libitpp-dev 4.2-4 / usr / include / itpp / comm / galois.h

/*!
 * \file
 * \brief Definitions of Galois Field algebra classes and functions
 * \author Tony Ottosson
 *
 * -------------------------------------------------------------------------
 *
 * Copyright (C) 1995-2010  (see AUTHORS file for a list of contributors)
 *
 * This file is part of IT++ - a C++ library of mathematical, signal
 * processing, speech processing, and communications classes and functions.
 *
 * IT++ is free software: you can redistribute it and/or modify it under the
 * terms of the GNU General Public License as published by the Free Software
 * Foundation, either version 3 of the License, or (at your option) any
 * later version.
 *
 * IT++ is distributed in the hope that it will be useful, but WITHOUT ANY
 * WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
 * FOR A PARTICULAR PURPOSE.  See the GNU General Public License for more
 * details.
 *
 * You should have received a copy of the GNU General Public License along
 * with IT++.  If not, see <http://www.gnu.org/licenses/>.
 *
 * -------------------------------------------------------------------------
 */

#ifndef GALOIS_H
#define GALOIS_H

#include <itpp/base/vec.h>
#include <itpp/base/array.h>
#include <itpp/base/binary.h>
#include <itpp/base/converters.h>


namespace itpp
{

/*!
  \brief Galois Field GF(q).
  \author Tony Ottosson

  Galois field GF(q), where \a q = 2^m. Possible \a m values is \a m = 1,2,...,16.
  Elements are given as exponents of the primitive element \a alpha.
  Observe that the zeroth element are given as "-1". ( log(0)=-Inf ).
  <h3> The following primitve polynomials are used to construct the fields:</h3>
  <ul>
  <li> GF(4): 1+x+x^2 </li>
  <li> GF(8): 1+x+x^3 </li>
  <li> GF(16): 1+x+x^4 </li>
  <li> GF(32): 1+x^2+x^5 </li>
  <li> GF(64): 1+x^2+x^6 </li>
  <li> GF(128): 1+x^3+x^7 </li>
  <li> GF(256): 1+x^2+x^3+x^4+x^8 </li>
  <li> GF(512): 1+x^4+x^9 </li>
  <li> GF(1024): 1+x^3+x^10 </li>
  <li> GF(2^11): 1+x^2+x^11 </li>
  <li> GF(2^12): 1+x+x^4+x^12 </li>
  <li> GF(2^13): 1+x+x^3+x^4+x^13 </li>
  <li> GF(2^14): 1+x+x^3+x^5+x^14 </li>
  <li> GF(2^15): 1+x+x^15 </li>
  <li> GF(2^16): 1+x+x^3+x^12+x^16 </li>
  </ul>

  As indicated it is possible to use this class for binary elements, that is GF(2).
  However, this is less efficient
  in storage (each element take 5 bytes of memory) and in speed.
  If possible use the class BIN instead.
  Observe, also that the element "0" is called "-1" and "1" called "0".
*/
class GF
{
public:
  //! Constructor
  GF() { m = 0; }
  //! Constructor
  GF(int qvalue) {
    m = 0;
    if (qvalue == 0) // qvalue==0 gives the zeroth element
      value = -1;
    else set_size(qvalue);
  }
  //! Constructor
  GF(int qvalue, int inexp) { m = 0; set(qvalue, inexp); }
  //! Copy constructor
  GF(const GF &ingf) { m = ingf.m; value = ingf.value; }

  //! GF(q) equals \a alpha ^ \a inexp
  void set(int qvalue, int inexp) {
    set_size(qvalue);
    it_assert_debug(inexp >= -1 && inexp < qvalue - 1, "GF::set, out of range");
    value = inexp;
  }
  /*!
    \brief GF(q) equals the element that corresponds to the given vector space.

    The format is (...,c,b,a), where the element x is given as x=...+c*alpha^2+b*alpha+a.
  */
  void set(int qvalue, const bvec &vectorspace);
  //! set q=2^mvalue
  void set_size(int qvalue);
  //! Return q.
  int get_size() const { return ((m != 0) ? q[m] : 0); }
  /*!
    \brief Returns the vector space representation of GF(q).

