_query.size() == qap_wit.degree() - 1);
    assert(pk.L_query.size() == qap_wit.num_variables() - qap_wit.num_inputs());
#endif
 
#ifdef MULTICORE
    const size_t chunks =
        omp_get_max_threads(); // to override, set OMP_NUM_THREADS env var or
                               // call omp_set_num_threads()
#else
    const size_t chunks = 1;
#endif
 
    libff::enter_block("Compute the proof");
 
    libff::enter_block("Compute evaluation to A-query", false);
    // TODO: sort out indexing
    libff::Fr_vector<ppT> const_padded_assignment(1, libff::Fr<ppT>::one());
    const_padded_assignment.insert(
        const_padded_assignment.end(),
        qap_wit.coefficients_for_ABCs.begin(),
        qap_wit.coefficients_for_ABCs.end());
 
    libff::G1<ppT> evaluation_At = libff::multi_exp_filter_one_zero<
        libff::G1<ppT>,
        libff::Fr<ppT>,
        Method,
        BaseForm>(
        pk.A_query.begin(),
        pk.A_query.begin() + qap_wit.num_variables() + 1,
        const_padded_assignment.begin(),
        const_padded_assignment.begin() + qap_wit.num_variables() + 1,
        chunks);
    libff::leave_block("Compute evaluation to A-query", false);
 
    libff::enter_block("Compute evaluation to B-query", false);
    knowledge_commitment<libff::G2<ppT>, libff::G1<ppT>> evaluation_Bt =
        kc_multi_exp_with_mixed_addition<
            libff::G2<ppT>,
            libff::G1<ppT>,
            libff::Fr<ppT>,
            Method,
            BaseForm>(
            pk.B_query,
            0,
            qap_wit.num_variables() + 1,
            const_padded_assignment.begin(),
            const_padded_assignment.begin() + qap_wit.num_variables() + 1,
            chunks);
    libff::leave_block("Compute evaluation to B-query", false);
 
    libff::enter_block("Compute evaluation to H-query", false);
    libff::G1<ppT> evaluation_Ht =
        libff::multi_exp<libff::G1<ppT>, libff::Fr<ppT>, Method, BaseForm>(
            pk.H_query.begin(),
            pk.H_query.begin() + (qap_wit.degree() - 1),
            qap_wit.coefficients_for_H.begin(),
            qap_wit.coefficients_for_H.begin() + (qap_wit.degree() - 1),
            chunks);
    libff::leave_block("Compute evaluation to H-query", false);
 
    libff::enter_block("Compute evaluation to L-query", false);
    libff::G1<ppT> evaluation_Lt = libff::multi_exp_filter_one_zero<
        libff::G1<ppT>,
        libff::Fr<ppT>,
        Method,
        BaseForm>(
        pk.L_query.begin(),
        pk.L_query.end(),
        const_padded_assignment.begin() + qap_wit.num_inputs() + 1,
        const_padded_assignment.begin() + qap_wit.num_variables() + 1,
        chunks);
    libff::leave_block("Compute evaluation to L-query", false);
 
    /* A = alpha + sum_i(a_i*A_i(t)) + r*delta */
    libff::G1<ppT> g1_A = pk.alpha_g1 + evaluation_At + r * pk.delta_g1;
 
    /* B = beta + sum_i(a_i*B_i(t)) + s*delta */
    libff::G1<ppT> g1_B = pk.beta_g1 + evaluation_Bt.h + s * pk.delta_g1;
    libff::G2<ppT> g2_B = pk.beta_g2 + evaluation_Bt.g + s * pk.delta_g2;
 
    /* C = sum_i(a_i*((beta*A_i(t) + alpha*B_i(t) + C_i(t)) + H(t)*Z(t))/delta)
     * + A*s + r*b - r*s*delta */
    libff::G1<ppT> g1_C = evaluation_Ht + evaluation_Lt + s * g1_A + r * g1_B -
                          (r * s) * pk.delta_g1;
 
    libff::leave_block("Compute the proof");
 
    libff::leave_block("Call to r1cs_gg_ppzksnark_prover");
 
    r1cs_gg_ppzksnark_proof<ppT> proof = r1cs_gg_ppzksnark_proof<ppT>(
        std::move(g1_A), std::move(g2_B), std::move(g1_C));
    proof.print_size();
 
    return proof;
}
 
template<typename ppT>
r1cs_gg_ppzksnark_processed_verification_key<ppT>
r1cs_gg_ppzksnark_verifier_process_vk(
    const r1cs_gg_ppzksnark_verification_key<ppT> &vk)
{
    libff::enter_block("Call to r1cs_gg_ppzksnark_verifier_process_vk");
 
