Crypto++ 8.0 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 the smallest
30 /// non-negative integer in that class.
31 /// \details <tt>const Element&</tt> returned by member functions are references
32 /// to internal data members. Since each object may have only
33 /// one such data member for holding results, the following code
34 /// will produce incorrect results:
35 /// <pre> abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre>
36 /// But this should be fine:
37 /// <pre> abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre>
38 class CRYPTOPP_DLL ModularArithmetic : public AbstractRing<Integer>
39 {
40 public:
41
42 typedef int RandomizationParameter;
43 typedef Integer Element;
44
45 virtual ~ModularArithmetic() {}
46
47 /// \brief Construct a ModularArithmetic
48 /// \param modulus congruence class modulus
49 ModularArithmetic(const Integer &modulus = Integer::One())
50 : AbstractRing<Integer>(), m_modulus(modulus), m_result(static_cast<word>(0), modulus.reg.size()) {}
51
52 /// \brief Copy construct a ModularArithmetic
53 /// \param ma other ModularArithmetic
54 ModularArithmetic(const ModularArithmetic &ma)
55 : AbstractRing<Integer>(), m_modulus(ma.m_modulus), m_result(static_cast<word>(0), ma.m_modulus.reg.size()) {}
56
57 /// \brief Construct a ModularArithmetic
58 /// \param bt BER encoded ModularArithmetic
59 ModularArithmetic(BufferedTransformation &bt); // construct from BER encoded parameters
60
61 /// \brief Clone a ModularArithmetic
62 /// \returns pointer to a new ModularArithmetic
63 /// \details Clone effectively copy constructs a new ModularArithmetic. The caller is
64 /// responsible for deleting the pointer returned from this method.
65 virtual ModularArithmetic * Clone() const {return new ModularArithmetic(*this);}
66
67 /// \brief Encodes in DER format
68 /// \param bt BufferedTransformation object
69 void DEREncode(BufferedTransformation &bt) const;
70
71 /// \brief Encodes element in DER format
72 /// \param out BufferedTransformation object
73 /// \param a Element to encode
74 void DEREncodeElement(BufferedTransformation &out, const Element &a) const;
75
76 /// \brief Decodes element in DER format
77 /// \param in BufferedTransformation object
78 /// \param a Element to decode
79 void BERDecodeElement(BufferedTransformation &in, Element &a) const;
80
81 /// \brief Retrieves the modulus
82 /// \returns the modulus
83 const Integer& GetModulus() const {return m_modulus;}
84
85 /// \brief Sets the modulus
86 /// \param newModulus the new modulus
87 void SetModulus(const Integer &newModulus)
88 {m_modulus = newModulus; m_result.reg.resize(m_modulus.reg.size());}
89
90 /// \brief Retrieves the representation
91 /// \returns true if the if the modulus is in Montgomery form for multiplication, false otherwise
92 virtual bool IsMontgomeryRepresentation() const {return false;}
93
94 /// \brief Reduces an element in the congruence class
95 /// \param a element to convert
96 /// \returns the reduced element
97 /// \details ConvertIn is useful for derived classes, like MontgomeryRepresentation, which
98 /// must convert between representations.
99 virtual Integer ConvertIn(const Integer &a) const
100 {return a%m_modulus;}
101
102 /// \brief Reduces an element in the congruence class
103 /// \param a element to convert
104 /// \returns the reduced element
105 /// \details ConvertOut is useful for derived classes, like MontgomeryRepresentation, which
106 /// must convert between representations.
