nbtheory.cpp

研读AxCrypt对加解密的处理方法

// nbtheory.cpp - written and placed in the public domain by Wei Dai

#include "pch.h"

#ifndef CRYPTOPP_IMPORTS

#include "nbtheory.h"
#include "modarith.h"
#include "algparam.h"

#include <math.h>
#include <vector>

NAMESPACE_BEGIN(CryptoPP)

const word s_lastSmallPrime = 32719;

struct NewPrimeTable
{
	std::vector<word16> * operator()() const
	{
		const unsigned int maxPrimeTableSize = 3511;

		std::auto_ptr<std::vector<word16> > pPrimeTable(new std::vector<word16>);
		std::vector<word16> &primeTable = *pPrimeTable;
		primeTable.reserve(maxPrimeTableSize);

		primeTable.push_back(2);
		unsigned int testEntriesEnd = 1;
		
		for (unsigned int p=3; p<=s_lastSmallPrime; p+=2)
		{
			unsigned int j;
			for (j=1; j<testEntriesEnd; j++)
				if (p%primeTable[j] == 0)
					break;
			if (j == testEntriesEnd)
			{
				primeTable.push_back(p);
				testEntriesEnd = STDMIN((size_t)54U, primeTable.size());
			}
		}

		return pPrimeTable.release();
	}
};

const word16 * GetPrimeTable(unsigned int &size)
{
	const std::vector<word16> &primeTable = Singleton<std::vector<word16>, NewPrimeTable>().Ref();
	size = primeTable.size();
	return &primeTable[0];
}

bool IsSmallPrime(const Integer &p)
{
	unsigned int primeTableSize;
	const word16 * primeTable = GetPrimeTable(primeTableSize);

	if (p.IsPositive() && p <= primeTable[primeTableSize-1])
		return std::binary_search(primeTable, primeTable+primeTableSize, (word16)p.ConvertToLong());
	else
		return false;
}

bool TrialDivision(const Integer &p, unsigned bound)
{
	unsigned int primeTableSize;
	const word16 * primeTable = GetPrimeTable(primeTableSize);

	assert(primeTable[primeTableSize-1] >= bound);

	unsigned int i;
	for (i = 0; primeTable[i]<bound; i++)
		if ((p % primeTable[i]) == 0)
			return true;

	if (bound == primeTable[i])
		return (p % bound == 0);
	else
		return false;
}

bool SmallDivisorsTest(const Integer &p)
{
	unsigned int primeTableSize;
	const word16 * primeTable = GetPrimeTable(primeTableSize);
	return !TrialDivision(p, primeTable[primeTableSize-1]);
}

bool IsFermatProbablePrime(const Integer &n, const Integer &b)
{
	if (n <= 3)
		return n==2 || n==3;

	assert(n>3 && b>1 && b<n-1);
	return a_exp_b_mod_c(b, n-1, n)==1;
}

bool IsStrongProbablePrime(const Integer &n, const Integer &b)
{
	if (n <= 3)
		return n==2 || n==3;

	assert(n>3 && b>1 && b<n-1);

	if ((n.IsEven() && n!=2) || GCD(b, n) != 1)
		return false;

	Integer nminus1 = (n-1);
	unsigned int a;

	// calculate a = largest power of 2 that divides (n-1)
	for (a=0; ; a++)
		if (nminus1.GetBit(a))
			break;
	Integer m = nminus1>>a;

	Integer z = a_exp_b_mod_c(b, m, n);
	if (z==1 || z==nminus1)
		return true;
	for (unsigned j=1; j<a; j++)
	{
		z = z.Squared()%n;
		if (z==nminus1)
			return true;
		if (z==1)
			return false;
	}
	return false;
}

bool RabinMillerTest(RandomNumberGenerator &rng, const Integer &n, unsigned int rounds)
{
	if (n <= 3)
		return n==2 || n==3;

	assert(n>3);

	Integer b;
	for (unsigned int i=0; i<rounds; i++)
	{
		b.Randomize(rng, 2, n-2);
		if (!IsStrongProbablePrime(n, b))
			return false;
	}
	return true;
}

bool IsLucasProbablePrime(const Integer &n)
{
	if (n <= 1)
		return false;

	if (n.IsEven())
		return n==2;

	assert(n>2);