    The format is (...,c,b,a), where the element x is given as x=...+c*alpha^2+b*alpha+a.
  */
  bvec get_vectorspace() const;
  //! Returns the alpha exponent
  int  get_value() const;
  //! Equality check
  int operator==(const GF &ingf) const;
  //! Not-equality check
  int operator!=(const GF &ingf) const;

  //! GF(q) equals ingf
  void operator=(const GF &ingf);
  //! GF(q) equals alpha^inexp
  void operator=(const int inexp);
  //! sum of two GF(q)
  void operator+=(const GF &ingf);
  //! sum of two GF(q)
  GF operator+(const GF &ingf) const;
  //! Difference of two GF(q), same as sum for q=2^m.
  void operator-=(const GF &ingf);
  //! Difference of two GF(q), same as sum for q=2^m.
  GF operator-(const GF &ingf) const;
  //! product of two GF(q)
  void operator*=(const GF &ingf);
  //! product of two GF(q)
  GF operator*(const GF &ingf) const;
  //! division of two GF(q)
  void operator/=(const GF &ingf);
  //! product of two GF(q)
  GF operator/(const GF &ingf) const;
  //! Output stream for GF(q)
  friend std::ostream &operator<<(std::ostream &os, const GF &ingf);
protected:
private:
  char m;
  int value;
  static Array<Array<int> > alphapow, logalpha;
  static ivec q;
};

class GFX;

//! Multiplication of GF and GFX
GFX  operator*(const GF &ingf, const GFX &ingfx);
//! Multiplication of GFX and GF
GFX  operator*(const GFX &ingfx, const GF &ingf);
//! Division of GFX by GF
GFX  operator/(const GFX &ingfx, const GF &ingf);

/*!
  \brief Polynomials over GF(q)[x], where q=2^m, m=1,...,16
*/
class GFX
{
public:
  //! Constructor
  GFX();
  //! Constructor
  GFX(int qvalue);
  //! Constructor
  GFX(int qvalue, int indegree);
  //! Constructor
  GFX(int qvalue, const ivec &invalues);
  //! Constructor
  GFX(int qvalue, char *invalues);
  //! Constructor
  GFX(int qvalue, std::string invalues);
  //! Copy constructor
  GFX(const GFX &ingfx);
  //! Return q.
  int get_size() const;
  //! Return degree of GF(q)[x]
  int get_degree() const;
  /*!
    \brief Resize the polynomial to the given \c indegree. If \c copy is set to true, the old polynomial's coefficients are kept in the new polynomial, otherwise they are set to zero.
  */
  void set_degree(int indegree, bool copy = false);
  //! Return true degree of GF(q)[x]
  int get_true_degree() const;
  //! Set the GF(q)[x] polynomial
  void set(int qvalue, const char *invalues);
  //! Set the GF(q)[x] polynomial
  void set(int qvalue, const std::string invalues);
  //! Set the GF(q)[x] polynomial
  void set(int qvalue, const ivec &invalues);
  //! Set all coefficients to zero.
  void clear();
  //! Acces to individual element in the GF(q)[x] polynomial
  GF operator[](int index) const {
    it_assert_debug(index<=degree, "GFX::op[], out of range");
    return coeffs(index);
  }
  //! Acces to individual element in the GF(q)[x] polynomial
  GF &operator[](int index) {
    it_assert_debug(index<=degree, "GFX::op[], out of range");
    return coeffs(index);
  }
  //! Copy
  void operator=(const GFX &ingfx);
  //! sum of two GF(q)[x]
  void operator+=(const GFX &ingfx);
  //! sum of two GF(q)[x]
  GFX operator+(const GFX &ingfx) const;
  //! Difference of two GF(q), same as sum for q=2^m.
  void operator-=(const GFX &ingfx);
  //! Difference of two GF(q), same as sum for q=2^m.
  GFX operator-(const GFX &ingfx) const;
  //! product of two GF(q)[x]
  void operator*=(const GFX &ingfx);
  //! product of two GF(q)[x]
  GFX operator*(const GFX &ingfx) const;
  //! Evaluate polynom at alpha^inexp
  GF operator()(const GF &ingf);
  //! Multiply a GF element with a GF(q)[x]
  friend GFX  operator*(const GF &ingf, const GFX &ingfx);
  //! Multiply a GF(q)[x] with a GF element
  friend GFX  operator*(const GFX &ingfx, const GF &ingf);
  //! Divide a GF(q)[x] with a GF element
  friend GFX  operator/(const GFX &ingfx, const GF &ingf);

  //! Output stream
  friend std::ostream &operator<<(std::ostream &os, const GFX &ingfx);
protected:
private:
  int degree, q;
  Array<GF> coeffs;
};

//-------------- Help Functions ------------------
/*!
  \relates GFX
  \brief Int division of GF[q](x) polynomials: m(x) = c(x)/g(x).

