578 lines
14 KiB
C++
578 lines
14 KiB
C++
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// polynomi.cpp - written and placed in the public domain by Wei Dai
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// Part of the code for polynomial evaluation and interpolation
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// originally came from Hal Finney's public domain secsplit.c.
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#include "pch.h"
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#include "polynomi.h"
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#include "secblock.h"
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#include <sstream>
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#include <iostream>
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NAMESPACE_BEGIN(CryptoPP)
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template <class T>
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void PolynomialOver<T>::Randomize(RandomNumberGenerator &rng, const RandomizationParameter ¶meter, const Ring &ring)
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{
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m_coefficients.resize(parameter.m_coefficientCount);
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for (unsigned int i=0; i<m_coefficients.size(); ++i)
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m_coefficients[i] = ring.RandomElement(rng, parameter.m_coefficientParameter);
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}
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template <class T>
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void PolynomialOver<T>::FromStr(const char *str, const Ring &ring)
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{
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std::istringstream in((char *)str);
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bool positive = true;
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CoefficientType coef;
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unsigned int power;
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while (in)
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{
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std::ws(in);
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if (in.peek() == 'x')
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coef = ring.MultiplicativeIdentity();
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else
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in >> coef;
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std::ws(in);
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if (in.peek() == 'x')
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{
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in.get();
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std::ws(in);
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if (in.peek() == '^')
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{
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in.get();
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in >> power;
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}
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else
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power = 1;
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}
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else
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power = 0;
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if (!positive)
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coef = ring.Inverse(coef);
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SetCoefficient(power, coef, ring);
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std::ws(in);
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switch (in.get())
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{
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case '+':
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positive = true;
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break;
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case '-':
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positive = false;
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break;
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default:
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return; // something's wrong with the input string
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}
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}
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}
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template <class T>
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unsigned int PolynomialOver<T>::CoefficientCount(const Ring &ring) const
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{
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unsigned count = m_coefficients.size();
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while (count && ring.Equal(m_coefficients[count-1], ring.Identity()))
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count--;
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const_cast<std::vector<CoefficientType> &>(m_coefficients).resize(count);
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return count;
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}
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template <class T>
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typename PolynomialOver<T>::CoefficientType PolynomialOver<T>::GetCoefficient(unsigned int i, const Ring &ring) const
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{
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return (i < m_coefficients.size()) ? m_coefficients[i] : ring.Identity();
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}
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template <class T>
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PolynomialOver<T>& PolynomialOver<T>::operator=(const PolynomialOver<T>& t)
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{
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if (this != &t)
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{
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m_coefficients.resize(t.m_coefficients.size());
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for (unsigned int i=0; i<m_coefficients.size(); i++)
