text: /***********************************************************************
   * Copyright (c) 2016 Andrew Poelstra                                 *
   * Distributed under the MIT software license, see the accompanying   *
   * file COPYING or http://www.opensource.org/licenses/mit-license.php.*
   **********************************************************************/
  
  #if defined HAVE_CONFIG_H
  #include "libsecp256k1-config.h"
  #endif
  
  #include <stdio.h>
  #include <stdlib.h>
  
  #include <time.h>
  
  #undef USE_ECMULT_STATIC_PRECOMPUTATION
  
  #ifndef EXHAUSTIVE_TEST_ORDER
  /* see group_impl.h for allowable values */
  #define EXHAUSTIVE_TEST_ORDER 13
  #define EXHAUSTIVE_TEST_LAMBDA 9   /* cube root of 1 mod 13 */
  #endif
  
  #include "secp256k1.h"
  #include "group.h"
  #include "secp256k1.c"
  #include "testrand_impl.h"
  
  #ifdef ENABLE_MODULE_RECOVERY
  #include "modules/recovery/main_impl.h"
  #include "secp256k1_recovery.h"
  #endif
  
  /** stolen from tests.c */
  void ge_equals_ge(const secp256k1_ge *a, const secp256k1_ge *b) {
      CHECK(a->infinity == b->infinity);
      if (a->infinity) {
          return;
      }
      CHECK(secp256k1_fe_equal_var(&a->x, &b->x));
      CHECK(secp256k1_fe_equal_var(&a->y, &b->y));
  }
  
  void ge_equals_gej(const secp256k1_ge *a, const secp256k1_gej *b) {
      secp256k1_fe z2s;
      secp256k1_fe u1, u2, s1, s2;
      CHECK(a->infinity == b->infinity);
      if (a->infinity) {
          return;
      }
      /* Check a.x * b.z^2 == b.x && a.y * b.z^3 == b.y, to avoid inverses. */
      secp256k1_fe_sqr(&z2s, &b->z);
      secp256k1_fe_mul(&u1, &a->x, &z2s);
      u2 = b->x; secp256k1_fe_normalize_weak(&u2);
      secp256k1_fe_mul(&s1, &a->y, &z2s); secp256k1_fe_mul(&s1, &s1, &b->z);
      s2 = b->y; secp256k1_fe_normalize_weak(&s2);
      CHECK(secp256k1_fe_equal_var(&u1, &u2));
      CHECK(secp256k1_fe_equal_var(&s1, &s2));
  }
  
  void random_fe(secp256k1_fe *x) {
      unsigned char bin[32];
      do {
          secp256k1_rand256(bin);
          if (secp256k1_fe_set_b32(x, bin)) {
              return;
          }
      } while(1);
  }
  /** END stolen from tests.c */
  
  int secp256k1_nonce_function_smallint(unsigned char *nonce32, const unsigned char *msg32,
                                        const unsigned char *key32, const unsigned char *algo16,
                                        void *data, unsigned int attempt) {
      secp256k1_scalar s;
      int *idata = data;
      (void)msg32;
      (void)key32;
      (void)algo16;
      /* Some nonces cannot be used because they'd cause s and/or r to be zero.
       * The signing function has retry logic here that just re-calls the nonce
       * function with an increased `attempt`. So if attempt > 0 this means we
       * need to change the nonce to avoid an infinite loop. */
      if (attempt > 0) {
          *idata = (*idata + 1) % EXHAUSTIVE_TEST_ORDER;
      }
      secp256k1_scalar_set_int(&s, *idata);
      secp256k1_scalar_get_b32(nonce32, &s);
      return 1;
  }
  