	Integer b=3;
	unsigned int i=0;
	int j;

	while ((j=Jacobi(b.Squared()-4, n)) == 1)
	{
		if (++i==64 && n.IsSquare())	// avoid infinite loop if n is a square
			return false;
		++b; ++b;
	}

	if (j==0) 
		return false;
	else
		return Lucas(n+1, b, n)==2;
}

bool IsStrongLucasProbablePrime(const Integer &n)
{
	if (n <= 1)
		return false;

	if (n.IsEven())
		return n==2;

	assert(n>2);

	Integer b=3;
	unsigned int i=0;
	int j;

	while ((j=Jacobi(b.Squared()-4, n)) == 1)
	{
		if (++i==64 && n.IsSquare())	// avoid infinite loop if n is a square
			return false;
		++b; ++b;
	}

	if (j==0) 
		return false;

	Integer n1 = n+1;
	unsigned int a;

	// calculate a = largest power of 2 that divides n1
	for (a=0; ; a++)
		if (n1.GetBit(a))
			break;
	Integer m = n1>>a;

	Integer z = Lucas(m, b, n);
	if (z==2 || z==n-2)
		return true;
	for (i=1; i<a; i++)
	{
		z = (z.Squared()-2)%n;
		if (z==n-2)
			return true;
		if (z==2)
			return false;
	}
	return false;
}

struct NewLastSmallPrimeSquared
{
	Integer * operator()() const
	{
		return new Integer(Integer(s_lastSmallPrime).Squared());
	}
};

bool IsPrime(const Integer &p)
{
	if (p <= s_lastSmallPrime)
		return IsSmallPrime(p);
	else if (p <= Singleton<Integer, NewLastSmallPrimeSquared>().Ref())
		return SmallDivisorsTest(p);
	else
		return SmallDivisorsTest(p) && IsStrongProbablePrime(p, 3) && IsStrongLucasProbablePrime(p);
}

bool VerifyPrime(RandomNumberGenerator &rng, const Integer &p, unsigned int level)
{
	bool pass = IsPrime(p) && RabinMillerTest(rng, p, 1);
	if (level >= 1)
		pass = pass && RabinMillerTest(rng, p, 10);
	return pass;
}

unsigned int PrimeSearchInterval(const Integer &max)
{
	return max.BitCount();
}

static inline bool FastProbablePrimeTest(const Integer &n)
{
	return IsStrongProbablePrime(n,2);
}

AlgorithmParameters<AlgorithmParameters<AlgorithmParameters<NullNameValuePairs, Integer::RandomNumberType>, Integer>, Integer>
	MakeParametersForTwoPrimesOfEqualSize(unsigned int productBitLength)
{
	if (productBitLength < 16)
		throw InvalidArgument("invalid bit length");

	Integer minP, maxP;

	if (productBitLength%2==0)
	{
		minP = Integer(182) << (productBitLength/2-8);
		maxP = Integer::Power2(productBitLength/2)-1;
	}
	else
	{
		minP = Integer::Power2((productBitLength-1)/2);
		maxP = Integer(181) << ((productBitLength+1)/2-8);
	}

	return MakeParameters("RandomNumberType", Integer::PRIME)("Min", minP)("Max", maxP);
}

class PrimeSieve
{
public:
	// delta == 1 or -1 means double sieve with p = 2*q + delta
	PrimeSieve(const Integer &first, const Integer &last, const Integer &step, signed int delta=0);
	bool NextCandidate(Integer &c);

	void DoSieve();
	static void SieveSingle(std::vector<bool> &sieve, word16 p, const Integer &first, const Integer &step, word16 stepInv);

	Integer m_first, m_last, m_step;
	signed int m_delta;
	word m_next;
	std::vector<bool> m_sieve;
};

