Chris@102: /* boost random/detail/polynomial.hpp header file
Chris@102:  *
Chris@102:  * Copyright Steven Watanabe 2014
Chris@102:  * Distributed under the Boost Software License, Version 1.0. (See
Chris@102:  * accompanying file LICENSE_1_0.txt or copy at
Chris@102:  * http://www.boost.org/LICENSE_1_0.txt)
Chris@102:  *
Chris@102:  * See http://www.boost.org for most recent version including documentation.
Chris@102:  *
Chris@102:  * $Id$
Chris@102:  */
Chris@102: 
Chris@102: #ifndef BOOST_RANDOM_DETAIL_POLYNOMIAL_HPP
Chris@102: #define BOOST_RANDOM_DETAIL_POLYNOMIAL_HPP
Chris@102: 
Chris@102: #include <cstddef>
Chris@102: #include <limits>
Chris@102: #include <vector>
Chris@102: #include <algorithm>
Chris@102: #include <boost/assert.hpp>
Chris@102: #include <boost/cstdint.hpp>
Chris@102: 
Chris@102: namespace boost {
Chris@102: namespace random {
Chris@102: namespace detail {
Chris@102: 
Chris@102: class polynomial_ops {
Chris@102: public:
Chris@102:     typedef unsigned long digit_t;
Chris@102: 
Chris@102:     static void add(std::size_t size, const digit_t * lhs,
Chris@102:                        const digit_t * rhs, digit_t * output)
Chris@102:     {
Chris@102:         for(std::size_t i = 0; i < size; ++i) {
Chris@102:             output[i] = lhs[i] ^ rhs[i];
Chris@102:         }
Chris@102:     }
Chris@102: 
Chris@102:     static void add_shifted_inplace(std::size_t size, const digit_t * lhs,
Chris@102:                                     digit_t * output, std::size_t shift)
Chris@102:     {
Chris@102:         if(shift == 0) {
Chris@102:             add(size, lhs, output, output);
Chris@102:             return;
Chris@102:         }
Chris@102:         std::size_t bits = std::numeric_limits<digit_t>::digits;
Chris@102:         digit_t prev = 0;
Chris@102:         for(std::size_t i = 0; i < size; ++i) {
Chris@102:             digit_t tmp = lhs[i];
Chris@102:             output[i] ^= (tmp << shift) | (prev >> (bits-shift));
Chris@102:             prev = tmp;
Chris@102:         }
Chris@102:         output[size] ^= (prev >> (bits-shift));
Chris@102:     }
Chris@102: 
Chris@102:     static void multiply_simple(std::size_t size, const digit_t * lhs,
Chris@102:                                    const digit_t * rhs, digit_t * output)
Chris@102:     {
Chris@102:         std::size_t bits = std::numeric_limits<digit_t>::digits;
Chris@102:         for(std::size_t i = 0; i < 2*size; ++i) {
Chris@102:             output[i] = 0;
Chris@102:         }
Chris@102:         for(std::size_t i = 0; i < size; ++i) {
Chris@102:             for(std::size_t j = 0; j < bits; ++j) {
Chris@102:                 if((lhs[i] & (digit_t(1) << j)) != 0) {
Chris@102:                     add_shifted_inplace(size, rhs, output + i, j);
Chris@102:                 }
Chris@102:             }
Chris@102:         }
Chris@102:     }
Chris@102: 
Chris@102:     // memory requirements: (size - cutoff) * 4 + next_smaller
Chris@102:     static void multiply_karatsuba(std::size_t size,
Chris@102:                                const digit_t * lhs, const digit_t * rhs,
Chris@102:                                digit_t * output)
Chris@102:     {
Chris@102:         if(size < 64) {
Chris@102:             multiply_simple(size, lhs, rhs, output);
Chris@102:             return;
Chris@102:         }
Chris@102:         // split in half
Chris@102:         std::size_t cutoff = size/2;
Chris@102:         multiply_karatsuba(cutoff, lhs, rhs, output);
Chris@102:         multiply_karatsuba(size - cutoff, lhs + cutoff, rhs + cutoff,