  #ifdef USE_ENDOMORPHISM
  void test_exhaustive_endomorphism(const secp256k1_ge *group, int order) {
      int i;
      for (i = 0; i < order; i++) {
          secp256k1_ge res;
          secp256k1_ge_mul_lambda(&res, &group[i]);
          ge_equals_ge(&group[i * EXHAUSTIVE_TEST_LAMBDA % EXHAUSTIVE_TEST_ORDER], &res);
      }
  }
  #endif
  
  void test_exhaustive_addition(const secp256k1_ge *group, const secp256k1_gej *groupj, int order) {
      int i, j;
  
      /* Sanity-check (and check infinity functions) */
      CHECK(secp256k1_ge_is_infinity(&group[0]));
      CHECK(secp256k1_gej_is_infinity(&groupj[0]));
      for (i = 1; i < order; i++) {
          CHECK(!secp256k1_ge_is_infinity(&group[i]));
          CHECK(!secp256k1_gej_is_infinity(&groupj[i]));
      }
  
      /* Check all addition formulae */
      for (j = 0; j < order; j++) {
          secp256k1_fe fe_inv;
          secp256k1_fe_inv(&fe_inv, &groupj[j].z);
          for (i = 0; i < order; i++) {
              secp256k1_ge zless_gej;
              secp256k1_gej tmp;
              /* add_var */
              secp256k1_gej_add_var(&tmp, &groupj[i], &groupj[j], NULL);
              ge_equals_gej(&group[(i + j) % order], &tmp);
              /* add_ge */
              if (j > 0) {
                  secp256k1_gej_add_ge(&tmp, &groupj[i], &group[j]);
                  ge_equals_gej(&group[(i + j) % order], &tmp);
              }
              /* add_ge_var */
              secp256k1_gej_add_ge_var(&tmp, &groupj[i], &group[j], NULL);
              ge_equals_gej(&group[(i + j) % order], &tmp);
              /* add_zinv_var */
              zless_gej.infinity = groupj[j].infinity;
              zless_gej.x = groupj[j].x;
              zless_gej.y = groupj[j].y;
              secp256k1_gej_add_zinv_var(&tmp, &groupj[i], &zless_gej, &fe_inv);
              ge_equals_gej(&group[(i + j) % order], &tmp);
          }
      }
  
      /* Check doubling */
      for (i = 0; i < order; i++) {
          secp256k1_gej tmp;
          if (i > 0) {
              secp256k1_gej_double_nonzero(&tmp, &groupj[i], NULL);
              ge_equals_gej(&group[(2 * i) % order], &tmp);
          }
          secp256k1_gej_double_var(&tmp, &groupj[i], NULL);
          ge_equals_gej(&group[(2 * i) % order], &tmp);
      }
  
      /* Check negation */
      for (i = 1; i < order; i++) {
          secp256k1_ge tmp;
          secp256k1_gej tmpj;
          secp256k1_ge_neg(&tmp, &group[i]);
          ge_equals_ge(&group[order - i], &tmp);
          secp256k1_gej_neg(&tmpj, &groupj[i]);
          ge_equals_gej(&group[order - i], &tmpj);
      }
  }
  
  void test_exhaustive_ecmult(const secp256k1_context *ctx, const secp256k1_ge *group, const secp256k1_gej *groupj, int order) {
      int i, j, r_log;
      for (r_log = 1; r_log < order; r_log++) {
          for (j = 0; j < order; j++) {
              for (i = 0; i < order; i++) {
                  secp256k1_gej tmp;
                  secp256k1_scalar na, ng;
                  secp256k1_scalar_set_int(&na, i);
                  secp256k1_scalar_set_int(&ng, j);
  
                  secp256k1_ecmult(&ctx->ecmult_ctx, &tmp, &groupj[r_log], &na, &ng);
                  ge_equals_gej(&group[(i * r_log + j) % order], &tmp);
  
                  if (i > 0) {
                      secp256k1_ecmult_const(&tmp, &group[i], &ng, 256);
                      ge_equals_gej(&group[(i * j) % order], &tmp);
                  }
              }
          }
      }
  }
  
  typedef struct {
      secp256k1_scalar sc[2];
      secp256k1_ge pt[2];
  } ecmult_multi_data;
  