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m_coefficients[i] = t.m_coefficients[i];
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}
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return *this;
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}
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template <class T>
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PolynomialOver<T>& PolynomialOver<T>::Accumulate(const PolynomialOver<T>& t, const Ring &ring)
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{
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unsigned int count = t.CoefficientCount(ring);
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if (count > CoefficientCount(ring))
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m_coefficients.resize(count, ring.Identity());
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for (unsigned int i=0; i<count; i++)
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ring.Accumulate(m_coefficients[i], t.GetCoefficient(i, ring));
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return *this;
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}
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template <class T>
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PolynomialOver<T>& PolynomialOver<T>::Reduce(const PolynomialOver<T>& t, const Ring &ring)
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{
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unsigned int count = t.CoefficientCount(ring);
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if (count > CoefficientCount(ring))
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m_coefficients.resize(count, ring.Identity());
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for (unsigned int i=0; i<count; i++)
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ring.Reduce(m_coefficients[i], t.GetCoefficient(i, ring));
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return *this;
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}
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template <class T>
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typename PolynomialOver<T>::CoefficientType PolynomialOver<T>::EvaluateAt(const CoefficientType &x, const Ring &ring) const
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{
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int degree = Degree(ring);
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if (degree < 0)
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return ring.Identity();
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CoefficientType result = m_coefficients[degree];
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for (int j=degree-1; j>=0; j--)
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{
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result = ring.Multiply(result, x);
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ring.Accumulate(result, m_coefficients[j]);
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}
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return result;
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}
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template <class T>
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PolynomialOver<T>& PolynomialOver<T>::ShiftLeft(unsigned int n, const Ring &ring)
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{
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unsigned int i = CoefficientCount(ring) + n;
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m_coefficients.resize(i, ring.Identity());
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while (i > n)
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{
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i--;
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m_coefficients[i] = m_coefficients[i-n];
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}
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while (i)
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{
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i--;
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m_coefficients[i] = ring.Identity();
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}
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return *this;
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}
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template <class T>
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PolynomialOver<T>& PolynomialOver<T>::ShiftRight(unsigned int n, const Ring &ring)
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{
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unsigned int count = CoefficientCount(ring);
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if (count > n)
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{
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for (unsigned int i=0; i<count-n; i++)
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m_coefficients[i] = m_coefficients[i+n];
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m_coefficients.resize(count-n, ring.Identity());
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}
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else
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m_coefficients.resize(0, ring.Identity());
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return *this;
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}
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template <class T>
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void PolynomialOver<T>::SetCoefficient(unsigned int i, const CoefficientType &value, const Ring &ring)
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{
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if (i >= m_coefficients.size())
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m_coefficients.resize(i+1, ring.Identity());
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m_coefficients[i] = value;
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}
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template <class T>
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void PolynomialOver<T>::Negate(const Ring &ring)
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{
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unsigned int count = CoefficientCount(ring);
