00001
00002
00003 #include "pch.h"
00004 #include "algebra.h"
00005 #include "integer.h"
00006
00007 #include <vector>
00008
00009 NAMESPACE_BEGIN(CryptoPP)
00010
00011 template <class T> const T& AbstractGroup<T>::Double(const Element &a) const
00012 {
00013 return Add(a, a);
00014 }
00015
00016 template <class T> const T& AbstractGroup<T>::Subtract(const Element &a, const Element &b) const
00017 {
00018
00019 Element a1(a);
00020 return Add(a1, Inverse(b));
00021 }
00022
00023 template <class T> T& AbstractGroup<T>::Accumulate(Element &a, const Element &b) const
00024 {
00025 return a = Add(a, b);
00026 }
00027
00028 template <class T> T& AbstractGroup<T>::Reduce(Element &a, const Element &b) const
00029 {
00030 return a = Subtract(a, b);
00031 }
00032
00033 template <class T> const T& AbstractRing<T>::Square(const Element &a) const
00034 {
00035 return Multiply(a, a);
00036 }
00037
00038 template <class T> const T& AbstractRing<T>::Divide(const Element &a, const Element &b) const
00039 {
00040
00041 Element a1(a);
00042 return Multiply(a1, MultiplicativeInverse(b));
00043 }
00044
00045 template <class T> const T& AbstractEuclideanDomain<T>::Mod(const Element &a, const Element &b) const
00046 {
00047 Element q;
00048 DivisionAlgorithm(result, q, a, b);
00049 return result;
00050 }
00051
00052 template <class T> const T& AbstractEuclideanDomain<T>::Gcd(const Element &a, const Element &b) const
00053 {
00054 Element g[3]={b, a};
00055 unsigned int i0=0, i1=1, i2=2;
00056
00057 while (!Equal(g[i1], this->Identity()))
00058 {
00059 g[i2] = Mod(g[i0], g[i1]);
00060 unsigned int t = i0; i0 = i1; i1 = i2; i2 = t;
00061 }
00062
00063 return result = g[i0];
00064 }
00065
00066 template <class T> const typename QuotientRing<T>::Element& QuotientRing<T>::MultiplicativeInverse(const Element &a) const
00067 {
00068 Element g[3]={m_modulus, a};
00069 #ifdef __BCPLUSPLUS__
00070
00071 Element v[3];
00072 v[0]=m_domain.Identity();
00073 v[1]=m_domain.MultiplicativeIdentity();
00074 #else
00075 Element v[3]={m_domain.Identity(), m_domain.MultiplicativeIdentity()};
00076 #endif
00077 Element y;
00078 unsigned int i0=0, i1=1, i2=2;
00079
00080 while (!Equal(g[i1], Identity()))
00081 {
00082
00083
00084 m_domain.DivisionAlgorithm(g[i2], y, g[i0], g[i1]);
00085
00086 v[i2] = m_domain.Subtract(v[i0], m_domain.Multiply(v[i1], y));
00087 unsigned int t = i0; i0 = i1; i1 = i2; i2 = t;
00088 }
00089
00090 return m_domain.IsUnit(g[i0]) ? m_domain.Divide(v[i0], g[i0]) : m_domain.Identity();
00091 }
00092
00093 template <class T> T AbstractGroup<T>::ScalarMultiply(const Element &base, const Integer &exponent) const
00094 {
00095 Element result;
00096 SimultaneousMultiply(&result, base, &exponent, 1);
00097 return result;
00098 }
00099
00100 template <class T> T AbstractGroup<T>::CascadeScalarMultiply(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const
00101 {
00102 const unsigned expLen = STDMAX(e1.BitCount(), e2.BitCount());
00103 if (expLen==0)
00104 return Identity();
00105
00106 const unsigned w = (expLen <= 46 ? 1 : (expLen <= 260 ? 2 : 3));
00107 const unsigned tableSize = 1<<w;
00108 std::vector<Element> powerTable(tableSize << w);
00109
00110 powerTable[1] = x;
00111 powerTable[tableSize] = y;
00112 if (w==1)
00113 powerTable[3] = Add(x,y);
00114 else
00115 {
00116 powerTable[2] = Double(x);
00117 powerTable[2*tableSize] = Double(y);