Chris@102:                               output + cutoff*2);
Chris@102:         std::vector<digit_t> local1(size - cutoff);
Chris@102:         std::vector<digit_t> local2(size - cutoff);
Chris@102:         // combine the digits for the inner multiply
Chris@102:         add(cutoff, lhs, lhs + cutoff, &local1[0]);
Chris@102:         if(size & 1) local1[cutoff] = lhs[size - 1];
Chris@102:         add(cutoff, rhs + cutoff, rhs, &local2[0]);
Chris@102:         if(size & 1) local2[cutoff] = rhs[size - 1];
Chris@102:         std::vector<digit_t> local3((size - cutoff) * 2);
Chris@102:         multiply_karatsuba(size - cutoff, &local1[0], &local2[0], &local3[0]);
Chris@102:         add(cutoff * 2, output, &local3[0], &local3[0]);
Chris@102:         add((size - cutoff) * 2, output + cutoff*2, &local3[0], &local3[0]);
Chris@102:         // Finally, add the inner result
Chris@102:         add((size - cutoff) * 2, output + cutoff, &local3[0], output + cutoff);
Chris@102:     }
Chris@102:     
Chris@102:     static void multiply_add_karatsuba(std::size_t size,
Chris@102:                                        const digit_t * lhs, const digit_t * rhs,
Chris@102:                                        digit_t * output)
Chris@102:     {
Chris@102:         std::vector<digit_t> buf(size * 2);
Chris@102:         multiply_karatsuba(size, lhs, rhs, &buf[0]);
Chris@102:         add(size * 2, &buf[0], output, output);
Chris@102:     }
Chris@102: 
Chris@102:     static void multiply(const digit_t * lhs, std::size_t lhs_size,
Chris@102:                          const digit_t * rhs, std::size_t rhs_size,
Chris@102:                          digit_t * output)
Chris@102:     {
Chris@102:         std::fill_n(output, lhs_size + rhs_size, digit_t(0));
Chris@102:         multiply_add(lhs, lhs_size, rhs, rhs_size, output);
Chris@102:     }
Chris@102: 
Chris@102:     static void multiply_add(const digit_t * lhs, std::size_t lhs_size,
Chris@102:                              const digit_t * rhs, std::size_t rhs_size,
Chris@102:                              digit_t * output)
Chris@102:     {
Chris@102:         // split into pieces that can be passed to
Chris@102:         // karatsuba multiply.
Chris@102:         while(lhs_size != 0) {
Chris@102:             if(lhs_size < rhs_size) {
Chris@102:                 std::swap(lhs, rhs);
Chris@102:                 std::swap(lhs_size, rhs_size);
Chris@102:             }
Chris@102:             
Chris@102:             multiply_add_karatsuba(rhs_size, lhs, rhs, output);
Chris@102:             
Chris@102:             lhs += rhs_size;
Chris@102:             lhs_size -= rhs_size;
Chris@102:             output += rhs_size;
Chris@102:         }
Chris@102:     }
Chris@102: 
Chris@102:     static void copy_bits(const digit_t * x, std::size_t low, std::size_t high,
Chris@102:                    digit_t * out)
Chris@102:     {
Chris@102:         const std::size_t bits = std::numeric_limits<digit_t>::digits;
Chris@102:         std::size_t offset = low/bits;
Chris@102:         x += offset;
Chris@102:         low -= offset*bits;
Chris@102:         high -= offset*bits;
Chris@102:         std::size_t n = (high-low)/bits;
Chris@102:         if(low == 0) {
Chris@102:             for(std::size_t i = 0; i < n; ++i) {
Chris@102:                 out[i] = x[i];
Chris@102:             }
Chris@102:         } else {
Chris@102:             for(std::size_t i = 0; i < n; ++i) {
Chris@102:                 out[i] = (x[i] >> low) | (x[i+1] << (bits-low));
