Crypto++ 8.9 Free C++ class library of cryptographic schemes

algebra.cpp
1 // algebra.cpp - originally written and placed in the public domain by Wei Dai
2
3 #include "pch.h"
4
5 #ifndef CRYPTOPP_ALGEBRA_CPP // SunCC workaround: compiler could cause this file to be included twice
6 #define CRYPTOPP_ALGEBRA_CPP
7
8 #include "algebra.h"
9 #include "integer.h"
10
11 #include <vector>
12
13 NAMESPACE_BEGIN(CryptoPP)
14
15 template <class T> const T& AbstractGroup<T>::Double(const Element &a) const
16 {
17 return this->Add(a, a);
18 }
19
20 template <class T> const T& AbstractGroup<T>::Subtract(const Element &a, const Element &b) const
21 {
22 // make copy of a in case Inverse() overwrites it
23 Element a1(a);
24 return this->Add(a1, Inverse(b));
25 }
26
27 template <class T> T& AbstractGroup<T>::Accumulate(Element &a, const Element &b) const
28 {
29 return a = this->Add(a, b);
30 }
31
32 template <class T> T& AbstractGroup<T>::Reduce(Element &a, const Element &b) const
33 {
34 return a = this->Subtract(a, b);
35 }
36
37 template <class T> const T& AbstractRing<T>::Square(const Element &a) const
38 {
39 return this->Multiply(a, a);
40 }
41
42 template <class T> const T& AbstractRing<T>::Divide(const Element &a, const Element &b) const
43 {
44 // make copy of a in case MultiplicativeInverse() overwrites it
45 Element a1(a);
46 return this->Multiply(a1, this->MultiplicativeInverse(b));
47 }
48
49 template <class T> const T& AbstractEuclideanDomain<T>::Mod(const Element &a, const Element &b) const
50 {
51 Element q;
52 this->DivisionAlgorithm(result, q, a, b);
53 return result;
54 }
55
56 template <class T> const T& AbstractEuclideanDomain<T>::Gcd(const Element &a, const Element &b) const
57 {
58 Element g[3]={b, a};
59 unsigned int i0=0, i1=1, i2=2;
60
61 while (!this->Equal(g[i1], this->Identity()))
62 {
63 g[i2] = this->Mod(g[i0], g[i1]);
64 unsigned int t = i0; i0 = i1; i1 = i2; i2 = t;
65 }
66
67 return result = g[i0];
68 }
69
70 template <class T> const typename QuotientRing<T>::Element& QuotientRing<T>::MultiplicativeInverse(const Element &a) const
71 {
72 Element g[3]={m_modulus, a};
73 Element v[3]={m_domain.Identity(), m_domain.MultiplicativeIdentity()};
74 Element y;
75 unsigned int i0=0, i1=1, i2=2;
76
77 while (!this->Equal(g[i1], this->Identity()))
78 {
79 // y = g[i0] / g[i1];
80 // g[i2] = g[i0] % g[i1];
81 m_domain.DivisionAlgorithm(g[i2], y, g[i0], g[i1]);
82 // v[i2] = v[i0] - (v[i1] * y);
83 v[i2] = m_domain.Subtract(v[i0], m_domain.Multiply(v[i1], y));
84 unsigned int t = i0; i0 = i1; i1 = i2; i2 = t;
85 }
86
87 return m_domain.IsUnit(g[i0]) ? m_domain.Divide(v[i0], g[i0]) : m_domain.Identity();
88 }
89
90 template <class T> T AbstractGroup<T>::ScalarMultiply(const Element &base, const Integer &exponent) const
91 {
92 Element result;
93 this->SimultaneousMultiply(&result, base, &exponent, 1);
94 return result;
95 }
96
97 template <class T> T AbstractGroup<T>::CascadeScalarMultiply(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const
98 {
99 const unsigned expLen = STDMAX(e1.BitCount(), e2.BitCount());
100 if (expLen==0)
101 return this->Identity();
102
103 const unsigned w = (expLen <= 46 ? 1 : (expLen <= 260 ? 2 : 3));
104 const unsigned tableSize = 1<<w;
105 std::vector<Element> powerTable(tableSize << w);
106
107 powerTable[1] = x;
108 powerTable[tableSize] = y;
109 if (w==1)
