internal static BigInteger pow2ModPow(BigInteger _base, BigInteger exponent, int j)
{
// PRE: (base > 0), (exponent > 0) and (j > 0)
BigInteger res = BigInteger.ONE;
BigInteger e = exponent.copy();
BigInteger baseMod2toN = _base.copy();
BigInteger res2;
/*
* If 'base' is odd then it's coprime with 2^j and phi(2^j) = 2^(j-1);
* so we can reduce reduce the exponent (mod 2^(j-1)).
*/
if (_base.testBit(0))
{
inplaceModPow2(e, j - 1);
}
inplaceModPow2(baseMod2toN, j);
for (int i = e.bitLength() - 1; i >= 0; i--)
{
res2 = res.copy();
inplaceModPow2(res2, j);
res = res.multiply(res2);
if (BitLevel.testBit(e, i))
{
res = res.multiply(baseMod2toN);
inplaceModPow2(res, j);
}
}
inplaceModPow2(res, j);
return(res);
}
public static BigInteger pow(BigInteger _base, int exponent)
{
// PRE: exp > 0
BigInteger res = BigInteger.ONE;
BigInteger acc = _base;
for (; exponent > 1; exponent >>= 1)
{
if ((exponent & 1) != 0)
{
// if odd, multiply one more time by acc
res = res.multiply(acc);
}
// acc = base^(2^i)
//a limit where karatsuba performs a faster square than the square algorithm
if (acc.numberLength == 1)
{
acc = acc.multiply(acc); // square
}
else
{
acc = new BigInteger(1, square(acc.digits, acc.numberLength, new int [acc.numberLength << 1]));
}
}
// exponent == 1, multiply one more time
res = res.multiply(acc);
return(res);
}
public static BigInteger powerOf10(long exp)
{
// PRE: exp >= 0
int intExp = (int)exp;
// "SMALL POWERS"
if (exp < bigTenPows.Length)
{
// The largest power that fit in 'long' type
return(bigTenPows[intExp]);
}
else if (exp <= 50)
{
// To calculate: 10^exp
return(BigInteger.TEN.pow(intExp));
}
else if (exp <= 1000)
{
// To calculate: 5^exp * 2^exp
return(bigFivePows[1].pow(intExp).shiftLeft(intExp));
}
if (exp <= Int32.MaxValue)
{
// To calculate: 5^exp * 2^exp
return(bigFivePows[1].pow(intExp).shiftLeft(intExp));
}
/*
* "HUGE POWERS"
*
* This branch probably won't be executed since the power of ten is too
* big.
*/
// To calculate: 5^exp
BigInteger powerOfFive = bigFivePows[1].pow(Int32.MaxValue);
BigInteger res = powerOfFive;
long longExp = exp - Int32.MaxValue;
intExp = (int)(exp % Int32.MaxValue);
while (longExp > Int32.MaxValue)
{
res = res.multiply(powerOfFive);
longExp -= Int32.MaxValue;
}
res = res.multiply(bigFivePows[1].pow(intExp));
// To calculate: 5^exp << exp
res = res.shiftLeft(Int32.MaxValue);
longExp = exp - Int32.MaxValue;
while (longExp > Int32.MaxValue)
{
res = res.shiftLeft(Int32.MaxValue);
longExp -= Int32.MaxValue;
}
res = res.shiftLeft(intExp);
return(res);
}
public static BigInteger multiplyByFivePow(BigInteger val, int exp)
{
// PRE: exp >= 0
if (exp < fivePows.Length)
{
return(multiplyByPositiveInt(val, fivePows[exp]));
}
else if (exp < bigFivePows.Length)
{
return(val.multiply(bigFivePows[exp]));
}
else // Large powers of five
{
return(val.multiply(bigFivePows[1].pow(exp)));
}
}
public static BigInteger multiplyByTenPow(BigInteger val, long exp)
{
// PRE: exp >= 0
return((exp < tenPows.Length)
? multiplyByPositiveInt(val, tenPows[(int)exp])
: val.multiply(powerOf10(exp)));
}
internal static BigInteger evenModPow(BigInteger _base, BigInteger exponent,
BigInteger modulus)
{
// PRE: (base > 0), (exponent > 0), (modulus > 0) and (modulus even)
// STEP 1: Obtain the factorization 'modulus'= q * 2^j.
int j = modulus.getLowestSetBit();
BigInteger q = modulus.shiftRight(j);
// STEP 2: Compute x1 := base^exponent (mod q).
BigInteger x1 = oddModPow(_base, exponent, q);
// STEP 3: Compute x2 := base^exponent (mod 2^j).
BigInteger x2 = pow2ModPow(_base, exponent, j);
// STEP 4: Compute q^(-1) (mod 2^j) and y := (x2-x1) * q^(-1) (mod 2^j)
BigInteger qInv = modPow2Inverse(q, j);
BigInteger y = (x2.subtract(x1)).multiply(qInv);
inplaceModPow2(y, j);
if (y.sign < 0)
{
y = y.add(BigInteger.getPowerOfTwo(j));
}
// STEP 5: Compute and return: x1 + q * y
return(x1.add(q.multiply(y)));
}
internal static BigInteger modPow2Inverse(BigInteger x, int n)
{
// PRE: (x > 0), (x is odd), and (n > 0)
BigInteger y = new BigInteger(1, new int[1 << n]);
y.numberLength = 1;
y.digits[0] = 1;
y.sign = 1;
for (int i = 1; i < n; i++)
{
if (BitLevel.testBit(x.multiply(y), i))
{
// Adding 2^i to y (setting the i-th bit)
y.digits[i >> 5] |= (1 << (i & 31));
}
}
return(y);
}
internal static BigInteger modPow2Inverse(BigInteger x, int n)
{
// PRE: (x > 0), (x is odd), and (n > 0)
BigInteger y = new BigInteger(1, new int[1 << n]);
y.numberLength = 1;
y.digits[0] = 1;
y.sign = 1;
for (int i = 1; i < n; i++) {
if (BitLevel.testBit(x.multiply(y), i)) {
// Adding 2^i to y (setting the i-th bit)
y.digits[i >> 5] |= (1 << (i & 31));
}
}
return y;
}
public static BigInteger multiplyByFivePow(BigInteger val, int exp)
{
// PRE: exp >= 0
if (exp < fivePows.Length) {
return multiplyByPositiveInt(val, fivePows[exp]);
} else if (exp < bigFivePows.Length) {
return val.multiply(bigFivePows[exp]);
} else {// Large powers of five
return val.multiply(bigFivePows[1].pow(exp));
}
}
public static BigInteger multiplyByTenPow(BigInteger val, long exp)
{
// PRE: exp >= 0
return ((exp < tenPows.Length)
? multiplyByPositiveInt(val, tenPows[(int)exp])
: val.multiply(powerOf10(exp)));
}