text: /* boost random/detail/polynomial.hpp header file
   *
   * Copyright Steven Watanabe 2014
   * Distributed under the Boost Software License, Version 1.0. (See
   * accompanying file LICENSE_1_0.txt or copy at
   * http://www.boost.org/LICENSE_1_0.txt)
   *
   * See http://www.boost.org for most recent version including documentation.
   *
   * $Id$
   */
  
  #ifndef BOOST_RANDOM_DETAIL_POLYNOMIAL_HPP
  #define BOOST_RANDOM_DETAIL_POLYNOMIAL_HPP
  
  #include <cstddef>
  #include <limits>
  #include <vector>
  #include <algorithm>
  #include <boost/assert.hpp>
  #include <boost/cstdint.hpp>
  
  namespace lslboost {
  namespace random {
  namespace detail {
  
  class polynomial_ops {
  public:
      typedef unsigned long digit_t;
  
      static void add(std::size_t size, const digit_t * lhs,
                         const digit_t * rhs, digit_t * output)
      {
          for(std::size_t i = 0; i < size; ++i) {
              output[i] = lhs[i] ^ rhs[i];
          }
      }
  
      static void add_shifted_inplace(std::size_t size, const digit_t * lhs,
                                      digit_t * output, std::size_t shift)
      {
          if(shift == 0) {
              add(size, lhs, output, output);
              return;
          }
          std::size_t bits = std::numeric_limits<digit_t>::digits;
          digit_t prev = 0;
          for(std::size_t i = 0; i < size; ++i) {
              digit_t tmp = lhs[i];
              output[i] ^= (tmp << shift) | (prev >> (bits-shift));
              prev = tmp;
          }
          output[size] ^= (prev >> (bits-shift));
      }
  
      static void multiply_simple(std::size_t size, const digit_t * lhs,
                                     const digit_t * rhs, digit_t * output)
      {
          std::size_t bits = std::numeric_limits<digit_t>::digits;
          for(std::size_t i = 0; i < 2*size; ++i) {
              output[i] = 0;
          }
          for(std::size_t i = 0; i < size; ++i) {
              for(std::size_t j = 0; j < bits; ++j) {
                  if((lhs[i] & (digit_t(1) << j)) != 0) {
                      add_shifted_inplace(size, rhs, output + i, j);
                  }
              }
          }
      }
  
      // memory requirements: (size - cutoff) * 4 + next_smaller
      static void multiply_karatsuba(std::size_t size,
                                 const digit_t * lhs, const digit_t * rhs,
                                 digit_t * output)
      {
          if(size < 64) {
              multiply_simple(size, lhs, rhs, output);
              return;
          }
          // split in half
          std::size_t cutoff = size/2;
          multiply_karatsuba(cutoff, lhs, rhs, output);
          multiply_karatsuba(size - cutoff, lhs + cutoff, rhs + cutoff,
                                output + cutoff*2);
          std::vector<digit_t> local1(size - cutoff);
          std::vector<digit_t> local2(size - cutoff);
          // combine the digits for the inner multiply
          add(cutoff, lhs, lhs + cutoff, &local1[0]);
          if(size & 1) local1[cutoff] = lhs[size - 1];
          add(cutoff, rhs + cutoff, rhs, &local2[0]);
          if(size & 1) local2[cutoff] = rhs[size - 1];
          std::vector<digit_t> local3((size - cutoff) * 2);
          multiply_karatsuba(size - cutoff, &local1[0], &local2[0], &local3[0]);
          add(cutoff * 2, output, &local3[0], &local3[0]);
          add((size - cutoff) * 2, output + cutoff*2, &local3[0], &local3[0]);
          // Finally, add the inner result
          add((size - cutoff) * 2, output + cutoff, &local3[0], output + cutoff);
      }
      
      static void multiply_add_karatsuba(std::size_t size,
                                         const digit_t * lhs, const digit_t * rhs,
                                         digit_t * output)
      {
          std::vector<digit_t> buf(size * 2);
          multiply_karatsuba(size, lhs, rhs, &buf[0]);
          add(size * 2, &buf[0], output, output);
      }
  
      static void multiply(const digit_t * lhs, std::size_t lhs_size,
                           const digit_t * rhs, std::size_t rhs_size,
                           digit_t * output)
      {
          std::fill_n(output, lhs_size + rhs_size, digit_t(0));
          multiply_add(lhs, lhs_size, rhs, rhs_size, output);
      }
  
      static void multiply_add(const digit_t * lhs, std::size_t lhs_size,
                               const digit_t * rhs, std::size_t rhs_size,
                               digit_t * output)
      {
          // split into pieces that can be passed to
          // karatsuba multiply.
          while(lhs_size != 0) {
              if(lhs_size < rhs_size) {
                  std::swap(lhs, rhs);
                  std::swap(lhs_size, rhs_size);
              }
              