110 powerTable[3] = this->Add(x,y);
111 else
112 {
113 powerTable[2] = this->Double(x);
114 powerTable[2*tableSize] = this->Double(y);
115
116 unsigned i, j;
117
118 for (i=3; i<tableSize; i+=2)
119 powerTable[i] = Add(powerTable[i-2], powerTable[2]);
120 for (i=1; i<tableSize; i+=2)
121 for (j=i+tableSize; j<(tableSize<<w); j+=tableSize)
122 powerTable[j] = Add(powerTable[j-tableSize], y);
123
124 for (i=3*tableSize; i<(tableSize<<w); i+=2*tableSize)
125 powerTable[i] = Add(powerTable[i-2*tableSize], powerTable[2*tableSize]);
126 for (i=tableSize; i<(tableSize<<w); i+=2*tableSize)
127 for (j=i+2; j<i+tableSize; j+=2)
128 powerTable[j] = Add(powerTable[j-1], x);
129 }
130
131 Element result;
132 unsigned power1 = 0, power2 = 0, prevPosition = expLen-1;
133 bool firstTime = true;
134
135 for (int i = expLen-1; i>=0; i--)
136 {
137 power1 = 2*power1 + e1.GetBit(i);
138 power2 = 2*power2 + e2.GetBit(i);
139
140 if (i==0 || 2*power1 >= tableSize || 2*power2 >= tableSize)
141 {
142 unsigned squaresBefore = prevPosition-i;
143 unsigned squaresAfter = 0;
144 prevPosition = i;
145 while ((power1 || power2) && power1%2 == 0 && power2%2==0)
146 {
147 power1 /= 2;
148 power2 /= 2;
149 squaresBefore--;
150 squaresAfter++;
151 }
152 if (firstTime)
153 {
154 result = powerTable[(power2<<w) + power1];
155 firstTime = false;
156 }
157 else
158 {
159 while (squaresBefore--)
160 result = this->Double(result);
161 if (power1 || power2)
162 Accumulate(result, powerTable[(power2<<w) + power1]);
163 }
164 while (squaresAfter--)
165 result = this->Double(result);
166 power1 = power2 = 0;
167 }
168 }
169 return result;
170 }
171
172 template <class Element, class Iterator> Element GeneralCascadeMultiplication(const AbstractGroup<Element> &group, Iterator begin, Iterator end)
173 {
174 if (end-begin == 1)
175 return group.ScalarMultiply(begin->base, begin->exponent);
176 else if (end-begin == 2)
177 return group.CascadeScalarMultiply(begin->base, begin->exponent, (begin+1)->base, (begin+1)->exponent);
178 else
179 {
180 Integer q, t;
181 Iterator last = end;
182 --last;
183
184 std::make_heap(begin, end);
185 std::pop_heap(begin, end);
186
187 while (!!begin->exponent)
188 {
189 // last->exponent is largest exponent, begin->exponent is next largest
190 t = last->exponent;
191 Integer::Divide(last->exponent, q, t, begin->exponent);
192
193 if (q == Integer::One())
194 group.Accumulate(begin->base, last->base); // avoid overhead of ScalarMultiply()
195 else
196 group.Accumulate(begin->base, group.ScalarMultiply(last->base, q));
197
198 std::push_heap(begin, end);
199 std::pop_heap(begin, end);
200 }
201
202 return group.ScalarMultiply(last->base, last->exponent);
203 }
204 }
205
206 struct WindowSlider
207 {
208 WindowSlider(const Integer &expIn, bool fastNegate, unsigned int windowSizeIn=0)
209 : exp(expIn), windowModulus(Integer::One()), windowSize(windowSizeIn), windowBegin(0), expWindow(0)
210 , fastNegate(fastNegate), negateNext(false), firstTime(true), finished(false)
211 {
212 if (windowSize == 0)
213 {
214 unsigned int expLen = exp.BitCount();
215 windowSize = expLen <= 17 ? 1 : (expLen <= 24 ? 2 : (expLen <= 70 ? 3 : (expLen <= 197 ? 4 : (expLen <= 539 ? 5 : (expLen <= 1434 ? 6 : 7)))));
216 }
217 windowModulus <<= windowSize;
218 }
219
220 void FindNextWindow()
221 {
222 unsigned int expLen = exp.WordCount() * WORD_BITS;
