Crypto++  8.8 Free C++ class library of cryptographic schemes
algebra.h
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1 // algebra.h - originally written and placed in the public domain by Wei Dai
2
3 /// \file algebra.h
4 /// \brief Classes for performing mathematics over different fields
5
6 #ifndef CRYPTOPP_ALGEBRA_H
7 #define CRYPTOPP_ALGEBRA_H
8
9 #include "config.h"
10 #include "integer.h"
11 #include "misc.h"
12
13 NAMESPACE_BEGIN(CryptoPP)
14
15 class Integer;
16
17 /// \brief Abstract group
18 /// \tparam T element class or type
19 /// \details <tt>const Element&</tt> returned by member functions are references
20 /// to internal data members. Since each object may have only
21 /// one such data member for holding results, the following code
22 /// will produce incorrect results:
24 /// But this should be fine:
26 template <class T> class CRYPTOPP_NO_VTABLE AbstractGroup
27 {
28 public:
29  typedef T Element;
30
31  virtual ~AbstractGroup() {}
32
33  /// \brief Compare two elements for equality
34  /// \param a first element
35  /// \param b second element
36  /// \return true if the elements are equal, false otherwise
37  /// \details Equal() tests the elements for equality using <tt>a==b</tt>
38  virtual bool Equal(const Element &a, const Element &b) const =0;
39
40  /// \brief Provides the Identity element
41  /// \return the Identity element
42  virtual const Element& Identity() const =0;
43
44  /// \brief Adds elements in the group
45  /// \param a first element
46  /// \param b second element
47  /// \return the sum of <tt>a</tt> and <tt>b</tt>
48  virtual const Element& Add(const Element &a, const Element &b) const =0;
49
50  /// \brief Inverts the element in the group
51  /// \param a first element
52  /// \return the inverse of the element
53  virtual const Element& Inverse(const Element &a) const =0;
54
55  /// \brief Determine if inversion is fast
56  /// \return true if inversion is fast, false otherwise
57  virtual bool InversionIsFast() const {return false;}
58
59  /// \brief Doubles an element in the group
60  /// \param a the element
61  /// \return the element doubled
62  virtual const Element& Double(const Element &a) const;
63
64  /// \brief Subtracts elements in the group
65  /// \param a first element
66  /// \param b second element
67  /// \return the difference of <tt>a</tt> and <tt>b</tt>. The element <tt>a</tt> must provide a Subtract member function.
68  virtual const Element& Subtract(const Element &a, const Element &b) const;
69
70  /// \brief TODO
71  /// \param a first element
72  /// \param b second element
74  virtual Element& Accumulate(Element &a, const Element &b) const;
75
76  /// \brief Reduces an element in the congruence class
77  /// \param a element to reduce
78  /// \param b the congruence class
79  /// \return the reduced element
80  virtual Element& Reduce(Element &a, const Element &b) const;
81
82  /// \brief Performs a scalar multiplication
83  /// \param a multiplicand
84  /// \param e multiplier
85  /// \return the product
86  virtual Element ScalarMultiply(const Element &a, const Integer &e) const;
87
88  /// \brief TODO
89  /// \param x first multiplicand
90  /// \param e1 the first multiplier
91  /// \param y second multiplicand
92  /// \param e2 the second multiplier
94  virtual Element CascadeScalarMultiply(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const;
95
96  /// \brief Multiplies a base to multiple exponents in a group
97  /// \param results an array of Elements
98  /// \param base the base to raise to the exponents
99  /// \param exponents an array of exponents
100  /// \param exponentsCount the number of exponents in the array
101  /// \details SimultaneousMultiply() multiplies the base to each exponent in the exponents array and stores the
102  /// result at the respective position in the results array.
103  /// \details SimultaneousMultiply() must be implemented in a derived class.
