mce_workfactor.cpp

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/*
 * (C) Copyright Projet SECRET, INRIA, Rocquencourt
 * (C) Bhaskar Biswas and Nicolas Sendrier
 * (C) 2014 Jack Lloyd
 *
 * Botan is released under the Simplified BSD License (see license.txt)
 *
 */
 
#include <botan/mceliece.h>
 
#include <botan/internal/bit_ops.h>
#include <cmath>
 
namespace Botan {
 
namespace {
 
double binomial(size_t n, size_t k) {
   double x = 1;
 
   for(size_t i = 0; i != k; ++i) {
      x *= n - i;
      x /= k - i;
   }
 
   return x;
}
 
double log_binomial(size_t n, size_t k) {
   double x = 0;
 
   for(size_t i = 0; i != k; ++i) {
      x += std::log(n - i);
      x -= std::log(k - i);
   }
 
   return x / std::log(2);
}
 
double nb_iter(size_t n, size_t k, size_t w, size_t p, size_t l) {
   double x = 2 * log_binomial(k / 2, p);
   x += log_binomial(n - k - l, w - 2 * p);
   x = log_binomial(n, w) - x;
   return x;
}
 
double cout_iter(size_t n, size_t k, size_t p, size_t l) {
   double x = binomial(k / 2, p);
   const size_t i = static_cast<size_t>(std::log(x) / std::log(2));
   double res = 2 * p * (n - k - l) * std::ldexp(x * x, -static_cast<int>(l));
 
   // x <- binomial(k/2,p)*2*(2*l+log[2](binomial(k/2,p)))
   x *= 2 * (2 * l + i);
 
   // res <- k*(n-k)/2 +
   // binomial(k/2,p)*2*(2*l+log[2](binomial(k/2,p))) +
   // 2*p*(n-k-l)*binomial(k/2,p)^2/2^l
   res += x + k * ((n - k) / 2.0);
 
   return std::log(res) / std::log(2);  // convert to bits
}
 
double cout_total(size_t n, size_t k, size_t w, size_t p, size_t l) {
   return nb_iter(n, k, w, p, l) + cout_iter(n, k, p, l);
}
 
double best_wf(size_t n, size_t k, size_t w, size_t p) {
   if(p >= k / 2) {
      return -1;
   }
 
   double min = cout_total(n, k, w, p, 0);
 
   for(size_t l = 1; l < n - k; ++l) {
      const double lwf = cout_total(n, k, w, p, l);
      if(lwf < min) {
         min = lwf;
      } else {
         break;
      }
   }
 
   return min;
}
 
}  // namespace
 
size_t mceliece_work_factor(size_t n, size_t t) {
   const size_t k = n - ceil_log2(n) * t;
 
   double min = cout_total(n, k, t, 0, 0);  // correspond a p=1
   for(size_t p = 0; p != t / 2; ++p) {
      double lwf = best_wf(n, k + 1, t, p);
      if(lwf < 0) {
         break;
      }
 
      min = std::min(min, lwf);
   }
 
   return static_cast<size_t>(min);
}
 
}  // namespace Botan