From b21162cf8e06f40baa1f58be6a8c17435cebc34d Mon Sep 17 00:00:00 2001
From: weidai <weidai@57ff6487-cd31-0410-9ec3-f628ee90f5f0>
Date: Fri, 4 Oct 2002 17:31:41 +0000
Subject: Initial revision

git-svn-id: svn://svn.code.sf.net/p/cryptopp/code/trunk/c5@2 57ff6487-cd31-0410-9ec3-f628ee90f5f0
---
 nbtheory.cpp | 1127 ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 1127 insertions(+)
 create mode 100644 nbtheory.cpp

(limited to 'nbtheory.cpp')

diff --git a/nbtheory.cpp b/nbtheory.cpp
new file mode 100644
index 0000000..852beb5
--- /dev/null
+++ b/nbtheory.cpp
@@ -0,0 +1,1127 @@
+// nbtheory.cpp - written and placed in the public domain by Wei Dai
+
+#include "pch.h"
+#include "nbtheory.h"
+#include "modarith.h"
+#include "algparam.h"
+
+#include <math.h>
+#include <vector>
+
+NAMESPACE_BEGIN(CryptoPP)
+
+const unsigned int maxPrimeTableSize = 3511;	// last prime 32719
+const word lastSmallPrime = 32719;
+unsigned int primeTableSize=552;
+
+word primeTable[maxPrimeTableSize] =
+	{2, 3, 5, 7, 11, 13, 17, 19,
+	23, 29, 31, 37, 41, 43, 47, 53,
+	59, 61, 67, 71, 73, 79, 83, 89,
+	97, 101, 103, 107, 109, 113, 127, 131,
+	137, 139, 149, 151, 157, 163, 167, 173,
+	179, 181, 191, 193, 197, 199, 211, 223,
+	227, 229, 233, 239, 241, 251, 257, 263,
+	269, 271, 277, 281, 283, 293, 307, 311,
+	313, 317, 331, 337, 347, 349, 353, 359,
+	367, 373, 379, 383, 389, 397, 401, 409,
+	419, 421, 431, 433, 439, 443, 449, 457,
+	461, 463, 467, 479, 487, 491, 499, 503,
+	509, 521, 523, 541, 547, 557, 563, 569,
+	571, 577, 587, 593, 599, 601, 607, 613,
+	617, 619, 631, 641, 643, 647, 653, 659,
+	661, 673, 677, 683, 691, 701, 709, 719,
+	727, 733, 739, 743, 751, 757, 761, 769,
+	773, 787, 797, 809, 811, 821, 823, 827,
+	829, 839, 853, 857, 859, 863, 877, 881,
+	883, 887, 907, 911, 919, 929, 937, 941,
+	947, 953, 967, 971, 977, 983, 991, 997,
+	1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049,
+	1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097,
+	1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163,
+	1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223,
+	1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283,
+	1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321,
+	1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423,
+	1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459,
+	1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511,
+	1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571,
+	1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619,
+	1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693,
+	1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747,
+	1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811,
+	1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877,
+	1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949,
+	1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003,
+	2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069,
+	2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129,
+	2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203,
+	2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267,
+	2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311,
+	2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377,
+	2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423,
+	2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503,
+	2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579,
+	2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657,
+	2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693,
+	2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741,
+	2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801,
+	2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861,
+	2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939,
+	2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011,
+	3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079,
+	3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167,