Chris@102:             }
Chris@102:         }
Chris@102:         if((high-low)%bits) {
Chris@102:             digit_t low_mask = (digit_t(1) << ((high-low)%bits)) - 1;
Chris@102:             digit_t result = (x[n] >> low);
Chris@102:             if(low != 0 && (n+1)*bits < high) {
Chris@102:                 result |= (x[n+1] << (bits-low));
Chris@102:             }
Chris@102:             out[n] = (result & low_mask);
Chris@102:         }
Chris@102:     }
Chris@102: 
Chris@102:     static void shift_left(digit_t * val, std::size_t size, std::size_t shift)
Chris@102:     {
Chris@102:         const std::size_t bits = std::numeric_limits<digit_t>::digits;
Chris@102:         BOOST_ASSERT(shift > 0);
Chris@102:         BOOST_ASSERT(shift < bits);
Chris@102:         digit_t prev = 0;
Chris@102:         for(std::size_t i = 0; i < size; ++i) {
Chris@102:             digit_t tmp = val[i];
Chris@102:             val[i] = (prev >> (bits - shift)) | (val[i] << shift);
Chris@102:             prev = tmp;
Chris@102:         }
Chris@102:     }
Chris@102: 
Chris@102:     static digit_t sqr(digit_t val) {
Chris@102:         const std::size_t bits = std::numeric_limits<digit_t>::digits;
Chris@102:         digit_t mask = (digit_t(1) << bits/2) - 1;
Chris@102:         for(std::size_t i = bits; i > 1; i /= 2) {
Chris@102:             val = ((val & ~mask) << i/2) | (val & mask);
Chris@102:             mask = mask & (mask >> i/4);
Chris@102:             mask = mask | (mask << i/2);
Chris@102:         }
Chris@102:         return val;
Chris@102:     }
Chris@102: 
Chris@102:     static void sqr(digit_t * val, std::size_t size)
Chris@102:     {
Chris@102:         const std::size_t bits = std::numeric_limits<digit_t>::digits;
Chris@102:         digit_t mask = (digit_t(1) << bits/2) - 1;
Chris@102:         for(std::size_t i = 0; i < size; ++i) {
Chris@102:             digit_t x = val[size - i - 1];
Chris@102:             val[(size - i - 1) * 2] = sqr(x & mask);
Chris@102:             val[(size - i - 1) * 2 + 1] = sqr(x >> bits/2);
Chris@102:         }
Chris@102:     }
Chris@102: 
Chris@102:     // optimized for the case when the modulus has few bits set.
Chris@102:     struct sparse_mod {
Chris@102:         sparse_mod(const digit_t * divisor, std::size_t divisor_bits)
Chris@102:         {
Chris@102:             const std::size_t bits = std::numeric_limits<digit_t>::digits;
Chris@102:             _remainder_bits = divisor_bits - 1;
Chris@102:             for(std::size_t i = 0; i < divisor_bits; ++i) {
Chris@102:                 if(divisor[i/bits] & (digit_t(1) << i%bits)) {
Chris@102:                     _bit_indices.push_back(i);
Chris@102:                 }
Chris@102:             }
Chris@102:             BOOST_ASSERT(_bit_indices.back() == divisor_bits - 1);
Chris@102:             _bit_indices.pop_back();
Chris@102:             if(_bit_indices.empty()) {
Chris@102:                 _block_bits = divisor_bits;
Chris@102:                 _lower_bits = 0;
Chris@102:             } else {
Chris@102:                 _block_bits = divisor_bits - _bit_indices.back() - 1;
Chris@102:                 _lower_bits = _bit_indices.back() + 1;
Chris@102:             }
Chris@102:             
Chris@102:             _partial_quotient.resize((_block_bits + bits - 1)/bits);
Chris@102:         }
Chris@102:         void operator()(digit_t * dividend, std::size_t dividend_bits)
Chris@102:         {
Chris@102:             const std::size_t bits = std::numeric_limits<digit_t>::digits;
