Crypto++  8.2
Free C++ class library of cryptographic schemes
modarith.h
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1 // modarith.h - originally written and placed in the public domain by Wei Dai
2 
3 /// \file modarith.h
4 /// \brief Class file for performing modular arithmetic.
5 
6 #ifndef CRYPTOPP_MODARITH_H
7 #define CRYPTOPP_MODARITH_H
8 
9 // implementations are in integer.cpp
10 
11 #include "cryptlib.h"
12 #include "integer.h"
13 #include "algebra.h"
14 #include "secblock.h"
15 #include "misc.h"
16 
17 #if CRYPTOPP_MSC_VERSION
18 # pragma warning(push)
19 # pragma warning(disable: 4231 4275)
20 #endif
21 
22 NAMESPACE_BEGIN(CryptoPP)
23 
24 CRYPTOPP_DLL_TEMPLATE_CLASS AbstractGroup<Integer>;
25 CRYPTOPP_DLL_TEMPLATE_CLASS AbstractRing<Integer>;
26 CRYPTOPP_DLL_TEMPLATE_CLASS AbstractEuclideanDomain<Integer>;
27 
28 /// \brief Ring of congruence classes modulo n
29 /// \details This implementation represents each congruence class as
30 /// the smallest non-negative integer in that class.
31 /// \details <tt>const Element&</tt> returned by member functions are
32 /// references to internal data members. Since each object may have
33 /// only one such data member for holding results, you should use the
34 /// class like this:
35 /// <pre> abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre>
36 /// The following code will produce incorrect results:
37 /// <pre> abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre>
38 /// \details If a ModularArithmetic is copied or assigned the modulus
39 /// is copied, but not the internal data members. The internal data
40 /// members are undefined after copy or assignment.
41 /// \sa <A HREF="https://cryptopp.com/wiki/Integer">Integer</A> on the
42 /// Crypto++ wiki.
43 class CRYPTOPP_DLL ModularArithmetic : public AbstractRing<Integer>
44 {
45 public:
46 
47  typedef int RandomizationParameter;
48  typedef Integer Element;
49 
50  virtual ~ModularArithmetic() {}
51 
52  /// \brief Construct a ModularArithmetic
53  /// \param modulus congruence class modulus
55  : m_modulus(modulus), m_result(static_cast<word>(0), modulus.reg.size()) {}
56 
57  /// \brief Copy construct a ModularArithmetic
58  /// \param ma other ModularArithmetic
60  : AbstractRing<Integer>(ma), m_modulus(ma.m_modulus), m_result(static_cast<word>(0), m_modulus.reg.size()) {}
61 
62  /// \brief Assign a ModularArithmetic
63  /// \param ma other ModularArithmetic
65  if (this != &ma)
66  {
67  m_modulus = ma.m_modulus;
68  m_result = Integer(static_cast<word>(0), m_modulus.reg.size());
69  }
70  return *this;
71  }
72 
73  /// \brief Construct a ModularArithmetic
74  /// \param bt BER encoded ModularArithmetic
75  ModularArithmetic(BufferedTransformation &bt); // construct from BER encoded parameters
76 
77  /// \brief Clone a ModularArithmetic
78  /// \returns pointer to a new ModularArithmetic
79  /// \details Clone effectively copy constructs a new ModularArithmetic. The caller is
80  /// responsible for deleting the pointer returned from this method.
81  virtual ModularArithmetic * Clone() const {return new ModularArithmetic(*this);}
82 
83  /// \brief Encodes in DER format
84  /// \param bt BufferedTransformation object
85  void DEREncode(BufferedTransformation &bt) const;
86 
87  /// \brief Encodes element in DER format
88  /// \param out BufferedTransformation object
89  /// \param a Element to encode
90  void DEREncodeElement(BufferedTransformation &out, const Element &a) const;
91 
92  /// \brief Decodes element in DER format
93  /// \param in BufferedTransformation object
94  /// \param a Element to decode
95  void BERDecodeElement(BufferedTransformation &in, Element &a) const;
96 
97  /// \brief Retrieves the modulus
98  /// \returns the modulus
99  const Integer& GetModulus() const {return m_modulus;}
100 
101  /// \brief Sets the modulus
102  /// \param newModulus the new modulus
103  void SetModulus(const Integer &newModulus)
104  {m_modulus = newModulus; m_result.reg.resize(m_modulus.reg.size());}
105 
106  /// \brief Retrieves the representation
107  /// \returns true if the if the modulus is in Montgomery form for multiplication, false otherwise
108  virtual bool IsMontgomeryRepresentation() const {return false;}
109 
110  /// \brief Reduces an element in the congruence class
111  /// \param a element to convert
112  /// \returns the reduced element
113  /// \details ConvertIn is useful for derived classes, like MontgomeryRepresentation, which
114  /// must convert between representations.