+	3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221,
+	3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301,
+	3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347,
+	3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413,
+	3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491,
+	3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541,
+	3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607,
+	3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671,
+	3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727,
+	3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797,
+	3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863,
+	3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923,
+	3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003};
+
+void BuildPrimeTable()
+{
+	unsigned int p=primeTable[primeTableSize-1];
+	for (unsigned int i=primeTableSize; i<maxPrimeTableSize; i++)
+	{
+		int j;
+		do
+		{
+			p+=2;
+			for (j=1; j<54; j++)
+				if (p%primeTable[j] == 0)
+					break;
+		} while (j!=54);
+		primeTable[i] = p;
+	}
+	primeTableSize = maxPrimeTableSize;
+	assert(primeTable[primeTableSize-1] == lastSmallPrime);
+}
+
+bool IsSmallPrime(const Integer &p)
+{
+	BuildPrimeTable();
+
+	if (p.IsPositive() && p <= primeTable[primeTableSize-1])
+		return std::binary_search(primeTable, primeTable+primeTableSize, (word)p.ConvertToLong());
+	else
+		return false;
+}
+
+bool TrialDivision(const Integer &p, unsigned bound)
+{
+	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)
+{
+	BuildPrimeTable();
+	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;
+}
+
+bool IsPrime(const Integer &p)
+{
+	static const Integer lastSmallPrimeSquared = Integer(lastSmallPrime).Squared();
+
+	if (p <= lastSmallPrime)
+		return IsSmallPrime(p);
+	else if (p <= lastSmallPrimeSquared)
+		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, word p, const Integer &first, const Integer &step, word 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, word p, const Integer &first, const Integer &step, word stepInv)
+{
+	if (stepInv)
+	{
+		unsigned int sieveSize = sieve.size();
+		word j = word((dword(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()
+{
+	BuildPrimeTable();
+
+	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)
+		{
+			word p = primeTable[i];
+			word stepInv = m_step.InverseMod(p);
+			SieveSingle(m_sieve, p, m_first, m_step, stepInv);
+
+			word 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))
+		{
+			p = gcd;
+			return true;
+		}
+		else
+			return false;
+	}
+
+	BuildPrimeTable();
+
+	if (p <= primeTable[primeTableSize-1])
+	{
+		word *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)
+			++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;
+
+	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(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;
+
+	BuildPrimeTable();
+	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)
+{
+	// isn't operator overloading great?
+	return p * (u * (xq-xp) % q) + xp;
+}
+
+Integer CRT(const Integer &xp, const Integer &p, const Integer &xq, const Integer &q)
+{
+	return CRT(xp, p, xq, q, EuclideanMultiplicativeInverse(p, q));
+}
+
+Integer ModularSquareRoot(const Integer &a, const Integer &p)
+{
+	if (p%4 == 3)
+		return a_exp_b_mod_c(a, (p+1)/4, p);
+
+	Integer q=p-1;
+	unsigned int r=0;
+	while (q.IsEven())
+	{
+		r++;
+		q >>= 1;
+	}
+
+	Integer n=2;
+	while (Jacobi(n, p) != -1)
+		++n;
+
+	Integer y = a_exp_b_mod_c(n, q, p);
+	Integer x = a_exp_b_mod_c(a, (q-1)/2, p);
+	Integer b = (x.Squared()%p)*a%p;
+	x = a*x%p;
+	Integer tempb, t;
+
+	while (b != 1)
+	{
+		unsigned m=0;
+		tempb = b;
+		do
+		{
+			m++;
+			b = b.Squared()%p;
+			if (m==r)
+				return Integer::Zero();
+		}
+		while (b != 1);
+
+		t = y;
+		for (unsigned i=0; i<r-m-1; i++)
+			t = t.Squared()%p;
+		y = t.Squared()%p;
+		r = m;
+		x = x*t%p;
+		b = tempb*y%p;
+	}
+
+	assert(x.Squared()%p == a);
+	return x;
+}
+
