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algebra.cpp

00001 // algebra.cpp - written and placed in the public domain by Wei Dai
00002 
00003 #include "pch.h"
00004 
00005 #include "algebra.h"
00006 #include "integer.h"
00007 
00008 #include <vector>
00009 
00010 NAMESPACE_BEGIN(CryptoPP)
00011 
00012 template <class T> const T& AbstractGroup<T>::Double(const Element &a) const
00013 {
00014         return Add(a, a);
00015 }
00016 
00017 template <class T> const T& AbstractGroup<T>::Subtract(const Element &a, const Element &b) const
00018 {
00019         // make copy of a in case Inverse() overwrites it
00020         Element a1(a);
00021         return Add(a1, Inverse(b));
00022 }
00023 
00024 template <class T> T& AbstractGroup<T>::Accumulate(Element &a, const Element &b) const
00025 {
00026         return a = Add(a, b);
00027 }
00028 
00029 template <class T> T& AbstractGroup<T>::Reduce(Element &a, const Element &b) const
00030 {
00031         return a = Subtract(a, b);
00032 }
00033 
00034 template <class T> const T& AbstractRing<T>::Square(const Element &a) const
00035 {
00036         return Multiply(a, a);
00037 }
00038 
00039 template <class T> const T& AbstractRing<T>::Divide(const Element &a, const Element &b) const
00040 {
00041         // make copy of a in case MultiplicativeInverse() overwrites it
00042         Element a1(a);
00043         return Multiply(a1, MultiplicativeInverse(b));
00044 }
00045 
00046 template <class T> const T& AbstractEuclideanDomain<T>::Mod(const Element &a, const Element &b) const
00047 {
00048         Element q;
00049         DivisionAlgorithm(result, q, a, b);
00050         return result;
00051 }
00052 
00053 template <class T> const T& AbstractEuclideanDomain<T>::Gcd(const Element &a, const Element &b) const
00054 {
00055         Element g[3]={b, a};
00056         unsigned int i0=0, i1=1, i2=2;
00057 
00058         while (!Equal(g[i1], Identity()))
00059         {
00060                 g[i2] = Mod(g[i0], g[i1]);
00061                 unsigned int t = i0; i0 = i1; i1 = i2; i2 = t;
00062         }
00063 
00064         return result = g[i0];
00065 }
00066 
00067 template <class T> const typename QuotientRing<T>::Element& QuotientRing<T>::MultiplicativeInverse(const Element &a) const
00068 {
00069         Element g[3]={m_modulus, a};
00070 #ifdef __BCPLUSPLUS__
00071     // BC++50 workaround          
00072         Element v[3];
00073     v[0]=m_domain.Identity();
00074     v[1]=m_domain.MultiplicativeIdentity();
00075 #else
00076         Element v[3]={m_domain.Identity(), m_domain.MultiplicativeIdentity()};
00077 #endif
00078         Element y;
00079         unsigned int i0=0, i1=1, i2=2;
00080 
00081         while (!Equal(g[i1], Identity()))
00082         {
00083                 // y = g[i0] / g[i1];
00084                 // g[i2] = g[i0] % g[i1];
00085                 m_domain.DivisionAlgorithm(g[i2], y, g[i0], g[i1]);
00086                 // v[i2] = v[i0] - (v[i1] * y);
00087                 v[i2] = m_domain.Subtract(v[i0], m_domain.Multiply(v[i1], y));
00088                 unsigned int t = i0; i0 = i1; i1 = i2; i2 = t;
00089         }
00090 
00091         return m_domain.IsUnit(g[i0]) ? m_domain.Divide(v[i0], g[i0]) : m_domain.Identity();
00092 }
00093 
00094 template <class T> T AbstractGroup<T>::ScalarMultiply(const Element &base, const Integer &exponent) const
00095 {
00096         Element result;
00097         SimultaneousMultiply(&result, base, &exponent, 1);
00098         return result;
00099 }
00100 
00101 template <class T> T AbstractGroup<T>::CascadeScalarMultiply(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const
00102 {
00103         const unsigned expLen = STDMAX(e1.BitCount(), e2.BitCount());
00104         if (expLen==0)
00105                 return Identity();
00106 
00107         const unsigned w = (expLen <= 46 ? 1 : (expLen <= 260 ? 2 : 3));
00108         const unsigned tableSize = 1<<w;
00109         std::vector<Element> powerTable(tableSize << w);
00110 
00111         powerTable[1] = x;
00112         powerTable[tableSize] = y;
00113         if (w==1)
00114                 powerTable[3] = Add(x,y);
00115         else
00116         {
00117                 powerTable[2] = Double(x);
00118                 powerTable[2*tableSize] = Double(y);
00119 
00120                 unsigned i, j;
00121 
00122                 for (i=3; i<tableSize; i+=2)
00123                         powerTable[i] = Add(powerTable[i-2], powerTable[2]);
00124                 for (i=1; i<tableSize; i+=2)
00125                         for (j=i+tableSize; j<(tableSize<<w); j+=tableSize)
00126                                 powerTable[j] = Add(powerTable[j-tableSize], y);
00127 
00128                 for (i=3*tableSize; i<(tableSize<<w); i+=2*tableSize)