Chris@102:             while(dividend_bits > _remainder_bits) {
Chris@102:                 std::size_t block_start = (std::max)(dividend_bits - _block_bits, _remainder_bits);
Chris@102:                 std::size_t block_size = (dividend_bits - block_start + bits - 1) / bits;
Chris@102:                 copy_bits(dividend, block_start, dividend_bits, &_partial_quotient[0]);
Chris@102:                 for(std::size_t i = 0; i < _bit_indices.size(); ++i) {
Chris@102:                     std::size_t pos = _bit_indices[i] + block_start - _remainder_bits;
Chris@102:                     add_shifted_inplace(block_size, &_partial_quotient[0], dividend + pos/bits, pos%bits);
Chris@102:                 }
Chris@102:                 add_shifted_inplace(block_size, &_partial_quotient[0], dividend + block_start/bits, block_start%bits);
Chris@102:                 dividend_bits = block_start;
Chris@102:             }
Chris@102:         }
Chris@102:         std::vector<digit_t> _partial_quotient;
Chris@102:         std::size_t _remainder_bits;
Chris@102:         std::size_t _block_bits;
Chris@102:         std::size_t _lower_bits;
Chris@102:         std::vector<std::size_t> _bit_indices;
Chris@102:     };
Chris@102: 
Chris@102:     // base should have the same number of bits as mod
Chris@102:     // base, and mod should both be able to hold a power
Chris@102:     // of 2 >= mod_bits.  out needs to be twice as large.
Chris@102:     static void mod_pow_x(boost::uintmax_t exponent, const digit_t * mod, std::size_t mod_bits, digit_t * out)
Chris@102:     {
Chris@102:         const std::size_t bits = std::numeric_limits<digit_t>::digits;
Chris@102:         const std::size_t n = (mod_bits + bits - 1) / bits;
Chris@102:         const std::size_t highbit = mod_bits - 1;
Chris@102:         if(exponent == 0) {
Chris@102:             out[0] = 1;
Chris@102:             std::fill_n(out + 1, n - 1, digit_t(0));
Chris@102:             return;
Chris@102:         }
Chris@102:         boost::uintmax_t i = std::numeric_limits<boost::uintmax_t>::digits - 1;
Chris@102:         while(((boost::uintmax_t(1) << i) & exponent) == 0) {
Chris@102:             --i;
Chris@102:         }
Chris@102:         out[0] = 2;
Chris@102:         std::fill_n(out + 1, n - 1, digit_t(0));
Chris@102:         sparse_mod m(mod, mod_bits);
Chris@102:         while(i--) {
Chris@102:             sqr(out, n);
Chris@102:             m(out, 2 * mod_bits - 1);
Chris@102:             if((boost::uintmax_t(1) << i) & exponent) {
Chris@102:                 shift_left(out, n, 1);
Chris@102:                 if(out[highbit / bits] & (digit_t(1) << highbit%bits))
Chris@102:                     add(n, out, mod, out);
Chris@102:             }
Chris@102:         }
Chris@102:     }
Chris@102: };
Chris@102: 
Chris@102: class polynomial
Chris@102: {
Chris@102:     typedef polynomial_ops::digit_t digit_t;
Chris@102: public:
Chris@102:     polynomial() : _size(0) {}
Chris@102:     class reference {
Chris@102:     public:
Chris@102:         reference(digit_t &value, int idx)
Chris@102:             : _value(value), _idx(idx) {}
Chris@102:         operator bool() const { return (_value & (digit_t(1) << _idx)) != 0; }
Chris@102:         reference& operator=(bool b)
Chris@102:         {
Chris@102:             if(b) {
Chris@102:                 _value |= (digit_t(1) << _idx);
Chris@102:             } else {
Chris@102:                 _value &= ~(digit_t(1) << _idx);
Chris@102:             }
Chris@102:             return *this;