  static int ecmult_multi_callback(secp256k1_scalar *sc, secp256k1_ge *pt, size_t idx, void *cbdata) {
      ecmult_multi_data *data = (ecmult_multi_data*) cbdata;
      *sc = data->sc[idx];
      *pt = data->pt[idx];
      return 1;
  }
  
  void test_exhaustive_ecmult_multi(const secp256k1_context *ctx, const secp256k1_ge *group, int order) {
      int i, j, k, x, y;
      secp256k1_scratch *scratch = secp256k1_scratch_create(&ctx->error_callback, 4096);
      for (i = 0; i < order; i++) {
          for (j = 0; j < order; j++) {
              for (k = 0; k < order; k++) {
                  for (x = 0; x < order; x++) {
                      for (y = 0; y < order; y++) {
                          secp256k1_gej tmp;
                          secp256k1_scalar g_sc;
                          ecmult_multi_data data;
  
                          secp256k1_scalar_set_int(&data.sc[0], i);
                          secp256k1_scalar_set_int(&data.sc[1], j);
                          secp256k1_scalar_set_int(&g_sc, k);
                          data.pt[0] = group[x];
                          data.pt[1] = group[y];
  
                          secp256k1_ecmult_multi_var(&ctx->ecmult_ctx, scratch, &tmp, &g_sc, ecmult_multi_callback, &data, 2);
                          ge_equals_gej(&group[(i * x + j * y + k) % order], &tmp);
                      }
                  }
              }
          }
      }
      secp256k1_scratch_destroy(scratch);
  }
  
  void r_from_k(secp256k1_scalar *r, const secp256k1_ge *group, int k) {
      secp256k1_fe x;
      unsigned char x_bin[32];
      k %= EXHAUSTIVE_TEST_ORDER;
      x = group[k].x;
      secp256k1_fe_normalize(&x);
      secp256k1_fe_get_b32(x_bin, &x);
      secp256k1_scalar_set_b32(r, x_bin, NULL);
  }
  
  void test_exhaustive_verify(const secp256k1_context *ctx, const secp256k1_ge *group, int order) {
      int s, r, msg, key;
      for (s = 1; s < order; s++) {
          for (r = 1; r < order; r++) {
              for (msg = 1; msg < order; msg++) {
                  for (key = 1; key < order; key++) {
                      secp256k1_ge nonconst_ge;
                      secp256k1_ecdsa_signature sig;
                      secp256k1_pubkey pk;
                      secp256k1_scalar sk_s, msg_s, r_s, s_s;
                      secp256k1_scalar s_times_k_s, msg_plus_r_times_sk_s;
                      int k, should_verify;
                      unsigned char msg32[32];
  
                      secp256k1_scalar_set_int(&s_s, s);
                      secp256k1_scalar_set_int(&r_s, r);
                      secp256k1_scalar_set_int(&msg_s, msg);
                      secp256k1_scalar_set_int(&sk_s, key);
  
                      /* Verify by hand */
                      /* Run through every k value that gives us this r and check that *one* works.
                       * Note there could be none, there could be multiple, ECDSA is weird. */
                      should_verify = 0;
                      for (k = 0; k < order; k++) {
                          secp256k1_scalar check_x_s;
                          r_from_k(&check_x_s, group, k);
                          if (r_s == check_x_s) {
                              secp256k1_scalar_set_int(&s_times_k_s, k);
                              secp256k1_scalar_mul(&s_times_k_s, &s_times_k_s, &s_s);
                              secp256k1_scalar_mul(&msg_plus_r_times_sk_s, &r_s, &sk_s);
                              secp256k1_scalar_add(&msg_plus_r_times_sk_s, &msg_plus_r_times_sk_s, &msg_s);
                              should_verify |= secp256k1_scalar_eq(&s_times_k_s, &msg_plus_r_times_sk_s);
                          }
                      }
                      /* nb we have a "high s" rule */
                      should_verify &= !secp256k1_scalar_is_high(&s_s);
  