00129                         powerTable[i] = Add(powerTable[i-2*tableSize], powerTable[2*tableSize]);
00130                 for (i=tableSize; i<(tableSize<<w); i+=2*tableSize)
00131                         for (j=i+2; j<i+tableSize; j+=2)
00132                                 powerTable[j] = Add(powerTable[j-1], x);
00133         }
00134 
00135         Element result;
00136         unsigned power1 = 0, power2 = 0, prevPosition = expLen-1;
00137         bool firstTime = true;
00138 
00139         for (int i = expLen-1; i>=0; i--)
00140         {
00141                 power1 = 2*power1 + e1.GetBit(i);
00142                 power2 = 2*power2 + e2.GetBit(i);
00143 
00144                 if (i==0 || 2*power1 >= tableSize || 2*power2 >= tableSize)
00145                 {
00146                         unsigned squaresBefore = prevPosition-i;
00147                         unsigned squaresAfter = 0;
00148                         prevPosition = i;
00149                         while ((power1 || power2) && power1%2 == 0 && power2%2==0)
00150                         {
00151                                 power1 /= 2;
00152                                 power2 /= 2;
00153                                 squaresBefore--;
00154                                 squaresAfter++;
00155                         }
00156                         if (firstTime)
00157                         {
00158                                 result = powerTable[(power2<<w) + power1];
00159                                 firstTime = false;
00160                         }
00161                         else
00162                         {
00163                                 while (squaresBefore--)
00164                                         result = Double(result);
00165                                 if (power1 || power2)
00166                                         Accumulate(result, powerTable[(power2<<w) + power1]);
00167                         }
00168                         while (squaresAfter--)
00169                                 result = Double(result);
00170                         power1 = power2 = 0;
00171                 }
00172         }
00173         return result;
00174 }
00175 
00176 template <class Element, class Iterator> Element GeneralCascadeMultiplication(const AbstractGroup<Element> &group, Iterator begin, Iterator end)
00177 {
00178         if (end-begin == 1)
00179                 return group.ScalarMultiply(begin->base, begin->exponent);
00180         else if (end-begin == 2)
00181                 return group.CascadeScalarMultiply(begin->base, begin->exponent, (begin+1)->base, (begin+1)->exponent);
00182         else
00183         {
00184                 Integer q, t;
00185                 Iterator last = end;
00186                 --last;
00187 
00188                 std::make_heap(begin, end);
00189                 std::pop_heap(begin, end);
00190 
00191                 while (!!begin->exponent)
00192                 {
00193                         // last->exponent is largest exponent, begin->exponent is next largest
00194                         t = last->exponent;
00195                         Integer::Divide(last->exponent, q, t, begin->exponent);
00196 
00197                         if (q == Integer::One())
00198                                 group.Accumulate(begin->base, last->base);      // avoid overhead of ScalarMultiply()
00199                         else
00200                                 group.Accumulate(begin->base, group.ScalarMultiply(last->base, q));
00201 
00202                         std::push_heap(begin, end);
00203                         std::pop_heap(begin, end);
00204                 }
00205 
00206                 return group.ScalarMultiply(last->base, last->exponent);
00207         }
00208 }
00209 
00210 struct WindowSlider
00211 {
00212         WindowSlider(const Integer &exp, bool fastNegate, unsigned int windowSizeIn=0)
00213                 : exp(exp), windowModulus(Integer::One()), windowSize(windowSizeIn), windowBegin(0), fastNegate(fastNegate), firstTime(true), finished(false)
00214         {
00215                 if (windowSize == 0)
00216                 {
00217                         unsigned int expLen = exp.BitCount();
00218                         windowSize = expLen <= 17 ? 1 : (expLen <= 24 ? 2 : (expLen <= 70 ? 3 : (expLen <= 197 ? 4 : (expLen <= 539 ? 5 : (expLen <= 1434 ? 6 : 7)))));
00219                 }
00220                 windowModulus <<= windowSize;
00221         }
00222 
00223         void FindNextWindow()
00224         {
00225                 unsigned int expLen = exp.WordCount() * WORD_BITS;
00226                 unsigned int skipCount = firstTime ? 0 : windowSize;
00227                 firstTime = false;
00228                 while (!exp.GetBit(skipCount))
00229                 {