104  /// \pre <tt>COUNTOF(results) == exponentsCount</tt>
105  /// \pre <tt>COUNTOF(exponents) == exponentsCount</tt>
106  virtual void SimultaneousMultiply(Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const;
107 };
108
109 /// \brief Abstract ring
110 /// \tparam T element class or type
111 /// \details <tt>const Element&</tt> returned by member functions are references
112 /// to internal data members. Since each object may have only
113 /// one such data member for holding results, the following code
114 /// will produce incorrect results:
116 /// But this should be fine:
118 template <class T> class CRYPTOPP_NO_VTABLE AbstractRing : public AbstractGroup<T>
119 {
120 public:
121  typedef T Element;
122
123  /// \brief Construct an AbstractRing
124  AbstractRing() {m_mg.m_pRing = this;}
125
126  /// \brief Copy construct an AbstractRing
127  /// \param source other AbstractRing
128  AbstractRing(const AbstractRing &source)
129  {CRYPTOPP_UNUSED(source); m_mg.m_pRing = this;}
130
131  /// \brief Assign an AbstractRing
132  /// \param source other AbstractRing
134  {CRYPTOPP_UNUSED(source); return *this;}
135
136  /// \brief Determines whether an element is a unit in the group
137  /// \param a the element
138  /// \return true if the element is a unit after reduction, false otherwise.
139  virtual bool IsUnit(const Element &a) const =0;
140
141  /// \brief Retrieves the multiplicative identity
142  /// \return the multiplicative identity
143  virtual const Element& MultiplicativeIdentity() const =0;
144
145  /// \brief Multiplies elements in the group
146  /// \param a the multiplicand
147  /// \param b the multiplier
148  /// \return the product of a and b
149  virtual const Element& Multiply(const Element &a, const Element &b) const =0;
150
151  /// \brief Calculate the multiplicative inverse of an element in the group
152  /// \param a the element
153  virtual const Element& MultiplicativeInverse(const Element &a) const =0;
154
155  /// \brief Square an element in the group
156  /// \param a the element
157  /// \return the element squared
158  virtual const Element& Square(const Element &a) const;
159
160  /// \brief Divides elements in the group
161  /// \param a the dividend
162  /// \param b the divisor
163  /// \return the quotient
164  virtual const Element& Divide(const Element &a, const Element &b) const;
165
166  /// \brief Raises a base to an exponent in the group
167  /// \param a the base
168  /// \param e the exponent
169  /// \return the exponentiation
170  virtual Element Exponentiate(const Element &a, const Integer &e) const;
171
172  /// \brief TODO
173  /// \param x first element
174  /// \param e1 first exponent
175  /// \param y second element
176  /// \param e2 second exponent
178  virtual Element CascadeExponentiate(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const;
179
180  /// \brief Exponentiates a base to multiple exponents in the Ring
181  /// \param results an array of Elements
182  /// \param base the base to raise to the exponents
183  /// \param exponents an array of exponents
184  /// \param exponentsCount the number of exponents in the array
185  /// \details SimultaneousExponentiate() raises the base to each exponent in the exponents array and stores the
186  /// result at the respective position in the results array.
187  /// \details SimultaneousExponentiate() must be implemented in a derived class.
188  /// \pre <tt>COUNTOF(results) == exponentsCount</tt>
189  /// \pre <tt>COUNTOF(exponents) == exponentsCount</tt>
190  virtual void SimultaneousExponentiate(Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const;
191
192  /// \brief Retrieves the multiplicative group
193  /// \return the multiplicative group
194  virtual const AbstractGroup<T>& MultiplicativeGroup() const
195  {return m_mg;}
196
197 private:
198  class MultiplicativeGroupT : public AbstractGroup<T>
199  {
200  public:
201  const AbstractRing<T>& GetRing() const
202  {return *m_pRing;}
203
204  bool Equal(const Element &a, const Element &b) const
205  {return GetRing().Equal(a, b);}
206
207  const Element& Identity() const
208  {return GetRing().MultiplicativeIdentity();}
209
210  const Element& Add(const Element &a, const Element &b) const
211  {return GetRing().Multiply(a, b);}
212
213  Element& Accumulate(Element &a, const Element &b) const
214  {return a = GetRing().Multiply(a, b);}
215