              multiply_add_karatsuba(rhs_size, lhs, rhs, output);
              
              lhs += rhs_size;
              lhs_size -= rhs_size;
              output += rhs_size;
          }
      }
  
      static void copy_bits(const digit_t * x, std::size_t low, std::size_t high,
                     digit_t * out)
      {
          const std::size_t bits = std::numeric_limits<digit_t>::digits;
          std::size_t offset = low/bits;
          x += offset;
          low -= offset*bits;
          high -= offset*bits;
          std::size_t n = (high-low)/bits;
          if(low == 0) {
              for(std::size_t i = 0; i < n; ++i) {
                  out[i] = x[i];
              }
          } else {
              for(std::size_t i = 0; i < n; ++i) {
                  out[i] = (x[i] >> low) | (x[i+1] << (bits-low));
              }
          }
          if((high-low)%bits) {
              digit_t low_mask = (digit_t(1) << ((high-low)%bits)) - 1;
              digit_t result = (x[n] >> low);
              if(low != 0 && (n+1)*bits < high) {
                  result |= (x[n+1] << (bits-low));
              }
              out[n] = (result & low_mask);
          }
      }
  
      static void shift_left(digit_t * val, std::size_t size, std::size_t shift)
      {
          const std::size_t bits = std::numeric_limits<digit_t>::digits;
          BOOST_ASSERT(shift > 0);
          BOOST_ASSERT(shift < bits);
          digit_t prev = 0;
          for(std::size_t i = 0; i < size; ++i) {
              digit_t tmp = val[i];
              val[i] = (prev >> (bits - shift)) | (val[i] << shift);
              prev = tmp;
          }
      }
  
      static digit_t sqr(digit_t val) {
          const std::size_t bits = std::numeric_limits<digit_t>::digits;
          digit_t mask = (digit_t(1) << bits/2) - 1;
          for(std::size_t i = bits; i > 1; i /= 2) {
              val = ((val & ~mask) << i/2) | (val & mask);
              mask = mask & (mask >> i/4);
              mask = mask | (mask << i/2);
          }
          return val;
      }
  
      static void sqr(digit_t * val, std::size_t size)
      {
          const std::size_t bits = std::numeric_limits<digit_t>::digits;
          digit_t mask = (digit_t(1) << bits/2) - 1;
          for(std::size_t i = 0; i < size; ++i) {
              digit_t x = val[size - i - 1];
              val[(size - i - 1) * 2] = sqr(x & mask);
              val[(size - i - 1) * 2 + 1] = sqr(x >> bits/2);
          }
      }
  
      // optimized for the case when the modulus has few bits set.
      struct sparse_mod {
          sparse_mod(const digit_t * divisor, std::size_t divisor_bits)
          {
              const std::size_t bits = std::numeric_limits<digit_t>::digits;
              _remainder_bits = divisor_bits - 1;
              for(std::size_t i = 0; i < divisor_bits; ++i) {
                  if(divisor[i/bits] & (digit_t(1) << i%bits)) {
                      _bit_indices.push_back(i);
                  }
              }
              BOOST_ASSERT(_bit_indices.back() == divisor_bits - 1);
              _bit_indices.pop_back();
              if(_bit_indices.empty()) {
                  _block_bits = divisor_bits;
                  _lower_bits = 0;
              } else {
                  _block_bits = divisor_bits - _bit_indices.back() - 1;
                  _lower_bits = _bit_indices.back() + 1;
              }
              
              _partial_quotient.resize((_block_bits + bits - 1)/bits);
          }
          void operator()(digit_t * dividend, std::size_t dividend_bits)
          {
              const std::size_t bits = std::numeric_limits<digit_t>::digits;
              while(dividend_bits > _remainder_bits) {
                  std::size_t block_start = (std::max)(dividend_bits - _block_bits, _remainder_bits);
                  std::size_t block_size = (dividend_bits - block_start + bits - 1) / bits;
                  copy_bits(dividend, block_start, dividend_bits, &_partial_quotient[0]);
                  for(std::size_t i = 0; i < _bit_indices.size(); ++i) {
                      std::size_t pos = _bit_indices[i] + block_start - _remainder_bits;
                      add_shifted_inplace(block_size, &_partial_quotient[0], dividend + pos/bits, pos%bits);
                  }
                  add_shifted_inplace(block_size, &_partial_quotient[0], dividend + block_start/bits, block_start%bits);
                  dividend_bits = block_start;
              }
          }
          std::vector<digit_t> _partial_quotient;
          std::size_t _remainder_bits;
          std::size_t _block_bits;
          std::size_t _lower_bits;
          std::vector<std::size_t> _bit_indices;
      };
  