+bool SolveModularQuadraticEquation(Integer &r1, Integer &r2, const Integer &a, const Integer &b, const Integer &c, const Integer &p)
+{
+	Integer D = (b.Squared() - 4*a*c) % p;
+	switch (Jacobi(D, p))
+	{
+	default:
+		assert(false);	// not reached
+		return false;
+	case -1:
+		return false;
+	case 0:
+		r1 = r2 = (-b*(a+a).InverseMod(p)) % p;
+		assert(((r1.Squared()*a + r1*b + c) % p).IsZero());
+		return true;
+	case 1:
+		Integer s = ModularSquareRoot(D, p);
+		Integer t = (a+a).InverseMod(p);
+		r1 = (s-b)*t % p;
+		r2 = (-s-b)*t % p;
+		assert(((r1.Squared()*a + r1*b + c) % p).IsZero());
+		assert(((r2.Squared()*a + r2*b + c) % p).IsZero());
+		return true;
+	}
+}
+
+Integer ModularRoot(const Integer &a, const Integer &dp, const Integer &dq,
+					const Integer &p, const Integer &q, const Integer &u)
+{
+	Integer p2 = ModularExponentiation((a % p), dp, p);
+	Integer q2 = ModularExponentiation((a % q), dq, q);
+	return CRT(p2, p, q2, q, u);
+}
+
+Integer ModularRoot(const Integer &a, const Integer &e,
+					const Integer &p, const Integer &q)
+{
+	Integer dp = EuclideanMultiplicativeInverse(e, p-1);
+	Integer dq = EuclideanMultiplicativeInverse(e, q-1);
+	Integer u = EuclideanMultiplicativeInverse(p, q);
+	assert(!!dp && !!dq && !!u);
+	return ModularRoot(a, dp, dq, p, q, u);
+}
+
+/*
+Integer GCDI(const Integer &x, const Integer &y)
+{
+	Integer a=x, b=y;
+	unsigned k=0;
+
+	assert(!!a && !!b);
+
+	while (a[0]==0 && b[0]==0)
+	{
+		a >>= 1;
+		b >>= 1;
+		k++;
+	}
+
+	while (a[0]==0)
+		a >>= 1;
+
+	while (b[0]==0)
+		b >>= 1;
+
+	while (1)
+	{
+		switch (a.Compare(b))
+		{
+			case -1:
+				b -= a;
+				while (b[0]==0)
+					b >>= 1;
+				break;
+
+			case 0:
+				return (a <<= k);
+
+			case 1:
+				a -= b;
+				while (a[0]==0)
+					a >>= 1;
+				break;
+
+			default:
+				assert(false);
+		}
+	}
+}
+
+Integer EuclideanMultiplicativeInverse(const Integer &a, const Integer &b)
+{
+	assert(b.Positive());
+
+	if (a.Negative())
+		return EuclideanMultiplicativeInverse(a%b, b);
+
+	if (b[0]==0)
+	{
+		if (!b || a[0]==0)
+			return Integer::Zero();       // no inverse
+		if (a==1)
+			return 1;
+		Integer u = EuclideanMultiplicativeInverse(b, a);
+		if (!u)
+			return Integer::Zero();       // no inverse
+		else
+			return (b*(a-u)+1)/a;
+	}
+
+	Integer u=1, d=a, v1=b, v3=b, t1, t3, b2=(b+1)>>1;
+
+	if (a[0])
+	{
+		t1 = Integer::Zero();
+		t3 = -b;
+	}
+	else
+	{
+		t1 = b2;
+		t3 = a>>1;
+	}
+
+	while (!!t3)
+	{
+		while (t3[0]==0)
+		{
+			t3 >>= 1;
+			if (t1[0]==0)
+				t1 >>= 1;
+			else
+			{
+				t1 >>= 1;
+				t1 += b2;
+			}
+		}
+		if (t3.Positive())
+		{
+			u = t1;
+			d = t3;
+		}
+		else
+		{
+			v1 = b-t1;
+			v3 = -t3;
+		}
+		t1 = u-v1;
+		t3 = d-v3;
+		if (t1.Negative())
+			t1 += b;
+	}
+	if (d==1)
+		return u;
+	else
+		return Integer::Zero();   // no inverse
+}
+*/
+
+int Jacobi(const Integer &aIn, const Integer &bIn)
+{
+	assert(bIn.IsOdd());
+
+	Integer b = bIn, a = aIn%bIn;
+	int result = 1;
+
+	while (!!a)
+	{
+		unsigned i=0;
+		while (a.GetBit(i)==0)
+			i++;
+		a>>=i;
+
+		if (i%2==1 && (b%8==3 || b%8==5))
+			result = -result;
+
+		if (a%4==3 && b%4==3)
+			result = -result;
+
+		std::swap(a, b);
+		a %= b;
+	}
+
+	return (b==1) ? result : 0;
+}
+
+Integer Lucas(const Integer &e, const Integer &pIn, const Integer &n)
+{
+	unsigned i = e.BitCount();
+	if (i==0)
+		return Integer::Two();
+
+	MontgomeryRepresentation m(n);
+	Integer p=m.ConvertIn(pIn%n), two=m.ConvertIn(Integer::Two());
+	Integer v=p, v1=m.Subtract(m.Square(p), two);
+
+	i--;
+	while (i--)
+	{
+		if (e.GetBit(i))
+		{
+			// v = (v*v1 - p) % m;
+			v = m.Subtract(m.Multiply(v,v1), p);
+			// v1 = (v1*v1 - 2) % m;
+			v1 = m.Subtract(m.Square(v1), two);
+		}
+		else
+		{
+			// v1 = (v*v1 - p) % m;
+			v1 = m.Subtract(m.Multiply(v,v1), p);
+			// v = (v*v - 2) % m;
+			v = m.Subtract(m.Square(v), two);
+		}
+	}
+	return m.ConvertOut(v);
+}
+
+// This is Peter Montgomery's unpublished Lucas sequence evalutation algorithm.