216  const Element& Inverse(const Element &a) const
217  {return GetRing().MultiplicativeInverse(a);}
218
219  const Element& Subtract(const Element &a, const Element &b) const
220  {return GetRing().Divide(a, b);}
221
222  Element& Reduce(Element &a, const Element &b) const
223  {return a = GetRing().Divide(a, b);}
224
225  const Element& Double(const Element &a) const
226  {return GetRing().Square(a);}
227
228  Element ScalarMultiply(const Element &a, const Integer &e) const
229  {return GetRing().Exponentiate(a, e);}
230
231  Element CascadeScalarMultiply(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const
232  {return GetRing().CascadeExponentiate(x, e1, y, e2);}
233
234  void SimultaneousMultiply(Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const
235  {GetRing().SimultaneousExponentiate(results, base, exponents, exponentsCount);}
236
237  const AbstractRing<T> *m_pRing;
238  };
239
240  MultiplicativeGroupT m_mg;
241 };
242
243 // ********************************************************
244
245 /// \brief Base and exponent
246 /// \tparam T base class or type
247 /// \tparam E exponent class or type
248 template <class T, class E = Integer>
250 {
251 public:
252  BaseAndExponent() {}
253  BaseAndExponent(const T &base, const E &exponent) : base(base), exponent(exponent) {}
254  bool operator<(const BaseAndExponent<T, E> &rhs) const {return exponent < rhs.exponent;}
255  T base;
256  E exponent;
257 };
258
259 // VC60 workaround: incomplete member template support
260 template <class Element, class Iterator>
261  Element GeneralCascadeMultiplication(const AbstractGroup<Element> &group, Iterator begin, Iterator end);
262 template <class Element, class Iterator>
263  Element GeneralCascadeExponentiation(const AbstractRing<Element> &ring, Iterator begin, Iterator end);
264
265 // ********************************************************
266
267 /// \brief Abstract Euclidean domain
268 /// \tparam T element class or type
269 /// \details <tt>const Element&</tt> returned by member functions are references
270 /// to internal data members. Since each object may have only
271 /// one such data member for holding results, the following code
272 /// will produce incorrect results:
274 /// But this should be fine:
276 template <class T> class CRYPTOPP_NO_VTABLE AbstractEuclideanDomain : public AbstractRing<T>
277 {
278 public:
279  typedef T Element;
280
281  /// \brief Performs the division algorithm on two elements in the ring
282  /// \param r the remainder
283  /// \param q the quotient
284  /// \param a the dividend
285  /// \param d the divisor
286  virtual void DivisionAlgorithm(Element &r, Element &q, const Element &a, const Element &d) const =0;
287
288  /// \brief Performs a modular reduction in the ring
289  /// \param a the element
290  /// \param b the modulus
291  /// \return the result of <tt>a%b</tt>.
292  virtual const Element& Mod(const Element &a, const Element &b) const =0;
293
294  /// \brief Calculates the greatest common denominator in the ring
295  /// \param a the first element
296  /// \param b the second element
297  /// \return the greatest common denominator of a and b.
298  virtual const Element& Gcd(const Element &a, const Element &b) const;
299
300 protected:
301  mutable Element result;
302 };
303
304 // ********************************************************
305
306 /// \brief Euclidean domain
307 /// \tparam T element class or type
308 /// \details <tt>const Element&</tt> returned by member functions are references
309 /// to internal data members. Since each object may have only
310 /// one such data member for holding results, the following code
311 /// will produce incorrect results:
313 /// But this should be fine:
315 template <class T> class EuclideanDomainOf : public AbstractEuclideanDomain<T>
316 {
317 public:
318  typedef T Element;
319
320  EuclideanDomainOf() {}
321
322  bool Equal(const Element &a, const Element &b) const
323  {return a==b;}
324
325  const Element& Identity() const
326  {return Element::Zero();}
327
328  const Element& Add(const Element &a, const Element &b) const
329  {return result = a+b;}
330
331  Element& Accumulate(Element &a, const Element &b) const
332  {return a+=b;}
333
334  const Element& Inverse(const Element &a) const
335  {return result = -a;}
336
337  const Element& Subtract(const Element &a, const Element &b) const
338  {return result = a-b;}
339