115  virtual Integer ConvertIn(const Integer &a) const
116  {return a%m_modulus;}
117 
118  /// \brief Reduces an element in the congruence class
119  /// \param a element to convert
120  /// \returns the reduced element
121  /// \details ConvertOut is useful for derived classes, like MontgomeryRepresentation, which
122  /// must convert between representations.
123  virtual Integer ConvertOut(const Integer &a) const
124  {return a;}
125 
126  /// \brief Divides an element by 2
127  /// \param a element to convert
128  const Integer& Half(const Integer &a) const;
129 
130  /// \brief Compare two elements for equality
131  /// \param a first element
132  /// \param b second element
133  /// \returns true if the elements are equal, false otherwise
134  /// \details Equal() tests the elements for equality using <tt>a==b</tt>
135  bool Equal(const Integer &a, const Integer &b) const
136  {return a==b;}
137 
138  /// \brief Provides the Identity element
139  /// \returns the Identity element
140  const Integer& Identity() const
141  {return Integer::Zero();}
142 
143  /// \brief Adds elements in the ring
144  /// \param a first element
145  /// \param b second element
146  /// \returns the sum of <tt>a</tt> and <tt>b</tt>
147  const Integer& Add(const Integer &a, const Integer &b) const;
148 
149  /// \brief TODO
150  /// \param a first element
151  /// \param b second element
152  /// \returns TODO
153  Integer& Accumulate(Integer &a, const Integer &b) const;
154 
155  /// \brief Inverts the element in the ring
156  /// \param a first element
157  /// \returns the inverse of the element
158  const Integer& Inverse(const Integer &a) const;
159 
160  /// \brief Subtracts elements in the ring
161  /// \param a first element
162  /// \param b second element
163  /// \returns the difference of <tt>a</tt> and <tt>b</tt>. The element <tt>a</tt> must provide a Subtract member function.
164  const Integer& Subtract(const Integer &a, const Integer &b) const;
165 
166  /// \brief TODO
167  /// \param a first element
168  /// \param b second element
169  /// \returns TODO
170  Integer& Reduce(Integer &a, const Integer &b) const;
171 
172  /// \brief Doubles an element in the ring
173  /// \param a the element
174  /// \returns the element doubled
175  /// \details Double returns <tt>Add(a, a)</tt>. The element <tt>a</tt> must provide an Add member function.
176  const Integer& Double(const Integer &a) const
177  {return Add(a, a);}
178 
179  /// \brief Retrieves the multiplicative identity
180  /// \returns the multiplicative identity
181  /// \details the base class implementations returns 1.
183  {return Integer::One();}
184 
185  /// \brief Multiplies elements in the ring
186  /// \param a the multiplicand
187  /// \param b the multiplier
188  /// \returns the product of a and b
189  /// \details Multiply returns <tt>a*b\%n</tt>.
190  const Integer& Multiply(const Integer &a, const Integer &b) const
191  {return m_result1 = a*b%m_modulus;}
192 
193  /// \brief Square an element in the ring
194  /// \param a the element
195  /// \returns the element squared
196  /// \details Square returns <tt>a*a\%n</tt>. The element <tt>a</tt> must provide a Square member function.
197  const Integer& Square(const Integer &a) const
198  {return m_result1 = a.Squared()%m_modulus;}
199 
200  /// \brief Determines whether an element is a unit in the ring
201  /// \param a the element
202  /// \returns true if the element is a unit after reduction, false otherwise.