107 virtual Integer ConvertOut(const Integer &a) const
108 {return a;}
109
110 /// \brief Divides an element by 2
111 /// \param a element to convert
112 const Integer& Half(const Integer &a) const;
113
114 /// \brief Compare two elements for equality
115 /// \param a first element
116 /// \param b second element
117 /// \returns true if the elements are equal, false otherwise
118 /// \details Equal() tests the elements for equality using <tt>a==b</tt>
119 bool Equal(const Integer &a, const Integer &b) const
120 {return a==b;}
121
122 /// \brief Provides the Identity element
123 /// \returns the Identity element
124 const Integer& Identity() const
125 {return Integer::Zero();}
126
127 /// \brief Adds elements in the ring
128 /// \param a first element
129 /// \param b second element
130 /// \returns the sum of <tt>a</tt> and <tt>b</tt>
131 const Integer& Add(const Integer &a, const Integer &b) const;
132
133 /// \brief TODO
134 /// \param a first element
135 /// \param b second element
136 /// \returns TODO
137 Integer& Accumulate(Integer &a, const Integer &b) const;
138
139 /// \brief Inverts the element in the ring
140 /// \param a first element
141 /// \returns the inverse of the element
142 const Integer& Inverse(const Integer &a) const;
143
144 /// \brief Subtracts elements in the ring
145 /// \param a first element
146 /// \param b second element
147 /// \returns the difference of <tt>a</tt> and <tt>b</tt>. The element <tt>a</tt> must provide a Subtract member function.
148 const Integer& Subtract(const Integer &a, const Integer &b) const;
149
150 /// \brief TODO
151 /// \param a first element
152 /// \param b second element
153 /// \returns TODO
154 Integer& Reduce(Integer &a, const Integer &b) const;
155
156 /// \brief Doubles an element in the ring
157 /// \param a the element
158 /// \returns the element doubled
159 /// \details Double returns <tt>Add(a, a)</tt>. The element <tt>a</tt> must provide an Add member function.
160 const Integer& Double(const Integer &a) const
161 {return Add(a, a);}
162
163 /// \brief Retrieves the multiplicative identity
164 /// \returns the multiplicative identity
165 /// \details the base class implementations returns 1.
166 const Integer& MultiplicativeIdentity() const
167 {return Integer::One();}
168
169 /// \brief Multiplies elements in the ring
170 /// \param a the multiplicand
171 /// \param b the multiplier
172 /// \returns the product of a and b
173 /// \details Multiply returns <tt>a*b\%n</tt>.
174 const Integer& Multiply(const Integer &a, const Integer &b) const
175 {return m_result1 = a*b%m_modulus;}
176
177 /// \brief Square an element in the ring
178 /// \param a the element
179 /// \returns the element squared
180 /// \details Square returns <tt>a*a\%n</tt>. The element <tt>a</tt> must provide a Square member function.
181 const Integer& Square(const Integer &a) const
182 {return m_result1 = a.Squared()%m_modulus;}
183
184 /// \brief Determines whether an element is a unit in the ring
185 /// \param a the element
186 /// \returns true if the element is a unit after reduction, false otherwise.
187 bool IsUnit(const Integer &a) const
188 {return Integer::Gcd(a, m_modulus).IsUnit();}
189
190 /// \brief Calculate the multiplicative inverse of an element in the ring
191 /// \param a the element
192 /// \details MultiplicativeInverse returns <tt>a<sup>-1</sup>\%n</tt>. The element <tt>a</tt> must
193 /// provide a InverseMod member function.
194 const Integer& MultiplicativeInverse(const Integer &a) const
195 {return m_result1 = a.InverseMod(m_modulus);}
196
197 /// \brief Divides elements in the ring
198 /// \param a the dividend
199 /// \param b the divisor
200 /// \returns the quotient
201 /// \details Divide returns <tt>a*b<sup>-1</sup>\%n</tt>.
202 const Integer& Divide(const Integer &a, const Integer &b) const
203 {return Multiply(a, MultiplicativeInverse(b));}
204
205 /// \brief TODO
206 /// \param x first element
207 /// \param e1 first exponent
208 /// \param y second element
209 /// \param e2 second exponent
210 /// \returns TODO
211 Integer CascadeExponentiate(const Integer &x, const Integer &e1, const Integer &y, const Integer &e2) const;
212
213 /// \brief Exponentiates a base to multiple exponents in the ring
214 /// \param results an array of Elements
215 /// \param base the base to raise to the exponents
216 /// \param exponents an array of exponents
217 /// \param exponentsCount the number of exponents in the array
218 /// \details SimultaneousExponentiate() raises the base to each exponent in the exponents array and stores the
219 /// result at the respective position in the results array.
220 /// \details SimultaneousExponentiate() must be implemented in a derived class.