+// The total number of multiplies and squares used is less than the binary
+// algorithm (see above).  Unfortunately I can't get it to run as fast as
+// the binary algorithm because of the extra overhead.
+/*
+Integer Lucas(const Integer &n, const Integer &P, const Integer &modulus)
+{
+	if (!n)
+		return 2;
+
+#define f(A, B, C)	m.Subtract(m.Multiply(A, B), C)
+#define X2(A) m.Subtract(m.Square(A), two)
+#define X3(A) m.Multiply(A, m.Subtract(m.Square(A), three))
+
+	MontgomeryRepresentation m(modulus);
+	Integer two=m.ConvertIn(2), three=m.ConvertIn(3);
+	Integer A=m.ConvertIn(P), B, C, p, d=n, e, r, t, T, U;
+
+	while (d!=1)
+	{
+		p = d;
+		unsigned int b = WORD_BITS * p.WordCount();
+		Integer alpha = (Integer(5)<<(2*b-2)).SquareRoot() - Integer::Power2(b-1);
+		r = (p*alpha)>>b;
+		e = d-r;
+		B = A;
+		C = two;
+		d = r;
+
+		while (d!=e)
+		{
+			if (d<e)
+			{
+				swap(d, e);
+				swap(A, B);
+			}
+
+			unsigned int dm2 = d[0], em2 = e[0];
+			unsigned int dm3 = d%3, em3 = e%3;
+
+//			if ((dm6+em6)%3 == 0 && d <= e + (e>>2))
+			if ((dm3+em3==0 || dm3+em3==3) && (t = e, t >>= 2, t += e, d <= t))
+			{
+				// #1
+//				t = (d+d-e)/3;
+//				t = d; t += d; t -= e; t /= 3;
+//				e = (e+e-d)/3;
+//				e += e; e -= d; e /= 3;
+//				d = t;
+
+//				t = (d+e)/3
+				t = d; t += e; t /= 3;
+				e -= t;
+				d -= t;
+
+				T = f(A, B, C);
+				U = f(T, A, B);
+				B = f(T, B, A);
+				A = U;
+				continue;
+			}
+
+//			if (dm6 == em6 && d <= e + (e>>2))
+			if (dm3 == em3 && dm2 == em2 && (t = e, t >>= 2, t += e, d <= t))
+			{
+				// #2
+//				d = (d-e)>>1;
+				d -= e; d >>= 1;
+				B = f(A, B, C);
+				A = X2(A);
+				continue;
+			}
+
+//			if (d <= (e<<2))
+			if (d <= (t = e, t <<= 2))
+			{
+				// #3
+				d -= e;
+				C = f(A, B, C);
+				swap(B, C);
+				continue;
+			}
+
+			if (dm2 == em2)
+			{
+				// #4
+//				d = (d-e)>>1;
+				d -= e; d >>= 1;
+				B = f(A, B, C);
+				A = X2(A);
+				continue;
+			}
+
+			if (dm2 == 0)
+			{
+				// #5
+				d >>= 1;
+				C = f(A, C, B);
+				A = X2(A);
+				continue;
+			}
+
+			if (dm3 == 0)
+			{
+				// #6
+//				d = d/3 - e;
+				d /= 3; d -= e;
+				T = X2(A);
+				C = f(T, f(A, B, C), C);
+				swap(B, C);
+				A = f(T, A, A);
+				continue;
+			}
+
+			if (dm3+em3==0 || dm3+em3==3)
+			{
+				// #7
+//				d = (d-e-e)/3;
+				d -= e; d -= e; d /= 3;
+				T = f(A, B, C);
+				B = f(T, A, B);
+				A = X3(A);
+				continue;
+			}
+
+			if (dm3 == em3)
+			{
+				// #8
+//				d = (d-e)/3;
+				d -= e; d /= 3;
+				T = f(A, B, C);
+				C = f(A, C, B);
+				B = T;
+				A = X3(A);
+				continue;
+			}
+
+			assert(em2 == 0);
+			// #9
+			e >>= 1;
+			C = f(C, B, A);
+			B = X2(B);
+		}
+
+		A = f(A, B, C);
+	}
+
+#undef f
+#undef X2
+#undef X3
+
+	return m.ConvertOut(A);
+}
+*/
+
+Integer InverseLucas(const Integer &e, const Integer &m, const Integer &p, const Integer &q, const Integer &u)
+{
+	Integer d = (m*m-4);
+	Integer p2 = p-Jacobi(d,p);
+	Integer q2 = q-Jacobi(d,q);
+	return CRT(Lucas(EuclideanMultiplicativeInverse(e,p2), m, p), p, Lucas(EuclideanMultiplicativeInverse(e,q2), m, q), q, u);
+}
+