PrimeSieve::PrimeSieve(const Integer &first, const Integer &last, const Integer &step, signed int delta)
	: m_first(first), m_last(last), m_step(step), m_delta(delta), m_next(0)
{
	DoSieve();
}

bool PrimeSieve::NextCandidate(Integer &c)
{
	m_next = std::find(m_sieve.begin()+m_next, m_sieve.end(), false) - m_sieve.begin();
	if (m_next == m_sieve.size())
	{
		m_first += m_sieve.size()*m_step;
		if (m_first > m_last)
			return false;
		else
		{
			m_next = 0;
			DoSieve();
			return NextCandidate(c);
		}
	}
	else
	{
		c = m_first + m_next*m_step;
		++m_next;
		return true;
	}
}

void PrimeSieve::SieveSingle(std::vector<bool> &sieve, word16 p, const Integer &first, const Integer &step, word16 stepInv)
{
	if (stepInv)
	{
		unsigned int sieveSize = sieve.size();
		word j = word((word32(p-(first%p))*stepInv) % p);
		// if the first multiple of p is p, skip it
		if (first.WordCount() <= 1 && first + step*j == p)
			j += p;
		for (; j < sieveSize; j += p)
			sieve[j] = true;
	}
}

void PrimeSieve::DoSieve()
{
	unsigned int primeTableSize;
	const word16 * primeTable = GetPrimeTable(primeTableSize);

	const unsigned int maxSieveSize = 32768;
	unsigned int sieveSize = STDMIN(Integer(maxSieveSize), (m_last-m_first)/m_step+1).ConvertToLong();

	m_sieve.clear();
	m_sieve.resize(sieveSize, false);

	if (m_delta == 0)
	{
		for (unsigned int i = 0; i < primeTableSize; ++i)
			SieveSingle(m_sieve, primeTable[i], m_first, m_step, m_step.InverseMod(primeTable[i]));
	}
	else
	{
		assert(m_step%2==0);
		Integer qFirst = (m_first-m_delta) >> 1;
		Integer halfStep = m_step >> 1;
		for (unsigned int i = 0; i < primeTableSize; ++i)
		{
			word16 p = primeTable[i];
			word16 stepInv = m_step.InverseMod(p);
			SieveSingle(m_sieve, p, m_first, m_step, stepInv);

			word16 halfStepInv = 2*stepInv < p ? 2*stepInv : 2*stepInv-p;
			SieveSingle(m_sieve, p, qFirst, halfStep, halfStepInv);
		}
	}
}

bool FirstPrime(Integer &p, const Integer &max, const Integer &equiv, const Integer &mod, const PrimeSelector *pSelector)
{
	assert(!equiv.IsNegative() && equiv < mod);

	Integer gcd = GCD(equiv, mod);
	if (gcd != Integer::One())
	{
		// the only possible prime p such that p%mod==equiv where GCD(mod,equiv)!=1 is GCD(mod,equiv)
		if (p <= gcd && gcd <= max && IsPrime(gcd) && (!pSelector || pSelector->IsAcceptable(gcd)))
		{
			p = gcd;
			return true;
		}
		else
			return false;
	}

	unsigned int primeTableSize;
	const word16 * primeTable = GetPrimeTable(primeTableSize);

	if (p <= primeTable[primeTableSize-1])
	{
		const word16 *pItr;

		--p;
		if (p.IsPositive())
			pItr = std::upper_bound(primeTable, primeTable+primeTableSize, (word)p.ConvertToLong());
		else
			pItr = primeTable;

		while (pItr < primeTable+primeTableSize && !(*pItr%mod == equiv && (!pSelector || pSelector->IsAcceptable(*pItr))))
			++pItr;

		if (pItr < primeTable+primeTableSize)
		{
			p = *pItr;
			return p <= max;
		}

		p = primeTable[primeTableSize-1]+1;
	}

	assert(p > primeTable[primeTableSize-1]);

	if (mod.IsOdd())
		return FirstPrime(p, max, CRT(equiv, mod, 1, 2, 1), mod<<1, pSelector);

	p += (equiv-p)%mod;

	if (p>max)
		return false;

	PrimeSieve sieve(p, max, mod);

	while (sieve.NextCandidate(p))
	{
		if ((!pSelector || pSelector->IsAcceptable(p)) && FastProbablePrimeTest(p) && IsPrime(p))
			return true;
	}

	return false;
}

// the following two functions are based on code and comments provided by Preda Mihailescu
static bool ProvePrime(const Integer &p, const Integer &q)
{
	assert(p < q*q*q);
	assert(p % q == 1);