221 /// \pre <tt>COUNTOF(results) == exponentsCount</tt>
222 /// \pre <tt>COUNTOF(exponents) == exponentsCount</tt>
223 void SimultaneousExponentiate(Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const;
224
225 /// \brief Provides the maximum bit size of an element in the ring
226 /// \returns maximum bit size of an element
227 unsigned int MaxElementBitLength() const
228 {return (m_modulus-1).BitCount();}
229
230 /// \brief Provides the maximum byte size of an element in the ring
231 /// \returns maximum byte size of an element
232 unsigned int MaxElementByteLength() const
233 {return (m_modulus-1).ByteCount();}
234
235 /// \brief Provides a random element in the ring
236 /// \param rng RandomNumberGenerator used to generate material
237 /// \param ignore_for_now unused
238 /// \returns a random element that is uniformly distributed
239 /// \details RandomElement constructs a new element in the range <tt>[0,n-1]</tt>, inclusive.
240 /// The element's class must provide a constructor with the signature <tt>Element(RandomNumberGenerator rng,
241 /// Element min, Element max)</tt>.
242 Element RandomElement(RandomNumberGenerator &rng , const RandomizationParameter &ignore_for_now = 0) const
243 // left RandomizationParameter arg as ref in case RandomizationParameter becomes a more complicated struct
244 {
245 CRYPTOPP_UNUSED(ignore_for_now);
246 return Element(rng, Integer::Zero(), m_modulus - Integer::One()) ;
247 }
248
249 /// \brief Compares two ModularArithmetic for equality
250 /// \param rhs other ModularArithmetic
251 /// \returns true if this is equal to the other, false otherwise
252 /// \details The operator tests for equality using <tt>this.m_modulus == rhs.m_modulus</tt>.
253 bool operator==(const ModularArithmetic &rhs) const
254 {return m_modulus == rhs.m_modulus;}
255
256 static const RandomizationParameter DefaultRandomizationParameter ;
257
258 protected:
259 Integer m_modulus;
260 mutable Integer m_result, m_result1;
261 };
262
263 // const ModularArithmetic::RandomizationParameter ModularArithmetic::DefaultRandomizationParameter = 0 ;
264
265 /// \brief Performs modular arithmetic in Montgomery representation for increased speed
266 /// \details The Montgomery representation represents each congruence class <tt>[a]</tt> as
267 /// <tt>a*r\%n</tt>, where <tt>r</tt> is a convenient power of 2.
268 /// \details <tt>const Element&</tt> returned by member functions are references to
269 /// internal data members. Since each object may have only one such data member for holding
270 /// results, the following code will produce incorrect results:
271 /// <pre> abcd = group.Add(group.Add(a,b), group.Add(c,d));</pre>
272 /// But this should be fine:
273 /// <pre> abcd = group.Add(a, group.Add(b, group.Add(c,d));</pre>
274 class CRYPTOPP_DLL MontgomeryRepresentation : public ModularArithmetic
275 {
276 public:
277 virtual ~MontgomeryRepresentation() {}
278
279 /// \brief Construct a MontgomeryRepresentation
280 /// \param modulus congruence class modulus
281 /// \note The modulus must be odd.
282 MontgomeryRepresentation(const Integer &modulus);
283
284 /// \brief Clone a MontgomeryRepresentation
285 /// \returns pointer to a new MontgomeryRepresentation
286 /// \details Clone effectively copy constructs a new MontgomeryRepresentation. The caller is
287 /// responsible for deleting the pointer returned from this method.
288 virtual ModularArithmetic * Clone() const {return new MontgomeryRepresentation(*this);}
289
290 bool IsMontgomeryRepresentation() const {return true;}
291
292 Integer ConvertIn(const Integer &a) const
293 {return (a<<(WORD_BITS*m_modulus.reg.size()))%m_modulus;}
294
295 Integer ConvertOut(const Integer &a) const;
296
297 const Integer& MultiplicativeIdentity() const
298 {return m_result1 = Integer::Power2(WORD_BITS*m_modulus.reg.size())%m_modulus;}
299
300 const Integer& Multiply(const Integer &a, const Integer &b) const;
301
302 const Integer& Square(const Integer &a) const;
303
304 const Integer& MultiplicativeInverse(const Integer &a) const;
305
306 Integer CascadeExponentiate(const Integer &x, const Integer &e1, const Integer &y, const Integer &e2) const
307 {return AbstractRing<Integer>::CascadeExponentiate(x, e1, y, e2);}
308
309 void SimultaneousExponentiate(Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const
310 {AbstractRing<Integer>::SimultaneousExponentiate(results, base, exponents, exponentsCount);}
311
312 private:
313 Integer m_u;
314 mutable IntegerSecBlock m_workspace;
315 };
316
317 NAMESPACE_END
318
319 #if CRYPTOPP_MSC_VERSION
320 # pragma warning(pop)
321 #endif
322
323 #endif
ModularArithmetic::Double
const Integer & Double(const Integer &a) const
Doubles an element in the ring.