+Integer InverseLucas(const Integer &e, const Integer &m, const Integer &p, const Integer &q)
+{
+	return InverseLucas(e, m, p, q, EuclideanMultiplicativeInverse(p, q));
+}
+
+unsigned int FactoringWorkFactor(unsigned int n)
+{
+	// extrapolated from the table in Odlyzko's "The Future of Integer Factorization"
+	// updated to reflect the factoring of RSA-130
+	if (n<5) return 0;
+	else return (unsigned int)(2.4 * pow((double)n, 1.0/3.0) * pow(log(double(n)), 2.0/3.0) - 5);
+}
+
+unsigned int DiscreteLogWorkFactor(unsigned int n)
+{
+	// assuming discrete log takes about the same time as factoring
+	if (n<5) return 0;
+	else return (unsigned int)(2.4 * pow((double)n, 1.0/3.0) * pow(log(double(n)), 2.0/3.0) - 5);
+}
+
+// ********************************************************
+
+void PrimeAndGenerator::Generate(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned int qbits)
+{
+	// no prime exists for delta = -1, qbits = 4, and pbits = 5
+	assert(qbits > 4);
+	assert(pbits > qbits);
+
+	if (qbits+1 == pbits)
+	{
+		Integer minP = Integer::Power2(pbits-1);
+		Integer maxP = Integer::Power2(pbits) - 1;
+		bool success = false;
+
+		while (!success)
+		{
+			p.Randomize(rng, minP, maxP, Integer::ANY, 6+5*delta, 12);
+			PrimeSieve sieve(p, STDMIN(p+PrimeSearchInterval(maxP)*12, maxP), 12, delta);
+
+			while (sieve.NextCandidate(p))
+			{
+				assert(IsSmallPrime(p) || SmallDivisorsTest(p));
+				q = (p-delta) >> 1;
+				assert(IsSmallPrime(q) || SmallDivisorsTest(q));
+				if (FastProbablePrimeTest(q) && FastProbablePrimeTest(p) && IsPrime(q) && IsPrime(p))
+				{
+					success = true;
+					break;
+				}
+			}
+		}
+
+		if (delta == 1)
+		{
+			// find g such that g is a quadratic residue mod p, then g has order q
+			// g=4 always works, but this way we get the smallest quadratic residue (other than 1)
+			for (g=2; Jacobi(g, p) != 1; ++g) {}
+			// contributed by Walt Tuvell: g should be the following according to the Law of Quadratic Reciprocity
+			assert((p%8==1 || p%8==7) ? g==2 : (p%12==1 || p%12==11) ? g==3 : g==4);
+		}
+		else
+		{
+			assert(delta == -1);
+			// find g such that g*g-4 is a quadratic non-residue, 
+			// and such that g has order q
+			for (g=3; ; ++g)
+				if (Jacobi(g*g-4, p)==-1 && Lucas(q, g, p)==2)
+					break;
+		}
+	}
+	else
+	{
+		Integer minQ = Integer::Power2(qbits-1);
+		Integer maxQ = Integer::Power2(qbits) - 1;
+		Integer minP = Integer::Power2(pbits-1);
+		Integer maxP = Integer::Power2(pbits) - 1;
+
+		do
+		{
+			q.Randomize(rng, minQ, maxQ, Integer::PRIME);
+		} while (!p.Randomize(rng, minP, maxP, Integer::PRIME, delta%q, q));
+
+		// find a random g of order q
+		if (delta==1)
+		{
+			do
+			{
+				Integer h(rng, 2, p-2, Integer::ANY);
+				g = a_exp_b_mod_c(h, (p-1)/q, p);
+			} while (g <= 1);
+			assert(a_exp_b_mod_c(g, q, p)==1);
+		}
+		else
+		{
+			assert(delta==-1);
+			do
+			{
+				Integer h(rng, 3, p-1, Integer::ANY);
+				if (Jacobi(h*h-4, p)==1)
+					continue;
+				g = Lucas((p+1)/q, h, p);
+			} while (g <= 2);
+			assert(Lucas(q, g, p) == 2);
+		}
+	}
+}
+
+NAMESPACE_END
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