  The reminder r(x) is not returned by this function.
*/
GFX divgfx(const GFX &c, const GFX &g);

/*!
  \relates GFX
  \brief Function that performs int division of gf[q](x) polynomials (a(x)/g(x)) and returns the reminder.
*/
GFX modgfx(const GFX &a, const GFX &b);


// --------------- Inlines ------------------------
// --------------- class GF -----------------------

inline void GF::set(int qvalue, const bvec &vectorspace)
{
  set_size(qvalue);
  it_assert_debug(vectorspace.length() == m, "GF::set, out of range");
  value = logalpha(m)(bin2dec(vectorspace));
}

inline bvec GF::get_vectorspace() const
{
  bvec temp(m);
  if (value == -1)
    temp = dec2bin(m, 0);
  else
    temp = dec2bin(m, alphapow(m)(value));
  return temp;
}

inline int  GF::get_value() const
{
  return value;
}

inline int GF::operator==(const GF &ingf) const
{
  if (value == -1 && ingf.value == -1)
    return true;
  if (m == ingf.m && value == ingf.value)
    return true;
  else
    return false;
}

inline int GF::operator!=(const GF &ingf) const
{
  GF tmp(*this);
  return !(tmp == ingf);
}

inline void GF::operator=(const GF &ingf)
{
  m = ingf.m;
  value = ingf.value;
}

inline void GF::operator=(const int inexp)
{
  it_assert_debug(m > 0 && inexp >= -1 && inexp < (q[m] - 1), "GF::op=, out of range");
  value = inexp;
}

inline void GF::operator+=(const GF &ingf)
{
  if (value == -1) {
    value = ingf.value;
    m = ingf.m;
  }
  else if (ingf.value != -1) {
    it_assert_debug(ingf.m == m, "GF::op+=, not same field");
    value = logalpha(m)(alphapow(m)(value) ^ alphapow(m)(ingf.value));
  }
}

inline GF GF::operator+(const GF &ingf) const
{
  GF tmp(*this);
  tmp += ingf;
  return tmp;
}

inline void GF::operator-=(const GF &ingf)
{
  (*this) += ingf;
}

inline GF GF::operator-(const GF &ingf) const
{
  GF tmp(*this);
  tmp -= ingf;
  return tmp;
}

inline void GF::operator*=(const GF &ingf)
{
  if (value == -1 || ingf.value == -1)
    value = -1;
  else {
    it_assert_debug(ingf.m == m, "GF::op+=, not same field");
    value = (value + ingf.value) % (q[m] - 1);
  }
}

inline GF GF::operator*(const GF &ingf) const
{
  GF tmp(*this);
  tmp *= ingf;
  return tmp;
}

inline void GF::operator/=(const GF &ingf)
{
  it_assert(ingf.value != -1, "GF::operator/: division by zero element"); // no division by the zeroth element
  if (value == -1)
    value = -1;
  else {
    it_assert_debug(ingf.m == m, "GF::op+=, not same field");
    value = (value - ingf.value + q[m] - 1) % (q[m] - 1);
  }
}

inline GF GF::operator/(const GF &ingf) const
{
  GF tmp(*this);
  tmp /= ingf;
  return tmp;
}

// ------------------ class GFX --------------------
inline GFX::GFX()
{
  degree = -1;
  q = 0;
}

inline GFX::GFX(int qvalue)
{
  it_assert_debug(qvalue >= 0, "GFX::GFX, out of range");
  q = qvalue;
}

inline void GFX::set(int qvalue, const ivec &invalues)
{
  it_assert_debug(qvalue > 0, "GFX::set, out of range");
  degree = invalues.size() - 1;
  coeffs.set_size(degree + 1, false);
  for (int i = 0;i < degree + 1;i++)
    coeffs(i).set(qvalue, invalues(i));
  q = qvalue;
}

inline void GFX::set(int qvalue, const char *invalues)
{
  set(qvalue, ivec(invalues));
}

inline void GFX::set(int qvalue, const std::string invalues)
{
  set(qvalue, invalues.c_str());
}

inline GFX::GFX(int qvalue, int indegree)
{
  it_assert_debug(qvalue > 0 && indegree >= 0, "GFX::GFX, out of range");
  q = qvalue;
  coeffs.set_size(indegree + 1, false);
  degree = indegree;
  for (int i = 0;i < degree + 1;i++)
    coeffs(i).set(q, -1);
}
inline GFX::GFX(int qvalue, const ivec &invalues)
{
  set(qvalue, invalues);
}