00230                         if (skipCount >= expLen)
00231                         {
00232                                 finished = true;
00233                                 return;
00234                         }
00235                         skipCount++;
00236                 }
00237 
00238                 exp >>= skipCount;
00239                 windowBegin += skipCount;
00240                 expWindow = exp % (1 << windowSize);
00241 
00242                 if (fastNegate && exp.GetBit(windowSize))
00243                 {
00244                         negateNext = true;
00245                         expWindow = (1 << windowSize) - expWindow;
00246                         exp += windowModulus;
00247                 }
00248                 else
00249                         negateNext = false;
00250         }
00251 
00252         Integer exp, windowModulus;
00253         unsigned int windowSize, windowBegin, expWindow;
00254         bool fastNegate, negateNext, firstTime, finished;
00255 };
00256 
00257 template <class T>
00258 void AbstractGroup<T>::SimultaneousMultiply(T *results, const T &base, const Integer *expBegin, unsigned int expCount) const
00259 {
00260         std::vector<std::vector<Element> > buckets(expCount);
00261         std::vector<WindowSlider> exponents;
00262         exponents.reserve(expCount);
00263         unsigned int i;
00264 
00265         for (i=0; i<expCount; i++)
00266         {
00267                 assert(expBegin->NotNegative());
00268                 exponents.push_back(WindowSlider(*expBegin++, InversionIsFast(), 0));
00269                 exponents[i].FindNextWindow();
00270                 buckets[i].resize(1<<(exponents[i].windowSize-1), Identity());
00271         }
00272 
00273         unsigned int expBitPosition = 0;
00274         Element g = base;
00275         bool notDone = true;
00276 
00277         while (notDone)
00278         {
00279                 notDone = false;
00280                 for (i=0; i<expCount; i++)
00281                 {
00282                         if (!exponents[i].finished && expBitPosition == exponents[i].windowBegin)
00283                         {
00284                                 Element &bucket = buckets[i][exponents[i].expWindow/2];
00285                                 if (exponents[i].negateNext)
00286                                         Accumulate(bucket, Inverse(g));
00287                                 else
00288                                         Accumulate(bucket, g);
00289                                 exponents[i].FindNextWindow();
00290                         }
00291                         notDone = notDone || !exponents[i].finished;
00292                 }
00293 
00294                 if (notDone)
00295                 {
00296                         g = Double(g);
00297                         expBitPosition++;
00298                 }
00299         }
00300 
00301         for (i=0; i<expCount; i++)
00302         {
00303                 Element &r = *results++;
00304                 r = buckets[i][buckets[i].size()-1];
00305                 if (buckets[i].size() > 1)
00306                 {
00307                         for (int j = buckets[i].size()-2; j >= 1; j--)
00308                         {
00309                                 Accumulate(buckets[i][j], buckets[i][j+1]);
00310                                 Accumulate(r, buckets[i][j]);
00311                         }
00312                         Accumulate(buckets[i][0], buckets[i][1]);
00313                         r = Add(Double(r), buckets[i][0]);
00314                 }
00315         }
00316 }
00317 
00318 template <class T> T AbstractRing<T>::Exponentiate(const Element &base, const Integer &exponent) const
00319 {
00320         Element result;
00321         SimultaneousExponentiate(&result, base, &exponent, 1);
00322         return result;
00323 }
00324 
00325 template <class T> T AbstractRing<T>::CascadeExponentiate(const Element &x, const Integer &e1, const Element &y, const Integer &e2) const
00326 {
00327         return MultiplicativeGroup().AbstractGroup<T>::CascadeScalarMultiply(x, e1, y, e2);
00328 }
00329 
00330 template <class Element, class Iterator> Element GeneralCascadeExponentiation(const AbstractRing<Element> &ring, Iterator begin, Iterator end)
00331 {
00332         return GeneralCascadeMultiplication<Element>(ring.MultiplicativeGroup(), begin, end);
00333 }
00334 
00335 template <class T>
00336 void AbstractRing<T>::SimultaneousExponentiate(T *results, const T &base, const Integer *exponents, unsigned int expCount) const
00337 {
00338         MultiplicativeGroup().AbstractGroup<T>::SimultaneousMultiply(results, base, exponents, expCount);
00339 }
00340 
00341 NAMESPACE_END

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