340  Element& Reduce(Element &a, const Element &b) const
341  {return a-=b;}
342
343  const Element& Double(const Element &a) const
344  {return result = a.Doubled();}
345
346  const Element& MultiplicativeIdentity() const
347  {return Element::One();}
348
349  const Element& Multiply(const Element &a, const Element &b) const
350  {return result = a*b;}
351
352  const Element& Square(const Element &a) const
353  {return result = a.Squared();}
354
355  bool IsUnit(const Element &a) const
356  {return a.IsUnit();}
357
358  const Element& MultiplicativeInverse(const Element &a) const
359  {return result = a.MultiplicativeInverse();}
360
361  const Element& Divide(const Element &a, const Element &b) const
362  {return result = a/b;}
363
364  const Element& Mod(const Element &a, const Element &b) const
365  {return result = a%b;}
366
367  void DivisionAlgorithm(Element &r, Element &q, const Element &a, const Element &d) const
368  {Element::Divide(r, q, a, d);}
369
370  bool operator==(const EuclideanDomainOf<T> &rhs) const
371  {CRYPTOPP_UNUSED(rhs); return true;}
372
373 private:
374  mutable Element result;
375 };
376
377 /// \brief Quotient ring
378 /// \tparam T element class or type
379 /// \details <tt>const Element&</tt> returned by member functions are references
380 /// to internal data members. Since each object may have only
381 /// one such data member for holding results, the following code
382 /// will produce incorrect results:
384 /// But this should be fine:
386 template <class T> class QuotientRing : public AbstractRing<typename T::Element>
387 {
388 public:
389  typedef T EuclideanDomain;
390  typedef typename T::Element Element;
391
392  QuotientRing(const EuclideanDomain &domain, const Element &modulus)
393  : m_domain(domain), m_modulus(modulus) {}
394
395  const EuclideanDomain & GetDomain() const
396  {return m_domain;}
397
398  const Element& GetModulus() const
399  {return m_modulus;}
400
401  bool Equal(const Element &a, const Element &b) const
402  {return m_domain.Equal(m_domain.Mod(m_domain.Subtract(a, b), m_modulus), m_domain.Identity());}
403
404  const Element& Identity() const
405  {return m_domain.Identity();}
406
407  const Element& Add(const Element &a, const Element &b) const
409
410  Element& Accumulate(Element &a, const Element &b) const
411  {return m_domain.Accumulate(a, b);}
412
413  const Element& Inverse(const Element &a) const
414  {return m_domain.Inverse(a);}
415
416  const Element& Subtract(const Element &a, const Element &b) const
417  {return m_domain.Subtract(a, b);}
418
419  Element& Reduce(Element &a, const Element &b) const
420  {return m_domain.Reduce(a, b);}
421
422  const Element& Double(const Element &a) const
423  {return m_domain.Double(a);}
424
425  bool IsUnit(const Element &a) const
426  {return m_domain.IsUnit(m_domain.Gcd(a, m_modulus));}
427
428  const Element& MultiplicativeIdentity() const
429  {return m_domain.MultiplicativeIdentity();}
430
431  const Element& Multiply(const Element &a, const Element &b) const
432  {return m_domain.Mod(m_domain.Multiply(a, b), m_modulus);}
433
434  const Element& Square(const Element &a) const
435  {return m_domain.Mod(m_domain.Square(a), m_modulus);}
436
437  const Element& MultiplicativeInverse(const Element &a) const;
438
439  bool operator==(const QuotientRing<T> &rhs) const
440  {return m_domain == rhs.m_domain && m_modulus == rhs.m_modulus;}
441
442 protected:
443  EuclideanDomain m_domain;
444  Element m_modulus;
445 };
446
447 NAMESPACE_END
448
449 #ifdef CRYPTOPP_MANUALLY_INSTANTIATE_TEMPLATES
450 #include "algebra.cpp"
451 #endif
452
453 #endif
Abstract Euclidean domain.
Definition: algebra.h:277
virtual void DivisionAlgorithm(Element &r, Element &q, const Element &a, const Element &d) const =0
Performs the division algorithm on two elements in the ring.
Abstract group.
Definition: algebra.h:27
virtual bool Equal(const Element &a, const Element &b) const =0
Compare two elements for equality.
virtual bool InversionIsFast() const
Determine if inversion is fast.
Definition: algebra.h:57
virtual const Element & Identity() const =0
Provides the Identity element.
virtual const Element & Add(const Element &a, const Element &b) const =0
virtual const Element & Inverse(const Element &a) const =0
Inverts the element in the group.
Abstract ring.
Definition: algebra.h:119
AbstractRing & operator=(const AbstractRing &source)
Assign an AbstractRing.
