Crypto++ 8.5 Free C++ class library of cryptographic schemes

algebra.h
Go to the documentation of this file.
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:
23 /// <pre> abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre>
24 /// But this should be fine:
25 /// <pre> abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre>
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
73 /// \return TODO
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
93 /// \return TODO
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:
115 /// <pre> abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre>
116 /// But this should be fine:
117 /// <pre> abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre>
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
133 AbstractRing& operator=(const AbstractRing &source)
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
177 /// \return TODO
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>
249 struct BaseAndExponent
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:
273 /// <pre> abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre>
274 /// But this should be fine:
275 /// <pre> abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre>
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:
312 /// <pre> abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre>
313 /// But this should be fine:
314 /// <pre> abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre>
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:
383 /// <pre> abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre>
384 /// But this should be fine:
385 /// <pre> abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre>
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
408 {return m_domain.Add(a, b);}
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
QuotientRing::Identity
const Element & Identity() const
Provides the Identity element.
Definition: algebra.h:404
QuotientRing::Subtract
const Element & Subtract(const Element &a, const Element &b) const
Subtracts elements in the group.
Definition: algebra.h:416
EuclideanDomainOf::Equal
bool Equal(const Element &a, const Element &b) const
Compare two elements for equality.
Definition: algebra.h:322
QuotientRing::Inverse
const Element & Inverse(const Element &a) const
Inverts the element in the group.
Definition: algebra.h:413
AbstractGroup
Abstract group.
Definition: algebra.h:26
QuotientRing::Double
const Element & Double(const Element &a) const
Doubles an element in the group.
Definition: algebra.h:422
QuotientRing
Quotient ring.
Definition: algebra.h:386
AbstractGroup::Equal
virtual bool Equal(const Element &a, const Element &b) const =0
Compare two elements for equality.
PolynomialMod2
Polynomial with Coefficients in GF(2)
Definition: gf2n.h:26
QuotientRing::IsUnit
bool IsUnit(const Element &a) const
Determines whether an element is a unit in the group.
Definition: algebra.h:425
QuotientRing::Reduce
Element & Reduce(Element &a, const Element &b) const
Reduces an element in the congruence class.
Definition: algebra.h:419
QuotientRing::Accumulate
Element & Accumulate(Element &a, const Element &b) const
TODO.
Definition: algebra.h:410
QuotientRing::MultiplicativeInverse
const Element & MultiplicativeInverse(const Element &a) const
Calculate the multiplicative inverse of an element in the group.
Definition: algebra.cpp:70
QuotientRing::Equal
bool Equal(const Element &a, const Element &b) const
Compare two elements for equality.
Definition: algebra.h:401
EuclideanDomainOf::Subtract
const Element & Subtract(const Element &a, const Element &b) const
Subtracts elements in the group.
Definition: algebra.h:337
Square
Square block cipher.
Definition: square.h:24
BaseAndExponent
Base and exponent.
Definition: algebra.h:249
EuclideanDomainOf::IsUnit
bool IsUnit(const Element &a) const
Determines whether an element is a unit in the group.
Definition: algebra.h:355
AbstractRing::AbstractRing
AbstractRing()
Construct an AbstractRing.
Definition: algebra.h:124
misc.h
Utility functions for the Crypto++ library.
EuclideanDomainOf
Euclidean domain.
Definition: algebra.h:315
QuotientRing::Square
const Element & Square(const Element &a) const
Square an element in the group.
Definition: algebra.h:434
EuclideanDomainOf::Inverse
const Element & Inverse(const Element &a) const
Inverts the element in the group.
Definition: algebra.h:334
EuclideanDomainOf::Accumulate
Element & Accumulate(Element &a, const Element &b) const
TODO.
Definition: algebra.h:331
AbstractRing::AbstractRing
AbstractRing(const AbstractRing &source)
Copy construct an AbstractRing.
Definition: algebra.h:128
AbstractRing::MultiplicativeGroup
virtual const AbstractGroup< T > & MultiplicativeGroup() const
Retrieves the multiplicative group.
Definition: algebra.h:194
AbstractEuclideanDomain
Abstract Euclidean domain.
Definition: algebra.h:276
EuclideanDomainOf::Add
const Element & Add(const Element &a, const Element &b) const
Adds elements in the group.
Definition: algebra.h:328
EuclideanDomainOf::Multiply
const Element & Multiply(const Element &a, const Element &b) const
Multiplies elements in the group.
Definition: algebra.h:349
EuclideanDomainOf::Divide
const Element & Divide(const Element &a, const Element &b) const
Divides elements in the group.
Definition: algebra.h:361
EuclideanDomainOf::Reduce
Element & Reduce(Element &a, const Element &b) const
Reduces an element in the congruence class.
Definition: algebra.h:340
EuclideanDomainOf::DivisionAlgorithm
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
QuotientRing::Multiply
const Element & Multiply(const Element &a, const Element &b) const
Multiplies elements in the group.
Definition: algebra.h:431
EuclideanDomainOf::MultiplicativeIdentity
const Element & MultiplicativeIdentity() const
Retrieves the multiplicative identity.
Definition: algebra.h:346
EuclideanDomainOf::MultiplicativeInverse
const Element & MultiplicativeInverse(const Element &a) const
Calculate the multiplicative inverse of an element in the group.
Definition: algebra.h:358
CryptoPP
Crypto++ library namespace.
EuclideanDomainOf::Mod
const Element & Mod(const Element &a, const Element &b) const
Performs a modular reduction in the ring.
Definition: algebra.h:364
config.h
Library configuration file.
AbstractRing
Abstract ring.
Definition: algebra.h:118
EuclideanDomainOf::Double
const Element & Double(const Element &a) const
Doubles an element in the group.
Definition: algebra.h:343
EuclideanDomainOf::Square
const Element & Square(const Element &a) const
Square an element in the group.
Definition: algebra.h:352
QuotientRing::Add
const Element & Add(const Element &a, const Element &b) const
Adds elements in the group.
Definition: algebra.h:407
integer.h
Multiple precision integer with arithmetic operations.
AbstractGroup::InversionIsFast
virtual bool InversionIsFast() const
Determine if inversion is fast.
Definition: algebra.h:57
Integer
Multiple precision integer with arithmetic operations.
Definition: integer.h:49
AbstractRing::operator=
AbstractRing & operator=(const AbstractRing &source)
Assign an AbstractRing.
Definition: algebra.h:133
EuclideanDomainOf::Identity
const Element & Identity() const
Provides the Identity element.
Definition: algebra.h:325