Definition: modarith.h:160
ModularArithmetic::Equal
bool Equal(const Integer &a, const Integer &b) const
Compare two elements for equality.
Definition: modarith.h:119
ModularArithmetic::Square
const Integer & Square(const Integer &a) const
Square an element in the ring.
Definition: modarith.h:181
MontgomeryRepresentation::IsMontgomeryRepresentation
bool IsMontgomeryRepresentation() const
Retrieves the representation.
Definition: modarith.h:290
ModularArithmetic::Divide
const Integer & Divide(const Integer &a, const Integer &b) const
Divides elements in the ring.
Definition: modarith.h:202
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 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:1031
ModularArithmetic::ConvertOut
virtual Integer ConvertOut(const Integer &a) const
Reduces an element in the congruence class.
Definition: modarith.h:107
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:194
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:232
ModularArithmetic
Ring of congruence classes modulo n.
Definition: modarith.h:38
RandomNumberGenerator
Interface for random number generators.
Definition: cryptlib.h:1383
ModularArithmetic::MaxElementBitLength
unsigned int MaxElementBitLength() const
Provides the maximum bit size of an element in the ring.
Definition: modarith.h:227
algebra.h
Classes for performing mathematics over different fields.
BufferedTransformation
Interface for buffered transformations.
Definition: cryptlib.h:1598
Integer::One
static const Integer & 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:292
ModularArithmetic::Identity
const Integer & Identity() const
Provides the Identity element.
Definition: modarith.h:124
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:288
ModularArithmetic::operator==
bool operator==(const ModularArithmetic &rhs) const
Compares two ModularArithmetic for equality.
Definition: modarith.h:253
ModularArithmetic::Multiply
const Integer & Multiply(const Integer &a, const Integer &b) const
Multiplies elements in the ring.
Definition: modarith.h:174
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 Power2(size_t e)
Exponentiates to a power of 2.
Definition: integer.cpp:3079
Integer
Multiple precision integer with arithmetic operations.
Definition: integer.h:49
ModularArithmetic::GetModulus
const Integer & GetModulus() const
Retrieves the modulus.
Definition: modarith.h:83
MontgomeryRepresentation::MultiplicativeIdentity
const Integer & MultiplicativeIdentity() const
Retrieves the multiplicative identity.
Definition: modarith.h:297
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:49
AbstractGroup< Integer >
ModularArithmetic::ConvertIn
virtual Integer ConvertIn(const Integer &a) const
Reduces an element in the congruence class.
Definition: modarith.h:99
ModularArithmetic::Clone
virtual ModularArithmetic * Clone() const
Clone a ModularArithmetic.
Definition: modarith.h:65
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:87
MontgomeryRepresentation
Performs modular arithmetic in Montgomery representation for increased speed.
Definition: modarith.h:274
ModularArithmetic::RandomElement
Element RandomElement(RandomNumberGenerator &rng, const RandomizationParameter &ignore_for_now=0) const
Provides a random element in the ring.
Definition: modarith.h:242
MontgomeryRepresentation::CascadeExponentiate
Integer CascadeExponentiate(const Integer &x, const Integer &e1, const Integer &y, const Integer &e2) const
TODO.
Definition: modarith.h:306
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 & 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:54
ModularArithmetic::MultiplicativeIdentity
const Integer & MultiplicativeIdentity() const
Retrieves the multiplicative identity.
Definition: modarith.h:166
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:309
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:92
ModularArithmetic::IsUnit
bool IsUnit(const Integer &a) const
Determines whether an element is a unit in the ring.
Definition: modarith.h:187