                      /* Verify by calling verify */
                      secp256k1_ecdsa_signature_save(&sig, &r_s, &s_s);
                      memcpy(&nonconst_ge, &group[sk_s], sizeof(nonconst_ge));
                      secp256k1_pubkey_save(&pk, &nonconst_ge);
                      secp256k1_scalar_get_b32(msg32, &msg_s);
                      CHECK(should_verify ==
                            secp256k1_ecdsa_verify(ctx, &sig, msg32, &pk));
                  }
              }
          }
      }
  }
  
  void test_exhaustive_sign(const secp256k1_context *ctx, const secp256k1_ge *group, int order) {
      int i, j, k;
  
      /* Loop */
      for (i = 1; i < order; i++) {  /* message */
          for (j = 1; j < order; j++) {  /* key */
              for (k = 1; k < order; k++) {  /* nonce */
                  const int starting_k = k;
                  secp256k1_ecdsa_signature sig;
                  secp256k1_scalar sk, msg, r, s, expected_r;
                  unsigned char sk32[32], msg32[32];
                  secp256k1_scalar_set_int(&msg, i);
                  secp256k1_scalar_set_int(&sk, j);
                  secp256k1_scalar_get_b32(sk32, &sk);
                  secp256k1_scalar_get_b32(msg32, &msg);
  
                  secp256k1_ecdsa_sign(ctx, &sig, msg32, sk32, secp256k1_nonce_function_smallint, &k);
  
                  secp256k1_ecdsa_signature_load(ctx, &r, &s, &sig);
                  /* Note that we compute expected_r *after* signing -- this is important
                   * because our nonce-computing function function might change k during
                   * signing. */
                  r_from_k(&expected_r, group, k);
                  CHECK(r == expected_r);
                  CHECK((k * s) % order == (i + r * j) % order ||
                        (k * (EXHAUSTIVE_TEST_ORDER - s)) % order == (i + r * j) % order);
  
                  /* Overflow means we've tried every possible nonce */
                  if (k < starting_k) {
                      break;
                  }
              }
          }
      }
  
      /* We would like to verify zero-knowledge here by counting how often every
       * possible (s, r) tuple appears, but because the group order is larger
       * than the field order, when coercing the x-values to scalar values, some
       * appear more often than others, so we are actually not zero-knowledge.
       * (This effect also appears in the real code, but the difference is on the
       * order of 1/2^128th the field order, so the deviation is not useful to a
       * computationally bounded attacker.)
       */
  }
  
  #ifdef ENABLE_MODULE_RECOVERY
  void test_exhaustive_recovery_sign(const secp256k1_context *ctx, const secp256k1_ge *group, int order) {
      int i, j, k;
  
      /* Loop */
      for (i = 1; i < order; i++) {  /* message */
          for (j = 1; j < order; j++) {  /* key */
              for (k = 1; k < order; k++) {  /* nonce */
                  const int starting_k = k;
                  secp256k1_fe r_dot_y_normalized;
                  secp256k1_ecdsa_recoverable_signature rsig;
                  secp256k1_ecdsa_signature sig;
                  secp256k1_scalar sk, msg, r, s, expected_r;
                  unsigned char sk32[32], msg32[32];
                  int expected_recid;
                  int recid;
                  secp256k1_scalar_set_int(&msg, i);
                  secp256k1_scalar_set_int(&sk, j);
                  secp256k1_scalar_get_b32(sk32, &sk);
                  secp256k1_scalar_get_b32(msg32, &msg);
  
                  secp256k1_ecdsa_sign_recoverable(ctx, &rsig, msg32, sk32, secp256k1_nonce_function_smallint, &k);
  