203  bool IsUnit(const Integer &a) const
204  {return Integer::Gcd(a, m_modulus).IsUnit();}
205 
206  /// \brief Calculate the multiplicative inverse of an element in the ring
207  /// \param a the element
208  /// \details MultiplicativeInverse returns <tt>a<sup>-1</sup>\%n</tt>. The element <tt>a</tt> must
209  /// provide a InverseMod member function.
210  const Integer& MultiplicativeInverse(const Integer &a) const
211  {return m_result1 = a.InverseMod(m_modulus);}
212 
213  /// \brief Divides elements in the ring
214  /// \param a the dividend
215  /// \param b the divisor
216  /// \returns the quotient
217  /// \details Divide returns <tt>a*b<sup>-1</sup>\%n</tt>.
218  const Integer& Divide(const Integer &a, const Integer &b) const
219  {return Multiply(a, MultiplicativeInverse(b));}
220 
221  /// \brief TODO
222  /// \param x first element
223  /// \param e1 first exponent
224  /// \param y second element
225  /// \param e2 second exponent
226  /// \returns TODO
227  Integer CascadeExponentiate(const Integer &x, const Integer &e1, const Integer &y, const Integer &e2) const;
228 
229  /// \brief Exponentiates a base to multiple exponents in the ring
230  /// \param results an array of Elements
231  /// \param base the base to raise to the exponents
232  /// \param exponents an array of exponents
233  /// \param exponentsCount the number of exponents in the array
234  /// \details SimultaneousExponentiate() raises the base to each exponent in the exponents array and stores the
235  /// result at the respective position in the results array.
236  /// \details SimultaneousExponentiate() must be implemented in a derived class.
237  /// \pre <tt>COUNTOF(results) == exponentsCount</tt>
238  /// \pre <tt>COUNTOF(exponents) == exponentsCount</tt>
239  void SimultaneousExponentiate(Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const;
240 
241  /// \brief Provides the maximum bit size of an element in the ring
242  /// \returns maximum bit size of an element
243  unsigned int MaxElementBitLength() const
244  {return (m_modulus-1).BitCount();}
245 
246  /// \brief Provides the maximum byte size of an element in the ring
247  /// \returns maximum byte size of an element
248  unsigned int MaxElementByteLength() const
249  {return (m_modulus-1).ByteCount();}
250 
251  /// \brief Provides a random element in the ring
252  /// \param rng RandomNumberGenerator used to generate material
253  /// \param ignore_for_now unused
254  /// \returns a random element that is uniformly distributed
255  /// \details RandomElement constructs a new element in the range <tt>[0,n-1]</tt>, inclusive.
256  /// The element's class must provide a constructor with the signature <tt>Element(RandomNumberGenerator rng,
257  /// Element min, Element max)</tt>.
258  Element RandomElement(RandomNumberGenerator &rng, const RandomizationParameter &ignore_for_now = 0) const
259  // left RandomizationParameter arg as ref in case RandomizationParameter becomes a more complicated struct
260  {
261  CRYPTOPP_UNUSED(ignore_for_now);
262  return Element(rng, Integer::Zero(), m_modulus - Integer::One()) ;
263  }
264 
265  /// \brief Compares two ModularArithmetic for equality
266  /// \param rhs other ModularArithmetic
267  /// \returns true if this is equal to the other, false otherwise
268  /// \details The operator tests for equality using <tt>this.m_modulus == rhs.m_modulus</tt>.
269  bool operator==(const ModularArithmetic &rhs) const
270  {return m_modulus == rhs.m_modulus;}
271 
272  static const RandomizationParameter DefaultRandomizationParameter;
273 
274 private:
275  // TODO: Clang on OS X needs a real operator=.
276  // Squash warning on missing assignment operator.
277  // ModularArithmetic& operator=(const ModularArithmetic &ma);
278 
279 protected:
280  Integer m_modulus;
281  mutable Integer m_result, m_result1;
282 };
283 
284 // const ModularArithmetic::RandomizationParameter ModularArithmetic::DefaultRandomizationParameter = 0 ;
285 
286 /// \brief Performs modular arithmetic in Montgomery representation for increased speed
287 /// \details The Montgomery representation represents each congruence class <tt>[a]</tt> as
288 /// <tt>a*r\%n</tt>, where <tt>r</tt> is a convenient power of 2.