Definition: algebra.h:133
virtual const Element & MultiplicativeIdentity() const =0
Retrieves the multiplicative identity.
virtual const Element & Multiply(const Element &a, const Element &b) const =0
Multiplies elements in the group.
virtual const Element & MultiplicativeInverse(const Element &a) const =0
Calculate the multiplicative inverse of an element in the group.
virtual bool IsUnit(const Element &a) const =0
Determines whether an element is a unit in the group.
AbstractRing(const AbstractRing &source)
Copy construct an AbstractRing.
Definition: algebra.h:128
virtual const AbstractGroup< T > & MultiplicativeGroup() const
Retrieves the multiplicative group.
Definition: algebra.h:194
AbstractRing()
Construct an AbstractRing.
Definition: algebra.h:124
Euclidean domain.
Definition: algebra.h:316
const Element & Inverse(const Element &a) const
Inverts the element in the group.
Definition: algebra.h:334
const Element & MultiplicativeInverse(const Element &a) const
Calculate the multiplicative inverse of an element in the group.
Definition: algebra.h:358
const Element & Divide(const Element &a, const Element &b) const
Divides elements in the group.
Definition: algebra.h:361
const Element & Multiply(const Element &a, const Element &b) const
Multiplies elements in the group.
Definition: algebra.h:349
bool Equal(const Element &a, const Element &b) const
Compare two elements for equality.
Definition: algebra.h:322
Element & Accumulate(Element &a, const Element &b) const
TODO.
Definition: algebra.h:331
const Element & Identity() const
Provides the Identity element.
Definition: algebra.h:325
const Element & Add(const Element &a, const Element &b) const
Definition: algebra.h:328
const Element & MultiplicativeIdentity() const
Retrieves the multiplicative identity.
Definition: algebra.h:346
Element & Reduce(Element &a, const Element &b) const
Reduces an element in the congruence class.
Definition: algebra.h:340
const Element & Subtract(const Element &a, const Element &b) const
Subtracts elements in the group.
Definition: algebra.h:337
const Element & Square(const Element &a) const
Square an element in the group.
Definition: algebra.h:352
bool IsUnit(const Element &a) const
Determines whether an element is a unit in the group.
Definition: algebra.h:355
const Element & Mod(const Element &a, const Element &b) const
Performs a modular reduction in the ring.
Definition: algebra.h:364
const Element & Double(const Element &a) const
Doubles an element in the group.
Definition: algebra.h:343
void DivisionAlgorithm(Element &r, Element &q, const Element &a, const Element &d) const
Performs the division algorithm on two elements in the ring.
Definition: algebra.h:367
Multiple precision integer with arithmetic operations.
Definition: integer.h:50
Quotient ring.
Definition: algebra.h:387
const Element & Subtract(const Element &a, const Element &b) const
Subtracts elements in the group.
Definition: algebra.h:416
const Element & MultiplicativeIdentity() const
Retrieves the multiplicative identity.
Definition: algebra.h:428
const Element & Double(const Element &a) const
Doubles an element in the group.
Definition: algebra.h:422
const Element & Square(const Element &a) const
Square an element in the group.
Definition: algebra.h:434
Element & Accumulate(Element &a, const Element &b) const
TODO.
Definition: algebra.h:410
const Element & Identity() const
Provides the Identity element.
Definition: algebra.h:404
bool Equal(const Element &a, const Element &b) const
Compare two elements for equality.
Definition: algebra.h:401
bool IsUnit(const Element &a) const
Determines whether an element is a unit in the group.
Definition: algebra.h:425
const Element & Inverse(const Element &a) const
Inverts the element in the group.
Definition: algebra.h:413
Element & Reduce(Element &a, const Element &b) const
Reduces an element in the congruence class.
Definition: algebra.h:419
const Element & MultiplicativeInverse(const Element &a) const
Calculate the multiplicative inverse of an element in the group.
Definition: algebra.cpp:70
const Element & Add(const Element &a, const Element &b) const
Definition: algebra.h:407
const Element & Multiply(const Element &a, const Element &b) const
Multiplies elements in the group.
Definition: algebra.h:431
Square block cipher.
Definition: square.h:25
Library configuration file.
Multiple precision integer with arithmetic operations.
Utility functions for the Crypto++ library.
Crypto++ library namespace.
const char * Identity()
ConstByteArrayParameter.
Definition: argnames.h:94
Base and exponent.
Definition: algebra.h:250