223 unsigned int skipCount = firstTime ? 0 : windowSize;
224 firstTime = false;
225 while (!exp.GetBit(skipCount))
226 {
227 if (skipCount >= expLen)
228 {
229 finished = true;
230 return;
231 }
232 skipCount++;
233 }
234
235 exp >>= skipCount;
236 windowBegin += skipCount;
237 expWindow = word32(exp % (word(1) << windowSize));
238
239 if (fastNegate && exp.GetBit(windowSize))
240 {
241 negateNext = true;
242 expWindow = (word32(1) << windowSize) - expWindow;
243 exp += windowModulus;
244 }
245 else
246 negateNext = false;
247 }
248
249 Integer exp, windowModulus;
250 unsigned int windowSize, windowBegin;
251 word32 expWindow;
252 bool fastNegate, negateNext, firstTime, finished;
253 };
254
255 template <class T>
256 void AbstractGroup<T>::SimultaneousMultiply(T *results, const T &base, const Integer *expBegin, unsigned int expCount) const
257 {
258 std::vector<std::vector<Element> > buckets(expCount);
259 std::vector<WindowSlider> exponents;
260 exponents.reserve(expCount);
261 unsigned int i;
262
263 for (i=0; expBegin && i<expCount; i++)
264 {
265 CRYPTOPP_ASSERT(expBegin->NotNegative());
266 exponents.push_back(WindowSlider(*expBegin++, InversionIsFast(), 0));
267 exponents[i].FindNextWindow();
268 buckets[i].resize(((size_t) 1) << (exponents[i].windowSize-1), Identity());
269 }
270
271 unsigned int expBitPosition = 0;
272 Element g = base;
273 bool notDone = true;
274
275 while (notDone)
276 {
277 notDone = false;
278 for (i=0; i<expCount; i++)
279 {
280 if (!exponents[i].finished && expBitPosition == exponents[i].windowBegin)
281 {
282 Element &bucket = buckets[i][exponents[i].expWindow/2];
283 if (exponents[i].negateNext)
284 Accumulate(bucket, Inverse(g));
285 else
286 Accumulate(bucket, g);
287 exponents[i].FindNextWindow();
288 }
289 notDone = notDone || !exponents[i].finished;
290 }
291
292 if (notDone)
293 {
294 g = Double(g);
295 expBitPosition++;
296 }
297 }
298
299 for (i=0; i<expCount; i++)
300 {
301 Element &r = *results++;
302 r = buckets[i][buckets[i].size()-1];
303 if (buckets[i].size() > 1)
304 {
305 for (int j = (int)buckets[i].size()-2; j >= 1; j--)
306 {
307 Accumulate(buckets[i][j], buckets[i][j+1]);
308 Accumulate(r, buckets[i][j]);
309 }
310 Accumulate(buckets[i][0], buckets[i][1]);
311 r = Add(Double(r), buckets[i][0]);
312 }
313 }
314 }
315
316 template <class T> T AbstractRing<T>::Exponentiate(const Element &base, const Integer &exponent) const
317 {
318 Element result;
319 SimultaneousExponentiate(&result, base, &exponent, 1);
320 return result;
321 }
322
323 template <class T> T AbstractRing<T>::CascadeExponentiate(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const
324 {
325 return MultiplicativeGroup().AbstractGroup<T>::CascadeScalarMultiply(x, e1, y, e2);
326 }
327
328 template <class Element, class Iterator> Element GeneralCascadeExponentiation(const AbstractRing<Element> &ring, Iterator begin, Iterator end)
329 {
330 return GeneralCascadeMultiplication<Element>(ring.MultiplicativeGroup(), begin, end);
331 }
332
333 template <class T>
334 void AbstractRing<T>::SimultaneousExponentiate(T *results, const T &base, const Integer *exponents, unsigned int expCount) const
335 {
336 MultiplicativeGroup().AbstractGroup<T>::SimultaneousMultiply(results, base, exponents, expCount);
337 }
338
339 NAMESPACE_END
340
341 #endif
algebra.h
Classes for performing mathematics over different fields.