inline GFX::GFX(int qvalue, char *invalues)
{
  set(qvalue, invalues);
}

inline GFX::GFX(int qvalue, std::string invalues)
{
  set(qvalue, invalues.c_str());
}

inline GFX::GFX(const GFX &ingfx)
{
  degree = ingfx.degree;
  coeffs = ingfx.coeffs;
  q = ingfx.q;
}

inline int GFX::get_size() const
{
  return q;
}

inline int GFX::get_degree() const
{
  return degree;
}

inline void GFX::set_degree(int indegree, bool copy)
{
  it_assert_debug(indegree >= -1, "GFX::set_degree, out of range");
  coeffs.set_size(indegree + 1, copy);
  degree = indegree;
}

inline int GFX::get_true_degree() const
{
  int i = degree;
  while (coeffs(i).get_value() == -1) {
    i--;
    if (i == -1)
      break;
  }
  return i;
}

inline void GFX::clear()
{
  it_assert_debug(degree >= 0 && q > 0, "GFX::clear, not set");
  for (int i = 0;i < degree + 1;i++)
    coeffs(i).set(q, -1);
}

inline void GFX::operator=(const GFX &ingfx)
{
  degree = ingfx.degree;
  coeffs = ingfx.coeffs;
  q = ingfx.q;
}

inline void GFX::operator+=(const GFX &ingfx)
{
  it_assert_debug(q == ingfx.q, "GFX::op+=, not same field");
  if (ingfx.degree > degree) {
    coeffs.set_size(ingfx.degree + 1, true);
    // set new coefficients to the zeroth element
    for (int j = degree + 1; j < coeffs.size(); j++) { coeffs(j).set(q, -1); }
    degree = ingfx.degree;
  }
  for (int i = 0;i < ingfx.degree + 1;i++) { coeffs(i) += ingfx.coeffs(i); }
}

inline GFX GFX::operator+(const GFX &ingfx) const
{
  GFX tmp(*this);
  tmp += ingfx;
  return tmp;
}

inline void GFX::operator-=(const GFX &ingfx)
{
  (*this) += ingfx;
}

inline GFX GFX::operator-(const GFX &ingfx) const
{
  GFX tmp(*this);
  tmp -= ingfx;
  return tmp;
}

inline void GFX::operator*=(const GFX &ingfx)
{
  it_assert_debug(q == ingfx.q, "GFX::op*=, Not same field");
  int i, j;
  Array<GF> tempcoeffs = coeffs;
  coeffs.set_size(degree + ingfx.degree + 1, false);
  for (j = 0; j < coeffs.size(); j++)
    coeffs(j).set(q, -1); // set coefficients to the zeroth element (log(0)=-Inf=-1)
  for (i = 0;i < degree + 1;i++)
    for (j = 0;j < ingfx.degree + 1;j++)
      coeffs(i + j) += tempcoeffs(i) * ingfx.coeffs(j);
  degree = coeffs.size() - 1;
}

inline GFX GFX::operator*(const GFX &ingfx) const
{
  GFX tmp(*this);
  tmp *= ingfx;
  return tmp;
}

inline GFX operator*(const GF &ingf, const GFX &ingfx)
{
  it_assert_debug(ingf.get_size() == ingfx.q, "GFX::op*, Not same field");
  GFX temp(ingfx);
  for (int i = 0;i < ingfx.degree + 1;i++)
    temp.coeffs(i) *= ingf;
  return temp;
}

inline GFX  operator*(const GFX &ingfx, const GF &ingf)
{
  return ingf*ingfx;
}

inline GFX  operator/(const GFX &ingfx, const GF &ingf)
{
  it_assert_debug(ingf.get_size() == ingfx.q, "GFX::op/, Not same field");
  GFX temp(ingfx);
  for (int i = 0;i < ingfx.degree + 1;i++)
    temp.coeffs(i) /= ingf;
  return temp;
}

inline GF GFX::operator()(const GF &ingf)
{
  it_assert_debug(q == ingf.get_size(), "GFX::op(), Not same field");
  GF temp(coeffs(0)), ingfpower(ingf);
  for (int i = 1; i < degree + 1; i++) {
    temp += coeffs(i) * ingfpower;
    ingfpower *= ingf;
  }
  return temp;
}

} // namespace itpp

#endif // #ifndef GALOIS_H