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for (unsigned int i=0; i<count; i++)
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m_coefficients[i] = ring.Inverse(m_coefficients[i]);
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}
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template <class T>
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void PolynomialOver<T>::swap(PolynomialOver<T> &t)
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{
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m_coefficients.swap(t.m_coefficients);
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}
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template <class T>
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bool PolynomialOver<T>::Equals(const PolynomialOver<T>& t, const Ring &ring) const
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{
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unsigned int count = CoefficientCount(ring);
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if (count != t.CoefficientCount(ring))
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return false;
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for (unsigned int i=0; i<count; i++)
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if (!ring.Equal(m_coefficients[i], t.m_coefficients[i]))
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return false;
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return true;
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}
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template <class T>
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PolynomialOver<T> PolynomialOver<T>::Plus(const PolynomialOver<T>& t, const Ring &ring) const
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{
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unsigned int i;
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unsigned int count = CoefficientCount(ring);
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unsigned int tCount = t.CoefficientCount(ring);
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if (count > tCount)
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{
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PolynomialOver<T> result(ring, count);
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for (i=0; i<tCount; i++)
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result.m_coefficients[i] = ring.Add(m_coefficients[i], t.m_coefficients[i]);
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for (; i<count; i++)
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result.m_coefficients[i] = m_coefficients[i];
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return result;
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}
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else
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{
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PolynomialOver<T> result(ring, tCount);
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for (i=0; i<count; i++)
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result.m_coefficients[i] = ring.Add(m_coefficients[i], t.m_coefficients[i]);
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for (; i<tCount; i++)
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result.m_coefficients[i] = t.m_coefficients[i];
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return result;
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}
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}
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template <class T>
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PolynomialOver<T> PolynomialOver<T>::Minus(const PolynomialOver<T>& t, const Ring &ring) const
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{
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unsigned int i;
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unsigned int count = CoefficientCount(ring);
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unsigned int tCount = t.CoefficientCount(ring);
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if (count > tCount)
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{
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PolynomialOver<T> result(ring, count);
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for (i=0; i<tCount; i++)
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result.m_coefficients[i] = ring.Subtract(m_coefficients[i], t.m_coefficients[i]);
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for (; i<count; i++)
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result.m_coefficients[i] = m_coefficients[i];
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return result;
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}
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else
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{
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PolynomialOver<T> result(ring, tCount);
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for (i=0; i<count; i++)
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result.m_coefficients[i] = ring.Subtract(m_coefficients[i], t.m_coefficients[i]);
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for (; i<tCount; i++)
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result.m_coefficients[i] = ring.Inverse(t.m_coefficients[i]);
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return result;
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}
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}
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template <class T>
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PolynomialOver<T> PolynomialOver<T>::Inverse(const Ring &ring) const
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{
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unsigned int count = CoefficientCount(ring);
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PolynomialOver<T> result(ring, count);
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for (unsigned int i=0; i<count; i++)
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result.m_coefficients[i] = ring.Inverse(m_coefficients[i]);
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return result;
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}
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template <class T>
