public BigInteger modPow(BigInteger exponent, BigInteger m)
{
if (m.sign <= 0)
{
throw new ArithmeticException("BigInteger: modulus not positive");
}
BigInteger _base = this;
if (m.isOne() | (exponent.sign > 0 & _base.sign == 0))
{
return(BigInteger.ZERO);
}
if (_base.sign == 0 && exponent.sign == 0)
{
return(BigInteger.ONE);
}
if (exponent.sign < 0)
{
_base = modInverse(m);
exponent = exponent.negate();
}
// From now on: (m > 0) and (exponent >= 0)
BigInteger res = (m.testBit(0)) ? Division.oddModPow(_base.abs(),
exponent, m) : Division.evenModPow(_base.abs(), exponent, m);
if ((_base.sign < 0) && exponent.testBit(0))
{
// -b^e mod m == ((-1 mod m) * (b^e mod m)) mod m
res = m.subtract(BigInteger.ONE).multiply(res).mod(m);
}
// else exponent is even, so base^exp is positive
return(res);
}
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 BigInteger modInverse(BigInteger m)
{
if (m.sign <= 0)
{
throw new ArithmeticException("BigInteger: modulus not positive");
}
// If both are even, no inverse exists
if (!(testBit(0) || m.testBit(0)))
{
throw new ArithmeticException("BigInteger not invertible.");
}
if (m.isOne())
{
return(ZERO);
}
// From now on: (m > 1)
BigInteger res = Division.modInverseMontgomery(abs().mod(m), m);
if (res.sign == 0)
{
throw new ArithmeticException("BigInteger not invertible.");
}
res = ((sign < 0) ? m.subtract(res) : res);
return(res);
}
private static int howManyIterations(BigInteger bi, int n)
{
int i = n - 1;
if (bi.sign > 0)
{
while (!bi.testBit(i))
{
i--;
}
return(n - 1 - i);
}
else
{
while (bi.testBit(i))
{
i--;
}
return(n - 1 - Math.Max(i, bi.getLowestSetBit()));
}
}
internal static bool isProbablePrime(BigInteger n, int certainty)
{
// PRE: n >= 0;
if ((certainty <= 0) || ((n.numberLength == 1) && (n.digits[0] == 2)))
{
return(true);
}
// To discard all even numbers
if (!n.testBit(0))
{
return(false);
}
// To check if 'n' exists in the table (it fit in 10 bits)
if ((n.numberLength == 1) && ((n.digits[0] & 0XFFFFFC00) == 0))
{
return(Array.BinarySearch(primes, n.digits[0]) >= 0);
}
// To check if 'n' is divisible by some prime of the table
for (int i = 1; i < primes.Length; i++)
{
if (Division.remainderArrayByInt(n.digits, n.numberLength,
primes[i]) == 0)
{
return(false);
}
}
// To set the number of iterations necessary for Miller-Rabin test
int ix;
int bitLength = n.bitLength();
for (ix = 2; bitLength < BITS[ix]; ix++)
{
;
}
certainty = Math.Min(ix, 1 + ((certainty - 1) >> 1));
return(millerRabin(n, certainty));
}
internal static bool isProbablePrime(BigInteger n, int certainty)
{
// PRE: n >= 0;
if ((certainty <= 0) || ((n.numberLength == 1) && (n.digits[0] == 2))) {
return true;
}
// To discard all even numbers
if (!n.testBit(0)) {
return false;
}
// To check if 'n' exists in the table (it fit in 10 bits)
if ((n.numberLength == 1) && ((n.digits[0] & 0XFFFFFC00) == 0)) {
return (Array.BinarySearch(primes, n.digits[0]) >= 0);
}
// To check if 'n' is divisible by some prime of the table
for (int i = 1; i < primes.Length; i++) {
if (Division.remainderArrayByInt(n.digits, n.numberLength,
primes[i]) == 0) {
return false;
}
}
// To set the number of iterations necessary for Miller-Rabin test
int ix;
int bitLength = n.bitLength();
for (ix = 2; bitLength < BITS[ix]; ix++) {
;
}
certainty = Math.Min(ix, 1 + ((certainty - 1) >> 1));
return millerRabin(n, certainty);
}
internal static BigInteger nextProbablePrime(BigInteger n)
{
// PRE: n >= 0
int i, j;
int certainty;
int gapSize = 1024; // for searching of the next probable prime number
int[] modules = new int[primes.Length];
bool[] isDivisible = new bool[gapSize];
BigInteger startPoint;
BigInteger probPrime;
// If n < "last prime of table" searches next prime in the table
if ((n.numberLength == 1) && (n.digits[0] >= 0)
&& (n.digits[0] < primes[primes.Length - 1])) {
for (i = 0; n.digits[0] >= primes[i]; i++) {
;
}
return BIprimes[i];
}
/*
* Creates a "N" enough big to hold the next probable prime Note that: N <
* "next prime" < 2*N
*/
startPoint = new BigInteger(1, n.numberLength,
new int[n.numberLength + 1]);
Array.Copy(n.digits, startPoint.digits, n.numberLength);
// To fix N to the "next odd number"
if (n.testBit(0)) {
Elementary.inplaceAdd(startPoint, 2);
} else {
startPoint.digits[0] |= 1;
}
// To set the improved certainly of Miller-Rabin
j = startPoint.bitLength();
for (certainty = 2; j < BITS[certainty]; certainty++) {
;
}
// To calculate modules: N mod p1, N mod p2, ... for first primes.
