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
  to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
  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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  SOFTWARE.
*/
 
# 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__) && !defined(_WIN32)
    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