      // base should have the same number of bits as mod
      // base, and mod should both be able to hold a power
      // of 2 >= mod_bits.  out needs to be twice as large.
      static void mod_pow_x(lslboost::uintmax_t exponent, const digit_t * mod, std::size_t mod_bits, digit_t * out)
      {
          const std::size_t bits = std::numeric_limits<digit_t>::digits;
          const std::size_t n = (mod_bits + bits - 1) / bits;
          const std::size_t highbit = mod_bits - 1;
          if(exponent == 0) {
              out[0] = 1;
              std::fill_n(out + 1, n - 1, digit_t(0));
              return;
          }
          lslboost::uintmax_t i = std::numeric_limits<lslboost::uintmax_t>::digits - 1;
          while(((lslboost::uintmax_t(1) << i) & exponent) == 0) {
              --i;
          }
          out[0] = 2;
          std::fill_n(out + 1, n - 1, digit_t(0));
          sparse_mod m(mod, mod_bits);
          while(i--) {
              sqr(out, n);
              m(out, 2 * mod_bits - 1);
              if((lslboost::uintmax_t(1) << i) & exponent) {
                  shift_left(out, n, 1);
                  if(out[highbit / bits] & (digit_t(1) << highbit%bits))
                      add(n, out, mod, out);
              }
          }
      }
  };
  
  class polynomial
  {
      typedef polynomial_ops::digit_t digit_t;
  public:
      polynomial() : _size(0) {}
      class reference {
      public:
          reference(digit_t &value, int idx)
              : _value(value), _idx(idx) {}
          operator bool() const { return (_value & (digit_t(1) << _idx)) != 0; }
          reference& operator=(bool b)
          {
              if(b) {
                  _value |= (digit_t(1) << _idx);
              } else {
                  _value &= ~(digit_t(1) << _idx);
              }
              return *this;
          }
          reference &operator^=(bool b)
          {
              _value ^= (digit_t(b) << _idx);
              return *this;
          }
  
          reference &operator=(const reference &other)
          {
              return *this = static_cast<bool>(other);
          }
      private:
          digit_t &_value;
          int _idx;
      };
      reference operator[](std::size_t i)
      {
          static const std::size_t bits = std::numeric_limits<digit_t>::digits;
          ensure_bit(i);
          return reference(_storage[i/bits], i%bits);
      }
      bool operator[](std::size_t i) const
      {
          static const std::size_t bits = std::numeric_limits<digit_t>::digits;
          if(i < size())
              return (_storage[i/bits] & (digit_t(1) << (i%bits))) != 0;
          else
              return false;
      }
      std::size_t size() const
      {
          return _size;
      }
      void resize(std::size_t n)
      {
          static const std::size_t bits = std::numeric_limits<digit_t>::digits;
          _storage.resize((n + bits - 1)/bits);
          // clear the high order bits in case we're shrinking.
          if(n%bits) {
              _storage.back() &= ((digit_t(1) << (n%bits)) - 1);
          }
          _size = n;
      }
      friend polynomial operator*(const polynomial &lhs, const polynomial &rhs);
      friend polynomial mod_pow_x(lslboost::uintmax_t exponent, polynomial mod);
  private:
      std::vector<polynomial_ops::digit_t> _storage;
      std::size_t _size;
      void ensure_bit(std::size_t i)
      {
          if(i >= size()) {
              resize(i + 1);
          }
      }
      void normalize()
      {
          while(size() && (*this)[size() - 1] == 0)
              resize(size() - 1);
      }
  };
  
  inline polynomial operator*(const polynomial &lhs, const polynomial &rhs)
  {
      polynomial result;
      result._storage.resize(lhs._storage.size() + rhs._storage.size());
      polynomial_ops::multiply(&lhs._storage[0], lhs._storage.size(),
                               &rhs._storage[0], rhs._storage.size(),
                               &result._storage[0]);
      result._size = lhs._size + rhs._size;
      return result;
  }
  
  inline polynomial mod_pow_x(lslboost::uintmax_t exponent, polynomial mod)
  {
      polynomial result;
      mod.normalize();
      std::size_t mod_size = mod.size();
      result._storage.resize(mod._storage.size() * 2);
      result._size = mod.size() * 2;
      polynomial_ops::mod_pow_x(exponent, &mod._storage[0], mod_size, &result._storage[0]);
      result.resize(mod.size() - 1);
      return result;
  }
  
  }
  }
  }
  
  #endif // BOOST_RANDOM_DETAIL_POLYNOMIAL_HPP
  