Chris@102:         }
Chris@102:         reference &operator^=(bool b)
Chris@102:         {
Chris@102:             _value ^= (digit_t(b) << _idx);
Chris@102:             return *this;
Chris@102:         }
Chris@102: 
Chris@102:         reference &operator=(const reference &other)
Chris@102:         {
Chris@102:             return *this = static_cast<bool>(other);
Chris@102:         }
Chris@102:     private:
Chris@102:         digit_t &_value;
Chris@102:         int _idx;
Chris@102:     };
Chris@102:     reference operator[](std::size_t i)
Chris@102:     {
Chris@102:         static const std::size_t bits = std::numeric_limits<digit_t>::digits;
Chris@102:         ensure_bit(i);
Chris@102:         return reference(_storage[i/bits], i%bits);
Chris@102:     }
Chris@102:     bool operator[](std::size_t i) const
Chris@102:     {
Chris@102:         static const std::size_t bits = std::numeric_limits<digit_t>::digits;
Chris@102:         if(i < size())
Chris@102:             return (_storage[i/bits] & (digit_t(1) << (i%bits))) != 0;
Chris@102:         else
Chris@102:             return false;
Chris@102:     }
Chris@102:     std::size_t size() const
Chris@102:     {
Chris@102:         return _size;
Chris@102:     }
Chris@102:     void resize(std::size_t n)
Chris@102:     {
Chris@102:         static const std::size_t bits = std::numeric_limits<digit_t>::digits;
Chris@102:         _storage.resize((n + bits - 1)/bits);
Chris@102:         // clear the high order bits in case we're shrinking.
Chris@102:         if(n%bits) {
Chris@102:             _storage.back() &= ((digit_t(1) << (n%bits)) - 1);
Chris@102:         }
Chris@102:         _size = n;
Chris@102:     }
Chris@102:     friend polynomial operator*(const polynomial &lhs, const polynomial &rhs);
Chris@102:     friend polynomial mod_pow_x(boost::uintmax_t exponent, polynomial mod);
Chris@102: private:
Chris@102:     std::vector<polynomial_ops::digit_t> _storage;
Chris@102:     std::size_t _size;
Chris@102:     void ensure_bit(std::size_t i)
Chris@102:     {
Chris@102:         if(i >= size()) {
Chris@102:             resize(i + 1);
Chris@102:         }
Chris@102:     }
Chris@102:     void normalize()
Chris@102:     {
Chris@102:         while(size() && (*this)[size() - 1] == 0)
Chris@102:             resize(size() - 1);
Chris@102:     }
Chris@102: };
Chris@102: 
Chris@102: inline polynomial operator*(const polynomial &lhs, const polynomial &rhs)
Chris@102: {
Chris@102:     polynomial result;
Chris@102:     result._storage.resize(lhs._storage.size() + rhs._storage.size());
Chris@102:     polynomial_ops::multiply(&lhs._storage[0], lhs._storage.size(),
Chris@102:                              &rhs._storage[0], rhs._storage.size(),
Chris@102:                              &result._storage[0]);
Chris@102:     result._size = lhs._size + rhs._size;
Chris@102:     return result;
Chris@102: }
Chris@102: 
Chris@102: inline polynomial mod_pow_x(boost::uintmax_t exponent, polynomial mod)
Chris@102: {
Chris@102:     polynomial result;
Chris@102:     mod.normalize();
Chris@102:     std::size_t mod_size = mod.size();
Chris@102:     result._storage.resize(mod._storage.size() * 2);
Chris@102:     result._size = mod.size() * 2;
Chris@102:     polynomial_ops::mod_pow_x(exponent, &mod._storage[0], mod_size, &result._storage[0]);
Chris@102:     result.resize(mod.size() - 1);
Chris@102:     return result;
Chris@102: }
Chris@102: 
Chris@102: }
Chris@102: }
Chris@102: }
Chris@102: 
Chris@102: #endif // BOOST_RANDOM_DETAIL_POLYNOMIAL_HPP