for (i = 0; i < primes.Length; i++) {
modules[i] = Division.remainder(startPoint, primes[i]) - gapSize;
}
while (true) {
// At this point, all numbers in the gap are initialized as
// probably primes
Array.Clear(isDivisible, 0, isDivisible.Length);
// To discard multiples of first primes
for (i = 0; i < primes.Length; i++) {
modules[i] = (modules[i] + gapSize) % primes[i];
j = (modules[i] == 0) ? 0 : (primes[i] - modules[i]);
for (; j < gapSize; j += primes[i]) {
isDivisible[j] = true;
}
}
// To execute Miller-Rabin for non-divisible numbers by all first
// primes
for (j = 0; j < gapSize; j++) {
if (!isDivisible[j]) {
probPrime = startPoint.copy();
Elementary.inplaceAdd(probPrime, j);
if (millerRabin(probPrime, certainty)) {
return probPrime;
}
}
}
Elementary.inplaceAdd(startPoint, gapSize);
}
}
internal static BigInteger nextProbablePrime(BigInteger n)
{
// PRE: n >= 0
int i, j;
int certainty;
int gapSize = 1024; // for searching of the next probable prime number
int[] modules = new int[primes.Length];
bool[] isDivisible = new bool[gapSize];
BigInteger startPoint;
BigInteger probPrime;
// If n < "last prime of table" searches next prime in the table
if ((n.numberLength == 1) && (n.digits[0] >= 0) &&
(n.digits[0] < primes[primes.Length - 1]))
{
for (i = 0; n.digits[0] >= primes[i]; i++)
{
;
}
return(BIprimes[i]);
}
/*
* Creates a "N" enough big to hold the next probable prime Note that: N <
* "next prime" < 2*N
*/
startPoint = new BigInteger(1, n.numberLength,
new int[n.numberLength + 1]);
Array.Copy(n.digits, startPoint.digits, n.numberLength);
// To fix N to the "next odd number"
if (n.testBit(0))
{
Elementary.inplaceAdd(startPoint, 2);
}
else
{
startPoint.digits[0] |= 1;
}
// To set the improved certainly of Miller-Rabin
j = startPoint.bitLength();
for (certainty = 2; j < BITS[certainty]; certainty++)
{
;
}
// To calculate modules: N mod p1, N mod p2, ... for first primes.