00118
00119 unsigned i, j;
00120
00121 for (i=3; i<tableSize; i+=2)
00122 powerTable[i] = Add(powerTable[i-2], powerTable[2]);
00123 for (i=1; i<tableSize; i+=2)
00124 for (j=i+tableSize; j<(tableSize<<w); j+=tableSize)
00125 powerTable[j] = Add(powerTable[j-tableSize], y);
00126
00127 for (i=3*tableSize; i<(tableSize<<w); i+=2*tableSize)
00128 powerTable[i] = Add(powerTable[i-2*tableSize], powerTable[2*tableSize]);
00129 for (i=tableSize; i<(tableSize<<w); i+=2*tableSize)
00130 for (j=i+2; j<i+tableSize; j+=2)
00131 powerTable[j] = Add(powerTable[j-1], x);
00132 }
00133
00134 Element result;
00135 unsigned power1 = 0, power2 = 0, prevPosition = expLen-1;
00136 bool firstTime = true;
00137
00138 for (int i = expLen-1; i>=0; i--)
00139 {
00140 power1 = 2*power1 + e1.GetBit(i);
00141 power2 = 2*power2 + e2.GetBit(i);
00142
00143 if (i==0 || 2*power1 >= tableSize || 2*power2 >= tableSize)
00144 {
00145 unsigned squaresBefore = prevPosition-i;
00146 unsigned squaresAfter = 0;
00147 prevPosition = i;
00148 while ((power1 || power2) && power1%2 == 0 && power2%2==0)
00149 {
00150 power1 /= 2;
00151 power2 /= 2;
00152 squaresBefore--;
00153 squaresAfter++;
00154 }
00155 if (firstTime)
00156 {
00157 result = powerTable[(power2<<w) + power1];
00158 firstTime = false;
00159 }
00160 else
00161 {
00162 while (squaresBefore--)
00163 result = Double(result);
00164 if (power1 || power2)
00165 Accumulate(result, powerTable[(power2<<w) + power1]);
00166 }
00167 while (squaresAfter--)
00168 result = Double(result);
00169 power1 = power2 = 0;
00170 }
00171 }
00172 return result;
00173 }
00174
00175 template <class Element, class Iterator> Element GeneralCascadeMultiplication(const AbstractGroup<Element> &group, Iterator begin, Iterator end)
00176 {
00177 if (end-begin == 1)
00178 return group.ScalarMultiply(begin->base, begin->exponent);
00179 else if (end-begin == 2)
00180 return group.CascadeScalarMultiply(begin->base, begin->exponent, (begin+1)->base, (begin+1)->exponent);
00181 else
00182 {
00183 Integer q, t;
00184 Iterator last = end;
00185 --last;
00186
00187 std::make_heap(begin, end);
00188 std::pop_heap(begin, end);
00189
00190 while (!!begin->exponent)
00191 {
00192
00193 t = last->exponent;
00194 Integer::Divide(last->exponent, q, t, begin->exponent);
00195
00196 if (q == Integer::One())
00197 group.Accumulate(begin->base, last->base);
00198 else
00199 group.Accumulate(begin->base, group.ScalarMultiply(last->base, q));
00200
00201 std::push_heap(begin, end);
00202 std::pop_heap(begin, end);
00203 }
00204
00205 return group.ScalarMultiply(last->base, last->exponent);
00206 }
00207 }
00208
00209 struct WindowSlider
00210 {
00211 WindowSlider(const Integer &exp, bool fastNegate, unsigned int windowSizeIn=0)
00212 : exp(exp), windowModulus(Integer::One()), windowSize(windowSizeIn), windowBegin(0), fastNegate(fastNegate), firstTime(true), finished(false)
00213 {
00214 if (windowSize == 0)
00215 {
00216 unsigned int expLen = exp.BitCount();
00217 windowSize = expLen <= 17 ? 1 : (expLen <= 24 ? 2 : (expLen <= 70 ? 3 : (expLen <= 197 ? 4 : (expLen <= 539 ? 5 : (expLen <= 1434 ? 6 : 7)))));
00218 }
00219 windowModulus <<= windowSize;
00220 }
00221
00222 void FindNextWindow()
00223 {
00224 unsigned int expLen = exp.WordCount() * WORD_BITS;
00225 unsigned int skipCount = firstTime ? 0 : windowSize;
00226 firstTime = false;
