Crypto++ 8.2 Free C++ class library of cryptographic schemes

modarith.h
Go to the documentation of this file.
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 <i>incorrect</i> 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
54 ModularArithmetic(const Integer &modulus = Integer::One())
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
59 ModularArithmetic(const ModularArithmetic &ma)
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
64 ModularArithmetic& operator=(const ModularArithmetic &ma) {
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 /// \return 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 /// \return 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 /// \return 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 /// \return 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 /// \return 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 /// \return 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 /// \return 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 /// \return 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 /// \return 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 /// \return 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 /// \return 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 /// \return 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 /// \return 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 /// \return the multiplicative identity
181 /// \details the base class implementations returns 1.
182 const Integer& MultiplicativeIdentity() const
183 {return Integer::One();}
184
185 /// \brief Multiplies elements in the ring
186 /// \param a the multiplicand
187 /// \param b the multiplier
188 /// \return 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 /// \return 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 /// \return 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 /// \return 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 /// \return 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 /// \return 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 /// \return 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 /// \return 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 /// \return 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 /// \return 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
318 const Integer& MultiplicativeIdentity() const
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
ModularArithmetic::Double
const Integer & Double(const Integer &a) const
Doubles an element in the ring.
Definition: modarith.h:176
ModularArithmetic::Equal
bool Equal(const Integer &a, const Integer &b) const
Compare two elements for equality.
Definition: modarith.h:135
ModularArithmetic::Square
const Integer & Square(const Integer &a) const
Square an element in the ring.
Definition: modarith.h:197
MontgomeryRepresentation::IsMontgomeryRepresentation
bool IsMontgomeryRepresentation() const
Retrieves the representation.
Definition: modarith.h:311
ModularArithmetic::Divide
const Integer & Divide(const Integer &a, const Integer &b) const
Divides elements in the ring.
Definition: modarith.h:218
misc.h
Utility functions for the Crypto++ library.
AbstractGroup::Subtract
virtual const Element & Subtract(const Element &a, const Element &b) const
Subtracts elements in the group.
Definition: algebra.cpp:20
AbstractRing::Multiply
virtual const Element & Multiply(const Element &a, const Element &b) const =0
Multiplies elements in the group.
Integer::Gcd
static Integer CRYPTOPP_API Gcd(const Integer &a, const Integer &n)
Calculate greatest common divisor.
Definition: integer.cpp:4416
SecBlock::resize
void resize(size_type newSize)
Change size and preserve contents.
Definition: secblock.h:1070
ModularArithmetic::ConvertOut
virtual Integer ConvertOut(const Integer &a) const
Reduces an element in the congruence class.
Definition: modarith.h:123
AbstractRing::CascadeExponentiate
virtual Element CascadeExponentiate(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const
TODO.
Definition: algebra.cpp:323
SecBlock< word, AllocatorWithCleanup< word, true > >
cryptlib.h
Abstract base classes that provide a uniform interface to this library.
ModularArithmetic::MultiplicativeInverse
const Integer & MultiplicativeInverse(const Integer &a) const
Calculate the multiplicative inverse of an element in the ring.
Definition: modarith.h:210
AbstractEuclideanDomain
Abstract Euclidean domain.
Definition: algebra.h:276
ModularArithmetic::MaxElementByteLength
unsigned int MaxElementByteLength() const
Provides the maximum byte size of an element in the ring.
Definition: modarith.h:248
ModularArithmetic
Ring of congruence classes modulo n.
Definition: modarith.h:43
RandomNumberGenerator
Interface for random number generators.
Definition: cryptlib.h:1413
ModularArithmetic::MaxElementBitLength
unsigned int MaxElementBitLength() const
Provides the maximum bit size of an element in the ring.
Definition: modarith.h:243
Integer::InverseMod
Integer InverseMod(const Integer &n) const
Calculate multiplicative inverse.
Definition: integer.cpp:4421
algebra.h
Classes for performing mathematics over different fields.
BufferedTransformation
Interface for buffered transformations.
Definition: cryptlib.h:1630
Integer::One
static const Integer &CRYPTOPP_API One()
Integer representing 1.
Definition: integer.cpp:4868
AbstractRing< Integer >
MontgomeryRepresentation::ConvertIn
Integer ConvertIn(const Integer &a) const
Reduces an element in the congruence class.
Definition: modarith.h:313
ModularArithmetic::Identity
const Integer & Identity() const
Provides the Identity element.