for (i = 0; i < primes.Length; i++)
{
modules[i] = Division.remainder(startPoint, primes[i]) - gapSize;
}
while (true)
{
// At this point, all numbers in the gap are initialized as
// probably primes
Array.Clear(isDivisible, 0, isDivisible.Length);
// To discard multiples of first primes
for (i = 0; i < primes.Length; i++)
{
modules[i] = (modules[i] + gapSize) % primes[i];
j = (modules[i] == 0) ? 0 : (primes[i] - modules[i]);
for (; j < gapSize; j += primes[i])
{
isDivisible[j] = true;
}
}
// To execute Miller-Rabin for non-divisible numbers by all first
// primes
for (j = 0; j < gapSize; j++)
{
if (!isDivisible[j])
{
probPrime = startPoint.copy();
Elementary.inplaceAdd(probPrime, j);
if (millerRabin(probPrime, certainty))
{
return(probPrime);
}
}
}
Elementary.inplaceAdd(startPoint, gapSize);
}
}
public BigInteger modPow(BigInteger exponent, BigInteger m)
{
if (m.sign <= 0) {
throw new ArithmeticException("BigInteger: modulus not positive");
}
BigInteger _base = this;
if (m.isOne() | (exponent.sign > 0 & _base.sign == 0)) {
return BigInteger.ZERO;
}
if (_base.sign == 0 && exponent.sign == 0) {
return BigInteger.ONE;
}
if (exponent.sign < 0) {
_base = modInverse(m);
exponent = exponent.negate();
}
// From now on: (m > 0) and (exponent >= 0)
BigInteger res = (m.testBit(0)) ? Division.oddModPow(_base.abs(),
exponent, m) : Division.evenModPow(_base.abs(), exponent, m);
if ((_base.sign < 0) && exponent.testBit(0)) {
// -b^e mod m == ((-1 mod m) * (b^e mod m)) mod m
res = m.subtract(BigInteger.ONE).multiply(res).mod(m);
}
// else exponent is even, so base^exp is positive
return res;
}
public BigInteger modInverse(BigInteger m)
{
if (m.sign <= 0) {
throw new ArithmeticException("BigInteger: modulus not positive");
}
// If both are even, no inverse exists
if (!(testBit(0) || m.testBit(0))) {
throw new ArithmeticException("BigInteger not invertible.");
}
if (m.isOne()) {
return ZERO;
}
// From now on: (m > 1)
BigInteger res = Division.modInverseMontgomery(abs().mod(m), m);
if (res.sign == 0) {
throw new ArithmeticException("BigInteger not invertible.");
}
res = ((sign < 0) ? m.subtract(res) : res);
return res;
}
private static int howManyIterations(BigInteger bi, int n)
{
int i = n - 1;
if (bi.sign > 0) {
while (!bi.testBit(i))
i--;
return n - 1 - i;
} else {
while (bi.testBit(i))
i--;
return n - 1 - Math.Max(i, bi.getLowestSetBit());
}
}
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;
}
internal static BigInteger modInverseMontgomery(BigInteger a, BigInteger p)
{
if (a.sign == 0){
// ZERO hasn't inverse
throw new ArithmeticException("BigInteger not invertible");
}
if (!p.testBit(0)){
// montgomery inverse require even modulo
return modInverseLorencz(a, p);
}
int m = p.numberLength * 32;
// PRE: a \in [1, p - 1]
BigInteger u, v, r, s;
u = p.copy(); // make copy to use inplace method
v = a.copy();
int max = Math.Max(v.numberLength, u.numberLength);
r = new BigInteger(1, 1, new int[max + 1]);
s = new BigInteger(1, 1, new int[max + 1]);
s.digits[0] = 1;
// s == 1 && v == 0
int k = 0;
int lsbu = u.getLowestSetBit();
int lsbv = v.getLowestSetBit();
int toShift;
if (lsbu > lsbv) {
BitLevel.inplaceShiftRight(u, lsbu);
BitLevel.inplaceShiftRight(v, lsbv);
BitLevel.inplaceShiftLeft(r, lsbv);
k += lsbu - lsbv;
} else {
BitLevel.inplaceShiftRight(u, lsbu);
BitLevel.inplaceShiftRight(v, lsbv);