289 /// \details <tt>const Element&</tt> returned by member functions are references to
290 /// internal data members. Since each object may have only one such data member for holding
291 /// results, the following code will produce incorrect results:
292 /// <pre> abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre>
293 /// But this should be fine:
294 /// <pre> abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre>
295 class CRYPTOPP_DLL MontgomeryRepresentation : public ModularArithmetic
296 {
297 public:
298  virtual ~MontgomeryRepresentation() {}
299 
300  /// \brief Construct a MontgomeryRepresentation
301  /// \param modulus congruence class modulus
302  /// \note The modulus must be odd.
303  MontgomeryRepresentation(const Integer &modulus);
304 
305  /// \brief Clone a MontgomeryRepresentation
306  /// \returns pointer to a new MontgomeryRepresentation
307  /// \details Clone effectively copy constructs a new MontgomeryRepresentation. The caller is
308  /// responsible for deleting the pointer returned from this method.
309  virtual ModularArithmetic * Clone() const {return new MontgomeryRepresentation(*this);}
310 
311  bool IsMontgomeryRepresentation() const {return true;}
312 
313  Integer ConvertIn(const Integer &a) const
314  {return (a<<(WORD_BITS*m_modulus.reg.size()))%m_modulus;}
315 
316  Integer ConvertOut(const Integer &a) const;
317 
319  {return m_result1 = Integer::Power2(WORD_BITS*m_modulus.reg.size())%m_modulus;}
320 
321  const Integer& Multiply(const Integer &a, const Integer &b) const;
322 
323  const Integer& Square(const Integer &a) const;
324 
325  const Integer& MultiplicativeInverse(const Integer &a) const;
326 
327  Integer CascadeExponentiate(const Integer &x, const Integer &e1, const Integer &y, const Integer &e2) const
328  {return AbstractRing<Integer>::CascadeExponentiate(x, e1, y, e2);}
329 
330  void SimultaneousExponentiate(Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const
331  {AbstractRing<Integer>::SimultaneousExponentiate(results, base, exponents, exponentsCount);}
332 
333 private:
334  Integer m_u;
335  mutable IntegerSecBlock m_workspace;
336 };
337 
338 NAMESPACE_END
339 
340 #if CRYPTOPP_MSC_VERSION
341 # pragma warning(pop)
342 #endif
343 
344 #endif
const Integer & Double(const Integer &a) const
Doubles an element in the ring.
Definition: modarith.h:176
bool Equal(const Integer &a, const Integer &b) const
Compare two elements for equality.
Definition: modarith.h:135
const Integer & Square(const Integer &a) const
Square an element in the ring.
Definition: modarith.h:197
bool IsMontgomeryRepresentation() const
Retrieves the representation.
Definition: modarith.h:311
const Integer & Divide(const Integer &a, const Integer &b) const
Divides elements in the ring.
Definition: modarith.h:218
Utility functions for the Crypto++ library.
virtual const Element & Subtract(const Element &a, const Element &b) const
Subtracts elements in the group.
Definition: algebra.cpp:20
virtual const Element & Multiply(const Element &a, const Element &b) const =0
Multiplies elements in the group.
static Integer CRYPTOPP_API Gcd(const Integer &a, const Integer &n)
Calculate greatest common divisor.
Definition: integer.cpp:4439
void resize(size_type newSize)
Change size and preserve contents.
Definition: secblock.h:1031
virtual Integer ConvertOut(const Integer &a) const
Reduces an element in the congruence class.
Definition: modarith.h:123
virtual Element CascadeExponentiate(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const
TODO.
Definition: algebra.cpp:323
Abstract base classes that provide a uniform interface to this library.
const Integer & MultiplicativeInverse(const Integer &a) const
Calculate the multiplicative inverse of an element in the ring.
Definition: modarith.h:210
Abstract Euclidean domain.
Definition: algebra.h:276
unsigned int MaxElementByteLength() const
Provides the maximum byte size of an element in the ring.