Definition: modarith.h:140
Integer::IsUnit
bool IsUnit() const
Determine if 1 or -1.
Definition: integer.cpp:4387
secblock.h
Classes and functions for secure memory allocations.
MontgomeryRepresentation::Clone
virtual ModularArithmetic * Clone() const
Clone a MontgomeryRepresentation.
Definition: modarith.h:309
ModularArithmetic::operator==
bool operator==(const ModularArithmetic &rhs) const
Compares two ModularArithmetic for equality.
Definition: modarith.h:269
ModularArithmetic::Multiply
const Integer & Multiply(const Integer &a, const Integer &b) const
Multiplies elements in the ring.
Definition: modarith.h:190
AbstractRing::MultiplicativeInverse
virtual const Element & MultiplicativeInverse(const Element &a) const =0
Calculate the multiplicative inverse of an element in the group.
Integer::Power2
static Integer CRYPTOPP_API Power2(size_t e)
Exponentiates to a power of 2.
Definition: integer.cpp:3087
Integer
Multiple precision integer with arithmetic operations.
Definition: integer.h:49
ModularArithmetic::GetModulus
const Integer & GetModulus() const
Retrieves the modulus.
Definition: modarith.h:99
ModularArithmetic::operator=
ModularArithmetic & operator=(const ModularArithmetic &ma)
Assign a ModularArithmetic.
Definition: modarith.h:64
MontgomeryRepresentation::MultiplicativeIdentity
const Integer & MultiplicativeIdentity() const
Retrieves the multiplicative identity.
Definition: modarith.h:318
AbstractGroup::Inverse
virtual const Element & Inverse(const Element &a) const =0
Inverts the element in the group.
ModularArithmetic::ModularArithmetic
ModularArithmetic(const Integer &modulus=Integer::One())
Construct a ModularArithmetic.
Definition: modarith.h:54
AbstractGroup< Integer >
ModularArithmetic::ConvertIn
virtual Integer ConvertIn(const Integer &a) const
Reduces an element in the congruence class.
Definition: modarith.h:115
ModularArithmetic::Clone
virtual ModularArithmetic * Clone() const
Clone a ModularArithmetic.
Definition: modarith.h:81
Integer::Squared
Integer Squared() const
Multiply this integer by itself.
Definition: integer.h:631
AbstractGroup::Accumulate
virtual Element & Accumulate(Element &a, const Element &b) const
TODO.
Definition: algebra.cpp:27
ModularArithmetic::SetModulus
void SetModulus(const Integer &newModulus)
Sets the modulus.
Definition: modarith.h:103
MontgomeryRepresentation
Performs modular arithmetic in Montgomery representation for increased speed.
Definition: modarith.h:295
ModularArithmetic::RandomElement
Element RandomElement(RandomNumberGenerator &rng, const RandomizationParameter &ignore_for_now=0) const
Provides a random element in the ring.
Definition: modarith.h:258
MontgomeryRepresentation::CascadeExponentiate
Integer CascadeExponentiate(const Integer &x, const Integer &e1, const Integer &y, const Integer &e2) const
TODO.
Definition: modarith.h:327
AbstractGroup::Add
virtual const Element & Add(const Element &a, const Element &b) const =0
Adds elements in the group.
integer.h
Multiple precision integer with arithmetic operations.
Integer::Zero
static const Integer &CRYPTOPP_API Zero()
Integer representing 0.
Definition: integer.cpp:4856
AbstractGroup::Reduce
virtual Element & Reduce(Element &a, const Element &b) const
Reduces an element in the congruence class.
Definition: algebra.cpp:32
CryptoPP
Crypto++ library namespace.
ModularArithmetic::ModularArithmetic
ModularArithmetic(const ModularArithmetic &ma)
Copy construct a ModularArithmetic.
Definition: modarith.h:59
ModularArithmetic::MultiplicativeIdentity
const Integer & MultiplicativeIdentity() const
Retrieves the multiplicative identity.
Definition: modarith.h:182
MontgomeryRepresentation::SimultaneousExponentiate
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
AbstractRing::SimultaneousExponentiate
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
ModularArithmetic::IsMontgomeryRepresentation
virtual bool IsMontgomeryRepresentation() const
Retrieves the representation.
Definition: modarith.h:108
ModularArithmetic::IsUnit
bool IsUnit(const Integer &a) const
Determines whether an element is a unit in the ring.
Definition: modarith.h:203