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PolynomialOver<T> PolynomialOver<T>::Times(const PolynomialOver<T>& t, const Ring &ring) const
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{
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if (IsZero(ring) || t.IsZero(ring))
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return PolynomialOver<T>();
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unsigned int count1 = CoefficientCount(ring), count2 = t.CoefficientCount(ring);
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PolynomialOver<T> result(ring, count1 + count2 - 1);
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for (unsigned int i=0; i<count1; i++)
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for (unsigned int j=0; j<count2; j++)
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ring.Accumulate(result.m_coefficients[i+j], ring.Multiply(m_coefficients[i], t.m_coefficients[j]));
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return result;
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}
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template <class T>
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PolynomialOver<T> PolynomialOver<T>::DividedBy(const PolynomialOver<T>& t, const Ring &ring) const
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{
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PolynomialOver<T> remainder, quotient;
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Divide(remainder, quotient, *this, t, ring);
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return quotient;
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}
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template <class T>
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PolynomialOver<T> PolynomialOver<T>::Modulo(const PolynomialOver<T>& t, const Ring &ring) const
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{
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PolynomialOver<T> remainder, quotient;
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Divide(remainder, quotient, *this, t, ring);
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return remainder;
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}
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template <class T>
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PolynomialOver<T> PolynomialOver<T>::MultiplicativeInverse(const Ring &ring) const
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{
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return Degree(ring)==0 ? ring.MultiplicativeInverse(m_coefficients[0]) : ring.Identity();
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}
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template <class T>
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bool PolynomialOver<T>::IsUnit(const Ring &ring) const
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{
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return Degree(ring)==0 && ring.IsUnit(m_coefficients[0]);
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}
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template <class T>
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std::istream& PolynomialOver<T>::Input(std::istream &in, const Ring &ring)
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{
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char c;
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unsigned int length = 0;
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SecBlock<char> str(length + 16);
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bool paren = false;
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std::ws(in);
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if (in.peek() == '(')
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{
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paren = true;
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in.get();
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}
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do
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{
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in.read(&c, 1);
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str[length++] = c;
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if (length >= str.size())
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str.Grow(length + 16);
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}
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// if we started with a left paren, then read until we find a right paren,
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// otherwise read until the end of the line
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while (in && ((paren && c != ')') || (!paren && c != '\n')));
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str[length-1] = '\0';
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*this = PolynomialOver<T>(str, ring);
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return in;
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}
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template <class T>
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std::ostream& PolynomialOver<T>::Output(std::ostream &out, const Ring &ring) const
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{
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unsigned int i = CoefficientCount(ring);
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if (i)
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{
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bool firstTerm = true;
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while (i--)
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{
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if (m_coefficients[i] != ring.Identity())
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{
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if (firstTerm)
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{
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firstTerm = false;
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if (!i || !ring.Equal(m_coefficients[i], ring.MultiplicativeIdentity()))
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out << m_coefficients[i];
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}
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else
