pollard_rho.H

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/*
                          Aleph_w
 
  Data structures & Algorithms
  version 2.0.0b
  https://github.com/lrleon/Aleph-w
 
  This file is part of Aleph-w library
 
  Copyright (c) 2002-2026 Leandro Rabindranath Leon
 
  Permission is hereby granted, free of charge, to any person obtaining a copy
  of this software and associated documentation files (the "Software"), to deal
  in the Software without restriction, including without limitation the rights
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  copies of the Software, and to permit persons to whom the Software is
  furnished to do so, subject to the following conditions:
 
  The above copyright notice and this permission notice shall be included in all
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  THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
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*/
 
# ifndef POLLARD_RHO_H
# define POLLARD_RHO_H
 
# include <cstdint>
# include <numeric>
# include <random>
 
# include <ah-errors.H>
# include <tpl_array.H>
# include <ahSort.H>
# include <primality.H>
# include <modular_arithmetic.H>
 
namespace Aleph
{
 
namespace detail
{
 
[[nodiscard]] inline uint64_t pollard_rho_step(const uint64_t n, const uint64_t seed,
                                               const uint64_t c)
{
  uint64_t x = seed, y = seed, d = 1;
  auto f = [&](const uint64_t val) {
    const uint64_t v2 = mod_mul(val, val, n);
# if defined(__SIZEOF_INT128__)
    return static_cast<uint64_t>((static_cast<__uint128_t>(v2) + c) % n);
# else
    // Safe modular addition
    uint64_t c_mod = c % n;
    if (n - v2 <= c_mod) return c_mod - (n - v2);
    return v2 + c_mod;
# endif
  };
 
  // Limit iterations to prevent infinite loops on degenerate cases.
  // Pollard's rho is expected to find a factor in O(n^1/4) iterations.
  // For 64-bit integers, 1,000,000 iterations is a very safe upper bound.
  constexpr size_t max_iterations = 1000000;
  for (size_t iter = 0; iter < max_iterations and d == 1; ++iter)
    {
      x = f(x);
      y = f(f(y));
      const uint64_t diff = x > y ? x - y : y - x;
      d = std::gcd(diff, n);
    }
 
  if (d == 1 or d == n)
    return n; // Failure for this seed/c combination or iteration limit reached
 
  return d;
}
 
[[nodiscard]] inline uint64_t find_any_factor(const uint64_t n)
{
  if (n % 2 == 0) return 2;
  if (n % 3 == 0) return 3;
  if (n % 5 == 0) return 5;
 
    // Use a thread_local engine to avoid expensive re-construction and re-seeding.
    thread_local std::mt19937_64 rng([]() {
        std::random_device rd;
        return rd();
    }());
 
    constexpr size_t max_attempts = 256;
    for (size_t attempt = 0; attempt < max_attempts; ++attempt)
      {
        const uint64_t seed = rng() % (n - 2) + 2;
        const uint64_t c = rng() % (n - 1) + 1;
        if (const uint64_t d = pollard_rho_step(n, seed, c); d != n)
          return d;
      }
 
    ah_runtime_error()
      << "pollard_rho: failed to find a factor of " << n
      << " after " << max_attempts << " attempts";
 
# if defined(__GNUC__) || defined(__clang__)
    __builtin_unreachable();
# elif defined(_MSC_VER)
    __assume(0);
# endif
}
 
inline void extract_prime_factors(const uint64_t n, Array<uint64_t> & factors)
{
  if (n <= 1)
    return;
  if (miller_rabin(n))
    {
      factors.append(n);
      return;
    }
 
  const uint64_t factor = find_any_factor(n);
  extract_prime_factors(factor, factors);
  extract_prime_factors(n / factor, factors);
}
 
} // namespace detail
 
[[nodiscard]] inline Array<uint64_t> pollard_rho(uint64_t n)
{
  ah_invalid_argument_if(n < 2) << "pollard_rho: n must be >= 2";
  Array<uint64_t> factors;
  detail::extract_prime_factors(n, factors);
  in_place_sort(factors);
  return factors;
}
 
} // namespace Aleph
 
# endif // POLLARD_RHO_H