AbstractEuclideanDomain::Gcd
virtual const Element & Gcd(const Element &a, const Element &b) const
Calculates the greatest common denominator in the ring.
Definition: algebra.cpp:56
AbstractEuclideanDomain::Mod
virtual const Element & Mod(const Element &a, const Element &b) const =0
Performs a modular reduction in the ring.
Definition: algebra.cpp:49
AbstractGroup
Abstract group.
Definition: algebra.h:27
AbstractGroup::Reduce
virtual Element & Reduce(Element &a, const Element &b) const
Reduces an element in the congruence class.
Definition: algebra.cpp:32
AbstractGroup::CascadeScalarMultiply
virtual Element CascadeScalarMultiply(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const
TODO.
Definition: algebra.cpp:97
AbstractGroup::SimultaneousMultiply
virtual void SimultaneousMultiply(Element *results, const Element &base, const Integer *exponents, unsigned int exponentsCount) const
Multiplies a base to multiple exponents in a group.
Definition: algebra.cpp:256
AbstractGroup::Subtract
virtual const Element & Subtract(const Element &a, const Element &b) const
Subtracts elements in the group.
Definition: algebra.cpp:20
AbstractGroup::Accumulate
virtual Element & Accumulate(Element &a, const Element &b) const
TODO.
Definition: algebra.cpp:27
AbstractGroup::ScalarMultiply
virtual Element ScalarMultiply(const Element &a, const Integer &e) const
Performs a scalar multiplication.
Definition: algebra.cpp:90
AbstractRing
Abstract ring.
Definition: algebra.h:119
AbstractRing::Exponentiate
virtual Element Exponentiate(const Element &a, const Integer &e) const
Raises a base to an exponent in the group.
Definition: algebra.cpp:316
AbstractRing::Square
virtual const Element & Square(const Element &a) const
Square an element in the group.
Definition: algebra.cpp:37
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
AbstractRing::Divide
virtual const Element & Divide(const Element &a, const Element &b) const
Divides elements in the group.
Definition: algebra.cpp:42
AbstractRing::CascadeExponentiate
virtual Element CascadeExponentiate(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const
TODO.
Definition: algebra.cpp:323
AbstractRing::MultiplicativeGroup
virtual const AbstractGroup< T > & MultiplicativeGroup() const
Retrieves the multiplicative group.
Definition: algebra.h:194
Integer
Multiple precision integer with arithmetic operations.
Definition: integer.h:50
Integer::Divide
static void Divide(Integer &r, Integer &q, const Integer &a, const Integer &d)
Extended Division.
Integer::GetBit
bool GetBit(size_t i) const
Provides the i-th bit of the Integer.
Integer::One
static const Integer & One()
Integer representing 1.
Integer::BitCount
unsigned int BitCount() const
Determines the number of bits required to represent the Integer.
Integer::NotNegative
bool NotNegative() const
Determines if the Integer is non-negative.
Definition: integer.h:344
Integer::WordCount
unsigned int WordCount() const
Determines the number of words required to represent the Integer.
QuotientRing::MultiplicativeInverse
const Element & MultiplicativeInverse(const Element &a) const
Calculate the multiplicative inverse of an element in the group.
Definition: algebra.cpp:70
word
word64 word
Full word used for multiprecision integer arithmetic.
Definition: config_int.h:192
WORD_BITS
const unsigned int WORD_BITS
Size of a platform word in bits.
Definition: config_int.h:260
word32
unsigned int word32
32-bit unsigned datatype
Definition: config_int.h:72
integer.h
Multiple precision integer with arithmetic operations.
STDMAX
const T & STDMAX(const T &a, const T &b)
Replacement function for std::max.
Definition: misc.h:668
CryptoPP
Crypto++ library namespace.
Name::Identity
const char * Identity()
ConstByteArrayParameter.
Definition: argnames.h:94
pch.h
Precompiled header file.
ECPPoint
Elliptical Curve Point over GF(p), where p is prime.
Definition: ecpoint.h:21
WindowSlider
Definition: algebra.cpp:207
CRYPTOPP_ASSERT
#define CRYPTOPP_ASSERT(exp)
Debugging and diagnostic assertion.
Definition: trap.h:68