                  /* Check directly */
                  secp256k1_ecdsa_recoverable_signature_load(ctx, &r, &s, &recid, &rsig);
                  r_from_k(&expected_r, group, k);
                  CHECK(r == expected_r);
                  CHECK((k * s) % order == (i + r * j) % order ||
                        (k * (EXHAUSTIVE_TEST_ORDER - s)) % order == (i + r * j) % order);
                  /* In computing the recid, there is an overflow condition that is disabled in
                   * scalar_low_impl.h `secp256k1_scalar_set_b32` because almost every r.y value
                   * will exceed the group order, and our signing code always holds out for r
                   * values that don't overflow, so with a proper overflow check the tests would
                   * loop indefinitely. */
                  r_dot_y_normalized = group[k].y;
                  secp256k1_fe_normalize(&r_dot_y_normalized);
                  /* Also the recovery id is flipped depending if we hit the low-s branch */
                  if ((k * s) % order == (i + r * j) % order) {
                      expected_recid = secp256k1_fe_is_odd(&r_dot_y_normalized) ? 1 : 0;
                  } else {
                      expected_recid = secp256k1_fe_is_odd(&r_dot_y_normalized) ? 0 : 1;
                  }
                  CHECK(recid == expected_recid);
  
                  /* Convert to a standard sig then check */
                  secp256k1_ecdsa_recoverable_signature_convert(ctx, &sig, &rsig);
                  secp256k1_ecdsa_signature_load(ctx, &r, &s, &sig);
                  /* Note that we compute expected_r *after* signing -- this is important
                   * because our nonce-computing function function might change k during
                   * signing. */
                  r_from_k(&expected_r, group, k);
                  CHECK(r == expected_r);
                  CHECK((k * s) % order == (i + r * j) % order ||
                        (k * (EXHAUSTIVE_TEST_ORDER - s)) % order == (i + r * j) % order);
  
                  /* Overflow means we've tried every possible nonce */
                  if (k < starting_k) {
                      break;
                  }
              }
          }
      }
  }
  
  void test_exhaustive_recovery_verify(const secp256k1_context *ctx, const secp256k1_ge *group, int order) {
      /* This is essentially a copy of test_exhaustive_verify, with recovery added */
      int s, r, msg, key;
      for (s = 1; s < order; s++) {
          for (r = 1; r < order; r++) {
              for (msg = 1; msg < order; msg++) {
                  for (key = 1; key < order; key++) {
                      secp256k1_ge nonconst_ge;
                      secp256k1_ecdsa_recoverable_signature rsig;
                      secp256k1_ecdsa_signature sig;
                      secp256k1_pubkey pk;
                      secp256k1_scalar sk_s, msg_s, r_s, s_s;
                      secp256k1_scalar s_times_k_s, msg_plus_r_times_sk_s;
                      int recid = 0;
                      int k, should_verify;
                      unsigned char msg32[32];
  
                      secp256k1_scalar_set_int(&s_s, s);
                      secp256k1_scalar_set_int(&r_s, r);
                      secp256k1_scalar_set_int(&msg_s, msg);
                      secp256k1_scalar_set_int(&sk_s, key);
                      secp256k1_scalar_get_b32(msg32, &msg_s);
  
                      /* Verify by hand */
                      /* Run through every k value that gives us this r and check that *one* works.
                       * Note there could be none, there could be multiple, ECDSA is weird. */
                      should_verify = 0;
                      for (k = 0; k < order; k++) {
                          secp256k1_scalar check_x_s;
                          r_from_k(&check_x_s, group, k);
                          if (r_s == check_x_s) {
                              secp256k1_scalar_set_int(&s_times_k_s, k);
                              secp256k1_scalar_mul(&s_times_k_s, &s_times_k_s, &s_s);
                              secp256k1_scalar_mul(&msg_plus_r_times_sk_s, &r_s, &sk_s);
                              secp256k1_scalar_add(&msg_plus_r_times_sk_s, &msg_plus_r_times_sk_s, &msg_s);
                              should_verify |= secp256k1_scalar_eq(&s_times_k_s, &msg_plus_r_times_sk_s);
                          }
                      }
                      /* nb we have a "high s" rule */
                      should_verify &= !secp256k1_scalar_is_high(&s_s);
  