BitLevel.inplaceShiftLeft(s, lsbu);
k += lsbv - lsbu;
}
r.sign = 1;
while (v.signum() > 0) {
// INV v >= 0, u >= 0, v odd, u odd (except last iteration when v is even (0))
while (u.compareTo(v) > BigInteger.EQUALS) {
Elementary.inplaceSubtract(u, v);
toShift = u.getLowestSetBit();
BitLevel.inplaceShiftRight(u, toShift);
Elementary.inplaceAdd(r, s);
BitLevel.inplaceShiftLeft(s, toShift);
k += toShift;
}
while (u.compareTo(v) <= BigInteger.EQUALS) {
Elementary.inplaceSubtract(v, u);
if (v.signum() == 0)
break;
toShift = v.getLowestSetBit();
BitLevel.inplaceShiftRight(v, toShift);
Elementary.inplaceAdd(s, r);
BitLevel.inplaceShiftLeft(r, toShift);
k += toShift;
}
}
if (!u.isOne()){
// in u is stored the gcd
throw new ArithmeticException("BigInteger not invertible.");
}
if (r.compareTo(p) >= BigInteger.EQUALS) {
Elementary.inplaceSubtract(r, p);
}
r = p.subtract(r);
// Have pair: ((BigInteger)r, (Integer)k) where r == a^(-1) * 2^k mod (module)
int n1 = calcN(p);
if (k > m) {
r = monPro(r, BigInteger.ONE, p, n1);
k = k - m;
}
r = monPro(r, BigInteger.getPowerOfTwo(m - k), p, n1);
return r;
}
internal static BigInteger modInverseLorencz(BigInteger a, BigInteger modulo)
{
int max = Math.Max(a.numberLength, modulo.numberLength);
int[] uDigits = new int[max + 1]; // enough place to make all the inplace operation
int[] vDigits = new int[max + 1];
Array.Copy(modulo.digits, uDigits, modulo.numberLength);
Array.Copy(a.digits, vDigits, a.numberLength);
BigInteger u = new BigInteger(modulo.sign, modulo.numberLength,
uDigits);
BigInteger v = new BigInteger(a.sign, a.numberLength, vDigits);
BigInteger r = new BigInteger(0, 1, new int[max + 1]); // BigInteger.ZERO;
BigInteger s = new BigInteger(1, 1, new int[max + 1]);
s.digits[0] = 1;
// r == 0 && s == 1, but with enough place
int coefU = 0, coefV = 0;
int n = modulo.bitLength();
int k;
while (!isPowerOfTwo(u, coefU) && !isPowerOfTwo(v, coefV)) {
// modification of original algorithm: I calculate how many times the algorithm will enter in the same branch of if
k = howManyIterations(u, n);
if (k != 0) {
BitLevel.inplaceShiftLeft(u, k);
if (coefU >= coefV) {
BitLevel.inplaceShiftLeft(r, k);
} else {
BitLevel.inplaceShiftRight(s, Math.Min(coefV - coefU, k));
if (k - ( coefV - coefU ) > 0) {
BitLevel.inplaceShiftLeft(r, k - coefV + coefU);
}
}
coefU += k;
}
k = howManyIterations(v, n);
if (k != 0) {
BitLevel.inplaceShiftLeft(v, k);
if (coefV >= coefU) {
BitLevel.inplaceShiftLeft(s, k);
} else {
BitLevel.inplaceShiftRight(r, Math.Min(coefU - coefV, k));
if (k - ( coefU - coefV ) > 0) {
BitLevel.inplaceShiftLeft(s, k - coefU + coefV);
}
}
coefV += k;
}
if (u.signum() == v.signum()) {
if (coefU <= coefV) {
Elementary.completeInPlaceSubtract(u, v);
Elementary.completeInPlaceSubtract(r, s);
} else {
Elementary.completeInPlaceSubtract(v, u);
Elementary.completeInPlaceSubtract(s, r);
}
} else {
if (coefU <= coefV) {
Elementary.completeInPlaceAdd(u, v);
Elementary.completeInPlaceAdd(r, s);
} else {
Elementary.completeInPlaceAdd(v, u);
Elementary.completeInPlaceAdd(s, r);
}
}
if (v.signum() == 0 || u.signum() == 0){
throw new ArithmeticException("BigInteger not invertible");
}
}
if (isPowerOfTwo(v, coefV)) {
r = s;
if (v.signum() != u.signum())
u = u.negate();
}
if (u.testBit(n)) {
if (r.signum() < 0) {
r = r.negate();
} else {
r = modulo.subtract(r);
