#include "pch.h"
#include "nbtheory.h"
#include "modarith.h"
#include <math.h>
#include <vector>
NAMESPACE_BEGIN(CryptoPP)
const unsigned int maxPrimeTableSize = 3511; 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;
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()) 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()) return false;
++b; ++b;
}
if (j==0)
return false;
Integer n1 = n+1;
unsigned int a;
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)
{
return IsPrime(p) && RabinMillerTest(rng, p, 10);
}
unsigned int PrimeSearchInterval(const Integer &max)
{
return max.BitCount();
}
static inline bool FastProbablePrimeTest(const Integer &n)
{
return IsStrongProbablePrime(n,2);
}
class PrimeSieve
{
public:
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 (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)
{
assert(!equiv.IsNegative() && equiv < mod);
Integer gcd = GCD(equiv, mod);
if (gcd != Integer::One())
{
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, 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);
p += (equiv-p)%mod;
if (p>max)
return false;
PrimeSieve sieve(p, max, mod);
while (sieve.NextCandidate(p))
{
if (FastProbablePrimeTest(p) && IsPrime(p))
return true;
}
return false;
}
static bool ProvePrime(const Integer &p, const Integer &q)
{
assert(p < q*q*q);
assert(p % q == 1);
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())
{
p.Randomize(rng, minP, maxP, Integer::PRIME);
return p;
}
unsigned int qbits = (pbits+2)/3 + 1 + rng.GetLong(0, pbits/36);
Integer q = MihailescuProvablePrime(rng, qbits);
Integer q2 = q<<1;
while (true)
{
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;
}
}
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.GetLong())/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)
{
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); 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);
}
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 2;
MontgomeryRepresentation m(n);
Integer p=m.ConvertIn(pIn%n), two=m.ConvertIn(2);
Integer v=p, v1=m.Subtract(m.Square(p), two);
i--;
while (i--)
{
if (e.GetBit(i))
{
v = m.Subtract(m.Multiply(v,v1), p);
v1 = m.Subtract(m.Square(v1), two);
}
else
{
v1 = m.Subtract(m.Multiply(v,v1), p);
v = m.Subtract(m.Square(v), two);
}
}
return m.ConvertOut(v);
}
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)
{
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)
{
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);
}
PrimeAndGenerator::PrimeAndGenerator(signed int delta, RandomNumberGenerator &rng, unsigned int pbits)
{
assert(pbits > 5);
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)
{
for (g=2; Jacobi(g, p) != 1; ++g);
assert((p%8==1 || p%8==7) ? g==2 : (p%12==1 || p%12==11) ? g==3 : g==4);
}
else
{
assert(delta == -1);
for (g=3; ; ++g)
if (Jacobi(g*g-4, p)==-1 && Lucas(q, g, p)==2)
break;
}
}
PrimeAndGenerator::PrimeAndGenerator(signed int delta, RandomNumberGenerator &rng, unsigned int pbits, unsigned int qbits)
{
assert(qbits > 4);
assert(pbits > qbits);
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));
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