Definition: modarith.h:248
Ring of congruence classes modulo n.
Definition: modarith.h:43
Interface for random number generators.
Definition: cryptlib.h:1383
unsigned int MaxElementBitLength() const
Provides the maximum bit size of an element in the ring.
Definition: modarith.h:243
Classes for performing mathematics over different fields.
Interface for buffered transformations.
Definition: cryptlib.h:1598
static const Integer &CRYPTOPP_API One()
Integer representing 1.
Definition: integer.cpp:4891
Integer ConvertIn(const Integer &a) const
Reduces an element in the congruence class.
Definition: modarith.h:313
const Integer & Identity() const
Provides the Identity element.
Definition: modarith.h:140
bool IsUnit() const
Determine if 1 or -1.
Definition: integer.cpp:4410
Classes and functions for secure memory allocations.
virtual ModularArithmetic * Clone() const
Clone a MontgomeryRepresentation.
Definition: modarith.h:309
bool operator==(const ModularArithmetic &rhs) const
Compares two ModularArithmetic for equality.
Definition: modarith.h:269
const Integer & Multiply(const Integer &a, const Integer &b) const
Multiplies elements in the ring.
Definition: modarith.h:190
virtual const Element & MultiplicativeInverse(const Element &a) const =0
Calculate the multiplicative inverse of an element in the group.
static Integer CRYPTOPP_API Power2(size_t e)
Exponentiates to a power of 2.
Definition: integer.cpp:3093
Multiple precision integer with arithmetic operations.
Definition: integer.h:49
const Integer & GetModulus() const
Retrieves the modulus.
Definition: modarith.h:99
ModularArithmetic & operator=(const ModularArithmetic &ma)
Assign a ModularArithmetic.
Definition: modarith.h:64
const Integer & MultiplicativeIdentity() const
Retrieves the multiplicative identity.
Definition: modarith.h:318
virtual const Element & Inverse(const Element &a) const =0
Inverts the element in the group.
ModularArithmetic(const Integer &modulus=Integer::One())
Construct a ModularArithmetic.
Definition: modarith.h:54
virtual Integer ConvertIn(const Integer &a) const
Reduces an element in the congruence class.
Definition: modarith.h:115
virtual ModularArithmetic * Clone() const
Clone a ModularArithmetic.
Definition: modarith.h:81
virtual Element & Accumulate(Element &a, const Element &b) const
TODO.
Definition: algebra.cpp:27
void SetModulus(const Integer &newModulus)
Sets the modulus.
Definition: modarith.h:103
Performs modular arithmetic in Montgomery representation for increased speed.
Definition: modarith.h:295
Element RandomElement(RandomNumberGenerator &rng, const RandomizationParameter &ignore_for_now=0) const
Provides a random element in the ring.
Definition: modarith.h:258
Integer CascadeExponentiate(const Integer &x, const Integer &e1, const Integer &y, const Integer &e2) const
TODO.
Definition: modarith.h:327
virtual const Element & Add(const Element &a, const Element &b) const =0
Adds elements in the group.
Multiple precision integer with arithmetic operations.
static const Integer &CRYPTOPP_API Zero()
Integer representing 0.
Definition: integer.cpp:4879
virtual Element & Reduce(Element &a, const Element &b) const
Reduces an element in the congruence class.
Definition: algebra.cpp:32
Crypto++ library namespace.
ModularArithmetic(const ModularArithmetic &ma)
Copy construct a ModularArithmetic.
Definition: modarith.h:59
const Integer & MultiplicativeIdentity() const
Retrieves the multiplicative identity.
Definition: modarith.h:182
void SimultaneousExponentiate(Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const
Exponentiates a base to multiple exponents in the ring.
Definition: modarith.h:330
virtual void SimultaneousExponentiate(Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const
Exponentiates a base to multiple exponents in the Ring.
Definition: algebra.cpp:334
virtual bool IsMontgomeryRepresentation() const
Retrieves the representation.
Definition: modarith.h:108
bool IsUnit(const Integer &a) const
Determines whether an element is a unit in the ring.
Definition: modarith.h:203