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;
}
Integer InvertibleRabinFunction::CalculateInverse(const Integer &in) const
{
DoQuickSanityCheck();
Integer cp=in%m_p, cq=in%m_q;
int jp = Jacobi(cp, m_p);
int jq = Jacobi(cq, m_q);
if (jq==-1)
{
cp = cp*EuclideanMultiplicativeInverse(m_r, m_p)%m_p;
cq = cq*EuclideanMultiplicativeInverse(m_r, m_q)%m_q;
}
if (jp==-1)
{
cp = cp*EuclideanMultiplicativeInverse(m_s, m_p)%m_p;
cq = cq*EuclideanMultiplicativeInverse(m_s, m_q)%m_q;
}
cp = ModularSquareRoot(cp, m_p);
cq = ModularSquareRoot(cq, m_q);
if (jp==-1)
cp = m_p-cp;
Integer out = CRT(cq, m_q, cp, m_p, m_u);
if ((jq==-1 && out.IsEven()) || (jq==1 && out.IsOdd()))
out = m_n-out;
return out;
}
Integer InvertibleRabinFunction::CalculateInverse(const Integer &in) const
{
Integer cp=in%p, cq=in%q;
int jp = Jacobi(cp, p);
int jq = Jacobi(cq, q);
if (jq==-1)
{
cp = cp*EuclideanMultiplicativeInverse(r, p)%p;
cq = cq*EuclideanMultiplicativeInverse(r, q)%q;
}
if (jp==-1)
{
cp = cp*EuclideanMultiplicativeInverse(s, p)%p;
cq = cq*EuclideanMultiplicativeInverse(s, q)%q;
}
cp = ModularSquareRoot(cp, p);
cq = ModularSquareRoot(cq, q);
if (jp==-1)
cp = p-cp;
Integer out = CRT(cq, q, cp, p, u);
if ((jq==-1 && out.IsEven()) || (jq==1 && out.IsOdd()))
out = n-out;
return out;
}
void InvertibleRSAFunction::Initialize(const Integer &n, const Integer &e, const Integer &d)
{
m_n = n;
m_e = e;
m_d = d;
Integer r = --(d*e);
while (r.IsEven())
r >>= 1;
ModularArithmetic modn(n);
for (Integer i = 2; ; ++i)
{
Integer a = modn.Exponentiate(i, r);
if (a == 1)
continue;
Integer b;
while (a != -1)
{
b = modn.Square(a);
if (b == 1)
{
m_p = GCD(a-1, n);
m_q = n/m_p;
m_dp = m_d % (m_p-1);
m_dq = m_d % (m_q-1);
m_u = m_q.InverseMod(m_p);
return;
}
a = b;
}
}
}
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;
}
void InvertibleRSAFunction::Initialize(const Integer &n, const Integer &e, const Integer &d)
{
if (n.IsEven() || e.IsEven() | d.IsEven())
throw InvalidArgument("InvertibleRSAFunction: input is not a valid RSA private key");
m_n = n;
m_e = e;
m_d = d;
Integer r = --(d*e);
unsigned int s = 0;
while (r.IsEven())
{
r >>= 1;
s++;
}
ModularArithmetic modn(n);
for (Integer i = 2; ; ++i)
{
Integer a = modn.Exponentiate(i, r);
if (a == 1)
continue;
Integer b;
unsigned int j = 0;
while (a != n-1)
{
b = modn.Square(a);
if (b == 1)
{
m_p = GCD(a-1, n);
m_q = n/m_p;
m_dp = m_d % (m_p-1);
m_dq = m_d % (m_q-1);
m_u = m_q.InverseMod(m_p);
return;
}
if (++j == s)
throw InvalidArgument("InvertibleRSAFunction: input is not a valid RSA private key");
a = b;
}
}
}
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;
}
Integer InvertibleRabinFunction::CalculateInverse(RandomNumberGenerator &rng, const Integer &in) const
{
DoQuickSanityCheck();
ModularArithmetic modn(m_n);
Integer r(rng, Integer::One(), m_n - Integer::One());
r = modn.Square(r);
Integer r2 = modn.Square(r);
Integer c = modn.Multiply(in, r2); // blind
Integer cp=c%m_p, cq=c%m_q;
int jp = Jacobi(cp, m_p);
int jq = Jacobi(cq, m_q);
if (jq==-1)
{
cp = cp*EuclideanMultiplicativeInverse(m_r, m_p)%m_p;
cq = cq*EuclideanMultiplicativeInverse(m_r, m_q)%m_q;
}
if (jp==-1)
{
cp = cp*EuclideanMultiplicativeInverse(m_s, m_p)%m_p;
cq = cq*EuclideanMultiplicativeInverse(m_s, m_q)%m_q;
}
cp = ModularSquareRoot(cp, m_p);
cq = ModularSquareRoot(cq, m_q);
if (jp==-1)
cp = m_p-cp;
Integer out = CRT(cq, m_q, cp, m_p, m_u);
out = modn.Divide(out, r); // unblind
if ((jq==-1 && out.IsEven()) || (jq==1 && out.IsOdd()))
out = m_n-out;
return out;
}
void InvertibleRSAFunction::Initialize(RandomNumberGenerator &rng, unsigned int keybits, const Integer &e)
{
GenerateRandom(rng, MakeParameters(Name::ModulusSize(), (int)keybits)(Name::PublicExponent(), e+e.IsEven()));
}