00227 while (!exp.GetBit(skipCount))
00228 {
00229 if (skipCount >= expLen)
00230 {
00231 finished = true;
00232 return;
00233 }
00234 skipCount++;
00235 }
00236
00237 exp >>= skipCount;
00238 windowBegin += skipCount;
00239 expWindow = exp % (1 << windowSize);
00240
00241 if (fastNegate && exp.GetBit(windowSize))
00242 {
00243 negateNext = true;
00244 expWindow = (1 << windowSize) - expWindow;
00245 exp += windowModulus;
00246 }
00247 else
00248 negateNext = false;
00249 }
00250
00251 Integer exp, windowModulus;
00252 unsigned int windowSize, windowBegin, expWindow;
00253 bool fastNegate, negateNext, firstTime, finished;
00254 };
00255
00256 template <class T>
00257 void AbstractGroup<T>::SimultaneousMultiply(T *results, const T &base, const Integer *expBegin, unsigned int expCount) const
00258 {
00259 std::vector<std::vector<Element> > buckets(expCount);
00260 std::vector<WindowSlider> exponents;
00261 exponents.reserve(expCount);
00262 unsigned int i;
00263
00264 for (i=0; i<expCount; i++)
00265 {
00266 assert(expBegin->NotNegative());
00267 exponents.push_back(WindowSlider(*expBegin++, InversionIsFast(), 0));
00268 exponents[i].FindNextWindow();
00269 buckets[i].resize(1<<(exponents[i].windowSize-1), Identity());
00270 }
00271
00272 unsigned int expBitPosition = 0;
00273 Element g = base;
00274 bool notDone = true;
00275
00276 while (notDone)
00277 {
00278 notDone = false;
00279 for (i=0; i<expCount; i++)
00280 {
00281 if (!exponents[i].finished && expBitPosition == exponents[i].windowBegin)
00282 {
00283 Element &bucket = buckets[i][exponents[i].expWindow/2];
00284 if (exponents[i].negateNext)
00285 Accumulate(bucket, Inverse(g));
00286 else
00287 Accumulate(bucket, g);
00288 exponents[i].FindNextWindow();
00289 }
00290 notDone = notDone || !exponents[i].finished;
00291 }
00292
00293 if (notDone)
00294 {
00295 g = Double(g);
00296 expBitPosition++;
00297 }
00298 }
00299
00300 for (i=0; i<expCount; i++)
00301 {
00302 Element &r = *results++;
00303 r = buckets[i][buckets[i].size()-1];
00304 if (buckets[i].size() > 1)
00305 {
00306 for (int j = buckets[i].size()-2; j >= 1; j--)
00307 {
00308 Accumulate(buckets[i][j], buckets[i][j+1]);
00309 Accumulate(r, buckets[i][j]);
00310 }
00311 Accumulate(buckets[i][0], buckets[i][1]);
00312 r = Add(Double(r), buckets[i][0]);
00313 }
00314 }
00315 }
00316
00317 template <class T> T AbstractRing<T>::Exponentiate(const Element &base, const Integer &exponent) const
00318 {
00319 Element result;
00320 SimultaneousExponentiate(&result, base, &exponent, 1);
00321 return result;
00322 }
00323
00324 template <class T> T AbstractRing<T>::CascadeExponentiate(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const
00325 {
00326 return MultiplicativeGroup().AbstractGroup<T>::CascadeScalarMultiply(x, e1, y, e2);
00327 }
00328
00329 template <class Element, class Iterator> Element GeneralCascadeExponentiation(const AbstractRing<Element> &ring, Iterator begin, Iterator end)
00330 {
00331 return GeneralCascadeMultiplication<Element>(ring.MultiplicativeGroup(), begin, end);
00332 }
00333
00334 template <class T>
00335 void AbstractRing<T>::SimultaneousExponentiate(T *results, const T &base, const Integer *exponents, unsigned int expCount) const
00336 {
00337 MultiplicativeGroup().AbstractGroup<T>::SimultaneousMultiply(results, base, exponents, expCount);
00338 }
00339
00340 NAMESPACE_END