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{
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CoefficientType inverse = ring.Inverse(m_coefficients[i]);
|
||
|
|
std::ostringstream pstr, nstr;
|
||
|
|
|
||
|
|
pstr << m_coefficients[i];
|
||
|
|
nstr << inverse;
|
||
|
|
|
||
|
|
if (pstr.str().size() <= nstr.str().size())
|
||
|
|
{
|
||
|
|
out << " + ";
|
||
|
|
if (!i || !ring.Equal(m_coefficients[i], ring.MultiplicativeIdentity()))
|
||
|
|
out << m_coefficients[i];
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
out << " - ";
|
||
|
|
if (!i || !ring.Equal(inverse, ring.MultiplicativeIdentity()))
|
||
|
|
out << inverse;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
switch (i)
|
||
|
|
{
|
||
|
|
case 0:
|
||
|
|
break;
|
||
|
|
case 1:
|
||
|
|
out << "x";
|
||
|
|
break;
|
||
|
|
default:
|
||
|
|
out << "x^" << i;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
out << ring.Identity();
|
||
|
|
}
|
||
|
|
return out;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T>
|
||
|
|
void PolynomialOver<T>::Divide(PolynomialOver<T> &r, PolynomialOver<T> &q, const PolynomialOver<T> &a, const PolynomialOver<T> &d, const Ring &ring)
|
||
|
|
{
|
||
|
|
unsigned int i = a.CoefficientCount(ring);
|
||
|
|
const int dDegree = d.Degree(ring);
|
||
|
|
|
||
|
|
if (dDegree < 0)
|
||
|
|
throw DivideByZero();
|
||
|
|
|
||
|
|
r = a;
|
||
|
|
q.m_coefficients.resize(STDMAX(0, int(i - dDegree)));
|
||
|
|
|
||
|
|
while (i > (unsigned int)dDegree)
|
||
|
|
{
|
||
|
|
--i;
|
||
|
|
q.m_coefficients[i-dDegree] = ring.Divide(r.m_coefficients[i], d.m_coefficients[dDegree]);
|
||
|
|
for (int j=0; j<=dDegree; j++)
|
||
|
|
ring.Reduce(r.m_coefficients[i-dDegree+j], ring.Multiply(q.m_coefficients[i-dDegree], d.m_coefficients[j]));
|
||
|
|
}
|
||
|
|
|
||
|
|
r.CoefficientCount(ring); // resize r.m_coefficients
|
||
|
|
}
|
||
|
|
|
||
|
|
// ********************************************************
|
||
|
|
|
||
|
|
// helper function for Interpolate() and InterpolateAt()
|
||
|
|
template <class T>
|
||
|
|
void RingOfPolynomialsOver<T>::CalculateAlpha(std::vector<CoefficientType> &alpha, const CoefficientType x[], const CoefficientType y[], unsigned int n) const
|
||
|
|
{
|
||
|
|
for (unsigned int j=0; j<n; ++j)
|
||
|
|
alpha[j] = y[j];
|
||
|
|
|
||
|
|
for (unsigned int k=1; k<n; ++k)
|
||
|
|
{
|
||
|
|
for (unsigned int j=n-1; j>=k; --j)
|
||
|
|
{
|
||
|
|
m_ring.Reduce(alpha[j], alpha[j-1]);
|
||
|
|
|
||
|
|
CoefficientType d = m_ring.Subtract(x[j], x[j-k]);
|
||
|
|
if (!m_ring.IsUnit(d))
|
||
|
|
throw InterpolationFailed();
|
||
|
|
alpha[j] = m_ring.Divide(alpha[j], d);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T>
|
||
|
|
typename RingOfPolynomialsOver<T>::Element RingOfPolynomialsOver<T>::Interpolate(const CoefficientType x[], const CoefficientType y[], unsigned int n) const
|
||
|
|
{
|
||
|
|
assert(n > 0);
|
||
|
|
|
||
|
|
std::vector<CoefficientType> alpha(n);
|
||
|
|
CalculateAlpha(alpha, x, y, n);
|
||
|
|
|
||
|
|
std::vector<CoefficientType> coefficients((size_t)n, m_ring.Identity());
|
||
|
|
coefficients[0] = alpha[n-1];
|
||
|
|
|
||
|
|
for (int j=n-2; j>=0; --j)
|
||
|
|
{
|
||
|
|
for (unsigned int i=n-j-1; i>0; i--)
|
||
|
|
coefficients[i] = m_ring.Subtract(coefficients[i-1], m_ring.Multiply(coefficients[i], x[j]));
|
||
|
|
|
||
|
|
coefficients[0] = m_ring.Subtract(alpha[j], m_ring.Multiply(coefficients[0], x[j]));
|
||
|
|
}
|
||
|
|
|
||
|
|
return PolynomialOver<T>(coefficients.begin(), coefficients.end());
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T>
|
||
|
|
typename RingOfPolynomialsOver<T>::CoefficientType RingOfPolynomialsOver<T>::InterpolateAt(const CoefficientType &position, const CoefficientType x[], const CoefficientType y[], unsigned int n) const
|
||
|
|
{
|
||
|
|
assert(n > 0);
|
||
|
|
|
||
|
|
std::vector<CoefficientType> alpha(n);
|
||
|
|
CalculateAlpha(alpha, x, y, n);
|
||
|
|
|
||
|
|
CoefficientType result = alpha[n-1];
|
||
|
|
for (int j=n-2; j>=0; --j)
|
||
|
|
{
|
||
|
|
result = m_ring.Multiply(result, m_ring.Subtract(position, x[j]));
|
||
|
|
m_ring.Accumulate(result, alpha[j]);
|
||
|
|
}
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class Ring, class Element>
|
||
|
|
void PrepareBulkPolynomialInterpolation(const Ring &ring, Element *w, const Element x[], unsigned int n)
|
||
|
|
{
|
||
|
|
for (unsigned int i=0; i<n; i++)
|
||
|
|
{
|
||
|
|
Element t = ring.MultiplicativeIdentity();
|
||
|
|
for (unsigned int j=0; j<n; j++)
|
||
|
|
if (i != j)
|
||
|
|
t = ring.Multiply(t, ring.Subtract(x[i], x[j]));
|
||
|
|
w[i] = ring.MultiplicativeInverse(t);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class Ring, class Element>
|
||
|
|
void PrepareBulkPolynomialInterpolationAt(const Ring &ring, Element *v, const Element &position, const Element x[], const Element w[], unsigned int n)
|
||
|
|
{
|
||
|
|
assert(n > 0);
|
||
|
|
|
||
|
|
std::vector<Element> a(2*n-1);
|
||
|
|
unsigned int i;
|
||
|
|
|
||
|
|
for (i=0; i<n; i++)
|
||
|
|
a[n-1+i] = ring.Subtract(position, x[i]);
|
||
|
|
|
||
|
|
for (i=n-1; i>1; i--)
|
||
|
|
a[i-1] = ring.Multiply(a[2*i], a[2*i-1]);
|
||
|
|
|
||
|
|
a[0] = ring.MultiplicativeIdentity();
|
||
|
|
|
||
|
|
for (i=0; i<n-1; i++)
|
||
|
|
{
|
||
|
|
std::swap(a[2*i+1], a[2*i+2]);
|
||
|
|
a[2*i+1] = ring.Multiply(a[i], a[2*i+1]);
|
||
|
|
a[2*i+2] = ring.Multiply(a[i], a[2*i+2]);
|
||
|
|
}
|
||
|
|
|
||
|
|
for (i=0; i<n; i++)
|
||
|
|
v[i] = ring.Multiply(a[n-1+i], w[i]);
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class Ring, class Element>
|
||
|
|
Element BulkPolynomialInterpolateAt(const Ring &ring, const Element y[], const Element v[], unsigned int n)
|
||
|
|
{
|
||
|
|
Element result = ring.Identity();
|
||
|
|
for (unsigned int i=0; i<n; i++)
|
||
|
|
ring.Accumulate(result, ring.Multiply(y[i], v[i]));
|
||
|
|
return result;
|
||
|
|
}
|
||
|
|
|
||
|
|
// ********************************************************
|
||
|
|
|
||
|
|
template <class T, int instance>
|
||
|
|
const PolynomialOverFixedRing<T, instance> &PolynomialOverFixedRing<T, instance>::Zero()
|
||
|
|
{
|
||
|
|
return Singleton<ThisType>().Ref();
|
||
|
|
}
|
||
|
|
|
||
|
|
template <class T, int instance>
|
||
|
|
const PolynomialOverFixedRing<T, instance> &PolynomialOverFixedRing<T, instance>::One()
|
||
|
|
{
|
||
|
|
return Singleton<ThisType, NewOnePolynomial>().Ref();
|
||
|
|
}
|
||
|
|
|
||
|
|
NAMESPACE_END
|