                      /* We would like to try recovering the pubkey and checking that it matches,
                       * but pubkey recovery is impossible in the exhaustive tests (the reason
                       * being that there are 12 nonzero r values, 12 nonzero points, and no
                       * overlap between the sets, so there are no valid signatures). */
  
                      /* Verify by converting to a standard signature and calling verify */
                      secp256k1_ecdsa_recoverable_signature_save(&rsig, &r_s, &s_s, recid);
                      secp256k1_ecdsa_recoverable_signature_convert(ctx, &sig, &rsig);
                      memcpy(&nonconst_ge, &group[sk_s], sizeof(nonconst_ge));
                      secp256k1_pubkey_save(&pk, &nonconst_ge);
                      CHECK(should_verify ==
                            secp256k1_ecdsa_verify(ctx, &sig, msg32, &pk));
                  }
              }
          }
      }
  }
  #endif
  
  int main(void) {
      int i;
      secp256k1_gej groupj[EXHAUSTIVE_TEST_ORDER];
      secp256k1_ge group[EXHAUSTIVE_TEST_ORDER];
  
      /* Build context */
      secp256k1_context *ctx = secp256k1_context_create(SECP256K1_CONTEXT_SIGN | SECP256K1_CONTEXT_VERIFY);
  
      /* TODO set z = 1, then do num_tests runs with random z values */
  
      /* Generate the entire group */
      secp256k1_gej_set_infinity(&groupj[0]);
      secp256k1_ge_set_gej(&group[0], &groupj[0]);
      for (i = 1; i < EXHAUSTIVE_TEST_ORDER; i++) {
          /* Set a different random z-value for each Jacobian point */
          secp256k1_fe z;
          random_fe(&z);
  
          secp256k1_gej_add_ge(&groupj[i], &groupj[i - 1], &secp256k1_ge_const_g);
          secp256k1_ge_set_gej(&group[i], &groupj[i]);
          secp256k1_gej_rescale(&groupj[i], &z);
  
          /* Verify against ecmult_gen */
          {
              secp256k1_scalar scalar_i;
              secp256k1_gej generatedj;
              secp256k1_ge generated;
  
              secp256k1_scalar_set_int(&scalar_i, i);
              secp256k1_ecmult_gen(&ctx->ecmult_gen_ctx, &generatedj, &scalar_i);
              secp256k1_ge_set_gej(&generated, &generatedj);
  
              CHECK(group[i].infinity == 0);
              CHECK(generated.infinity == 0);
              CHECK(secp256k1_fe_equal_var(&generated.x, &group[i].x));
              CHECK(secp256k1_fe_equal_var(&generated.y, &group[i].y));
          }
      }
  
      /* Run the tests */
  #ifdef USE_ENDOMORPHISM
      test_exhaustive_endomorphism(group, EXHAUSTIVE_TEST_ORDER);
  #endif
      test_exhaustive_addition(group, groupj, EXHAUSTIVE_TEST_ORDER);
      test_exhaustive_ecmult(ctx, group, groupj, EXHAUSTIVE_TEST_ORDER);
      test_exhaustive_ecmult_multi(ctx, group, EXHAUSTIVE_TEST_ORDER);
      test_exhaustive_sign(ctx, group, EXHAUSTIVE_TEST_ORDER);
      test_exhaustive_verify(ctx, group, EXHAUSTIVE_TEST_ORDER);
  
  #ifdef ENABLE_MODULE_RECOVERY
      test_exhaustive_recovery_sign(ctx, group, EXHAUSTIVE_TEST_ORDER);
      test_exhaustive_recovery_verify(ctx, group, EXHAUSTIVE_TEST_ORDER);
  #endif
  
      secp256k1_context_destroy(ctx);
      return 0;
  }
  
  