// this is the Quisquater test. Numbers p having passed the Lucas - Lehmer test
// for q and verifying p < q^3 can only be built up of two factors, both = 1 mod q,
// or be prime. The next two lines build the discriminant of a quadratic equation
// which holds iff p is built up of two factors (excercise ... )

	Integer r = (p-1)/q;
	if (((r%q).Squared()-4*(r/q)).IsSquare())
		return false;

	unsigned int primeTableSize;
	const word16 * primeTable = GetPrimeTable(primeTableSize);

	assert(primeTableSize >= 50);
	for (int i=0; i<50; i++) 
	{
		Integer b = a_exp_b_mod_c(primeTable[i], r, p);
		if (b != 1) 
			return a_exp_b_mod_c(b, q, p) == 1;
	}
	return false;
}

Integer MihailescuProvablePrime(RandomNumberGenerator &rng, unsigned int pbits)
{
	Integer p;
	Integer minP = Integer::Power2(pbits-1);
	Integer maxP = Integer::Power2(pbits) - 1;

	if (maxP <= Integer(s_lastSmallPrime).Squared())
	{
		// Randomize() will generate a prime provable by trial division
		p.Randomize(rng, minP, maxP, Integer::PRIME);
		return p;
	}

	unsigned int qbits = (pbits+2)/3 + 1 + rng.GenerateWord32(0, pbits/36);
	Integer q = MihailescuProvablePrime(rng, qbits);
	Integer q2 = q<<1;

	while (true)
	{
		// this initializes the sieve to search in the arithmetic
		// progression p = p_0 + \lambda * q2 = p_0 + 2 * \lambda * q,
		// with q the recursively generated prime above. We will be able
		// to use Lucas tets for proving primality. A trick of Quisquater
		// allows taking q > cubic_root(p) rather then square_root: this
		// decreases the recursion.

		p.Randomize(rng, minP, maxP, Integer::ANY, 1, q2);
		PrimeSieve sieve(p, STDMIN(p+PrimeSearchInterval(maxP)*q2, maxP), q2);

		while (sieve.NextCandidate(p))
		{
			if (FastProbablePrimeTest(p) && ProvePrime(p, q))
				return p;
		}
	}

	// not reached
	return p;
}

Integer MaurerProvablePrime(RandomNumberGenerator &rng, unsigned int bits)
{
	const unsigned smallPrimeBound = 29, c_opt=10;
	Integer p;

	unsigned int primeTableSize;
	const word16 * primeTable = GetPrimeTable(primeTableSize);

	if (bits < smallPrimeBound)
	{
		do
			p.Randomize(rng, Integer::Power2(bits-1), Integer::Power2(bits)-1, Integer::ANY, 1, 2);
		while (TrialDivision(p, 1 << ((bits+1)/2)));
	}
	else
	{
		const unsigned margin = bits > 50 ? 20 : (bits-10)/2;
		double relativeSize;
		do
			relativeSize = pow(2.0, double(rng.GenerateWord32())/0xffffffff - 1);
		while (bits * relativeSize >= bits - margin);

		Integer a,b;
		Integer q = MaurerProvablePrime(rng, unsigned(bits*relativeSize));
		Integer I = Integer::Power2(bits-2)/q;
		Integer I2 = I << 1;
		unsigned int trialDivisorBound = (unsigned int)STDMIN((unsigned long)primeTable[primeTableSize-1], (unsigned long)bits*bits/c_opt);
		bool success = false;
		while (!success)
		{
			p.Randomize(rng, I, I2, Integer::ANY);
			p *= q; p <<= 1; ++p;
			if (!TrialDivision(p, trialDivisorBound))
			{
				a.Randomize(rng, 2, p-1, Integer::ANY);
				b = a_exp_b_mod_c(a, (p-1)/q, p);
				success = (GCD(b-1, p) == 1) && (a_exp_b_mod_c(b, q, p) == 1);
			}
		}
	}
	return p;
}

Integer CRT(const Integer &xp, const Integer &p, const Integer &xq, const Integer &q, const Integer &u)