    r1cs_gg_ppzksnark_processed_verification_key<ppT> pvk;
    pvk.vk_alpha_g1_precomp = ppT::precompute_G1(vk.alpha_g1);
    pvk.vk_beta_g2_precomp = ppT::precompute_G2(vk.beta_g2);
    pvk.vk_generator_g2_precomp = ppT::precompute_G2(libff::G2<ppT>::one());
    pvk.vk_delta_g2_precomp = ppT::precompute_G2(vk.delta_g2);
    pvk.ABC_g1 = vk.ABC_g1;
 
    libff::leave_block("Call to r1cs_gg_ppzksnark_verifier_process_vk");
 
    return pvk;
}
 
template<typename ppT>
bool r1cs_gg_ppzksnark_online_verifier_weak_IC(
    const r1cs_gg_ppzksnark_processed_verification_key<ppT> &pvk,
    const r1cs_gg_ppzksnark_primary_input<ppT> &primary_input,
    const r1cs_gg_ppzksnark_proof<ppT> &proof)
{
    libff::enter_block("Call to r1cs_gg_ppzksnark_online_verifier_weak_IC");
    assert(pvk.ABC_g1.domain_size() >= primary_input.size());
 
    libff::enter_block("Accumulate input");
    const accumulation_vector<libff::G1<ppT>> accumulated_IC =
        pvk.ABC_g1.template accumulate_chunk<libff::Fr<ppT>>(
            primary_input.begin(), primary_input.end(), 0);
    const libff::G1<ppT> &acc = accumulated_IC.first;
    libff::leave_block("Accumulate input");
 
    bool result = true;
 
    libff::enter_block("Check if the proof is well-formed");
    if (!proof.is_well_formed()) {
        if (!libff::inhibit_profiling_info) {
            libff::print_indent();
            printf("At least one of the proof elements does not lie on the "
                   "curve.\n");
        }
        result = false;
    }
    libff::leave_block("Check if the proof is well-formed");
 
    libff::enter_block("Online pairing computations");
    libff::enter_block("Check QAP divisibility");
    const libff::G1_precomp<ppT> proof_g_A_precomp =
        ppT::precompute_G1(proof.g_A);
    const libff::G2_precomp<ppT> proof_g_B_precomp =
        ppT::precompute_G2(proof.g_B);
    const libff::G1_precomp<ppT> proof_g_C_precomp =
        ppT::precompute_G1(proof.g_C);
    const libff::G1_precomp<ppT> acc_precomp = ppT::precompute_G1(acc);
 
    const libff::Fqk<ppT> f =
        ppT::miller_loop(pvk.vk_alpha_g1_precomp, pvk.vk_beta_g2_precomp);
    const libff::GT<ppT> vk_alpha_g1_beta_g2 = ppT::final_exponentiation(f);
 
    const libff::Fqk<ppT> QAP1 =
        ppT::miller_loop(proof_g_A_precomp, proof_g_B_precomp);
    const libff::Fqk<ppT> QAP2 = ppT::double_miller_loop(
        acc_precomp,
        pvk.vk_generator_g2_precomp,
        proof_g_C_precomp,
        pvk.vk_delta_g2_precomp);
    const libff::GT<ppT> QAP =
        ppT::final_exponentiation(QAP1 * QAP2.unitary_inverse());
 
    if (QAP != vk_alpha_g1_beta_g2) {
        if (!libff::inhibit_profiling_info) {
            libff::print_indent();
            printf("QAP divisibility check failed.\n");
        }
        result = false;
    }
    libff::leave_block("Check QAP divisibility");
    libff::leave_block("Online pairing computations");
 
    libff::leave_block("Call to r1cs_gg_ppzksnark_online_verifier_weak_IC");
 
    return result;
}
 
template<typename ppT>
bool r1cs_gg_ppzksnark_verifier_weak_IC(
    const r1cs_gg_ppzksnark_verification_key<ppT> &vk,
    const r1cs_gg_ppzksnark_primary_input<ppT> &primary_input,
    const r1cs_gg_ppzksnark_proof<ppT> &proof)
{
    libff::enter_block("Call to r1cs_gg_ppzksnark_verifier_weak_IC");
    r1cs_gg_ppzksnark_processed_verification_key<ppT> pvk =
        r1cs_gg_ppzksnark_verifier_process_vk<ppT>(vk);
    bool result = r1cs_gg_ppzksnark_online_verifier_weak_IC<ppT>(
        pvk, primary_input, proof);
    libff::leave_block("Call to r1cs_gg_ppzksnark_verifier_weak_IC");
    return result;
}
 
template<typename ppT>
bool r1cs_gg_ppzksnark_online_verifier_strong_IC(
    const r1cs_gg_ppzksnark_processed_verification_key<ppT> &pvk,
    const r1cs_gg_ppzksnark_primary_input<ppT> &primary_input,
    const r1cs_gg_ppzksnark_proof<ppT> &proof)
{
    bool result = true;
    libff::enter_block("Call to r1cs_gg_ppzksnark_online_verifier_strong_IC");
 
    if (pvk.ABC_g1.domain_size() != primary_input.size()) {
        libff::print_indent();
        printf(
            "Input length differs from expected (got %zu, expected %zu).\n",
            primary_input.size(),
            pvk.ABC_g1.domain_size());
        result = false;
    } else {
        result = r1cs_gg_ppzksnark_online_verifier_weak_IC(
            pvk, primary_input, proof);
    }
 
    libff::leave_block("Call to r1cs_gg_ppzksnark_online_verifier_strong_IC");
    return result;
}
 
template<typename ppT>
bool r1cs_gg_ppzksnark_verifier_strong_IC(
    const r1cs_gg_ppzksnark_verification_key<ppT> &vk,
    const r1cs_gg_ppzksnark_primary_input<ppT> &primary_input,
    const r1cs_gg_ppzksnark_proof<ppT> &proof)
{
    libff::enter_block("Call to r1cs_gg_ppzksnark_verifier_strong_IC");
    r1cs_gg_ppzksnark_processed_verification_key<ppT> pvk =
        r1cs_gg_ppzksnark_verifier_process_vk<ppT>(vk);
    bool result = r1cs_gg_ppzksnark_online_verifier_strong_IC<ppT>(
        pvk, primary_input, proof);
    libff::leave_block("Call to r1cs_gg_ppzksnark_verifier_strong_IC");
    return result;
}
 
template<typename ppT>
bool r1cs_gg_ppzksnark_affine_verifier_weak_IC(
    const r1cs_gg_ppzksnark_verification_key<ppT> &vk,
    const r1cs_gg_ppzksnark_primary_input<ppT> &primary_input,
    const r1cs_gg_ppzksnark_proof<ppT> &proof)
{
    libff::enter_block("Call to r1cs_gg_ppzksnark_affine_verifier_weak_IC");
    assert(vk.ABC_g1.domain_size() >= primary_input.size());
 
    libff::affine_ate_G2_precomp<ppT> pvk_vk_generator_g2_precomp =
        ppT::affine_ate_precompute_G2(libff::G2<ppT>::one());
    libff::affine_ate_G2_precomp<ppT> pvk_vk_delta_g2_precomp =
        ppT::affine_ate_precompute_G2(vk.delta_g2);
 
    libff::enter_block("Accumulate input");
    const accumulation_vector<libff::G1<ppT>> accumulated_IC =
        vk.ABC_g1.template accumulate_chunk<libff::Fr<ppT>>(
            primary_input.begin(), primary_input.end(), 0);
    const libff::G1<ppT> &acc = accumulated_IC.first;
    libff::leave_block("Accumulate input");
 
    bool result = true;
 
    libff::enter_block("Check if the proof is well-formed");
    if (!proof.is_well_formed()) {
        if (!libff::inhibit_profiling_info) {
            libff::print_indent();
            printf("At least one of the proof elements does not lie on the "
                   "curve.\n");
        }
        result = false;
    }
    libff::leave_block("Check if the proof is well-formed");
 
    libff::enter_block("Check QAP divisibility");
    const libff::affine_ate_G1_precomp<ppT> proof_g_A_precomp =
        ppT::affine_ate_precompute_G1(proof.g_A);
    const libff::affine_ate_G2_precomp<ppT> proof_g_B_precomp =
        ppT::affine_ate_precompute_G2(proof.g_B);
    const libff::affine_ate_G1_precomp<ppT> proof_g_C_precomp =
        ppT::affine_ate_precompute_G1(proof.g_C);
    const libff::affine_ate_G1_precomp<ppT> acc_precomp =
        ppT::affine_ate_precompute_G1(acc);
 
    const libff::Fqk<ppT> QAP_miller =
        ppT::affine_ate_e_times_e_over_e_miller_loop(
            acc_precomp,
            pvk_vk_generator_g2_precomp,
            proof_g_C_precomp,
            pvk_vk_delta_g2_precomp,
            proof_g_A_precomp,
            proof_g_B_precomp);
    const libff::GT<ppT> QAP =
        ppT::final_exponentiation(QAP_miller.unitary_inverse());
    const libff::GT<ppT> vk_alpha_g1_beta_g2 =
        ppT::reduced_pairing(vk.alpha_g1, vk.beta_g2);
 
    if (QAP != vk_alpha_g1_beta_g2) {
        if (!libff::inhibit_profiling_info) {
            libff::print_indent();
            printf("QAP divisibility check failed.\n");
        }
        result = false;
    }
    libff::leave_block("Check QAP divisibility");
 
    libff::leave_block("Call to r1cs_gg_ppzksnark_affine_verifier_weak_IC");
 
    return result;
}
 
} // namespace libsnark
#endif // R1CS_GG_PPZKSNARK_TCC_