}
}
if (r.signum() < 0) {
r = r.add(modulo);
}
return r;
}
internal static BigInteger modInverseLorencz(BigInteger a, BigInteger modulo)
{
int max = Math.Max(a.numberLength, modulo.numberLength);
int[] uDigits = new int[max + 1]; // enough place to make all the inplace operation
int[] vDigits = new int[max + 1];
Array.Copy(modulo.digits, uDigits, modulo.numberLength);
Array.Copy(a.digits, vDigits, a.numberLength);
BigInteger u = new BigInteger(modulo.sign, modulo.numberLength,
uDigits);
BigInteger v = new BigInteger(a.sign, a.numberLength, vDigits);
BigInteger r = new BigInteger(0, 1, new int[max + 1]); // BigInteger.ZERO;
BigInteger s = new BigInteger(1, 1, new int[max + 1]);
s.digits[0] = 1;
// r == 0 && s == 1, but with enough place
int coefU = 0, coefV = 0;
int n = modulo.bitLength();
int k;
while (!isPowerOfTwo(u, coefU) && !isPowerOfTwo(v, coefV))
{
// modification of original algorithm: I calculate how many times the algorithm will enter in the same branch of if
k = howManyIterations(u, n);
if (k != 0)
{
BitLevel.inplaceShiftLeft(u, k);
if (coefU >= coefV)
{
BitLevel.inplaceShiftLeft(r, k);
}
else
{
BitLevel.inplaceShiftRight(s, Math.Min(coefV - coefU, k));
if (k - (coefV - coefU) > 0)
{
BitLevel.inplaceShiftLeft(r, k - coefV + coefU);
}
}
coefU += k;
}
k = howManyIterations(v, n);
if (k != 0)
{
BitLevel.inplaceShiftLeft(v, k);
if (coefV >= coefU)
{
BitLevel.inplaceShiftLeft(s, k);
}
else
{
BitLevel.inplaceShiftRight(r, Math.Min(coefU - coefV, k));
if (k - (coefU - coefV) > 0)
{
BitLevel.inplaceShiftLeft(s, k - coefU + coefV);
}
}
coefV += k;
}
if (u.signum() == v.signum())
{
if (coefU <= coefV)
{
Elementary.completeInPlaceSubtract(u, v);
Elementary.completeInPlaceSubtract(r, s);
}
else
{
Elementary.completeInPlaceSubtract(v, u);
Elementary.completeInPlaceSubtract(s, r);
}
}
else
{
if (coefU <= coefV)
{
Elementary.completeInPlaceAdd(u, v);
Elementary.completeInPlaceAdd(r, s);
}
else
{
Elementary.completeInPlaceAdd(v, u);
Elementary.completeInPlaceAdd(s, r);
}
}
if (v.signum() == 0 || u.signum() == 0)
{
throw new ArithmeticException("BigInteger not invertible");
}
}
if (isPowerOfTwo(v, coefV))
{
r = s;
if (v.signum() != u.signum())
{
u = u.negate();
}
}
if (u.testBit(n))
{
if (r.signum() < 0)
{
r = r.negate();
}
else
{
r = modulo.subtract(r);
}
}
if (r.signum() < 0)
{
r = r.add(modulo);
}
return(r);
}
internal static BigInteger modInverseMontgomery(BigInteger a, BigInteger p)
{
if (a.sign == 0)
{
// ZERO hasn't inverse
throw new ArithmeticException("BigInteger not invertible");
}
if (!p.testBit(0))
{
// montgomery inverse require even modulo
return(modInverseLorencz(a, p));
}
int m = p.numberLength * 32;
// PRE: a \in [1, p - 1]
BigInteger u, v, r, s;
u = p.copy(); // make copy to use inplace method
v = a.copy();
int max = Math.Max(v.numberLength, u.numberLength);
r = new BigInteger(1, 1, new int[max + 1]);
s = new BigInteger(1, 1, new int[max + 1]);
s.digits[0] = 1;
// s == 1 && v == 0
int k = 0;
int lsbu = u.getLowestSetBit();
int lsbv = v.getLowestSetBit();
int toShift;
if (lsbu > lsbv)
{
BitLevel.inplaceShiftRight(u, lsbu);
BitLevel.inplaceShiftRight(v, lsbv);
BitLevel.inplaceShiftLeft(r, lsbv);
k += lsbu - lsbv;
}
else
{
BitLevel.inplaceShiftRight(u, lsbu);
BitLevel.inplaceShiftRight(v, lsbv);
BitLevel.inplaceShiftLeft(s, lsbu);
k += lsbv - lsbu;
}
r.sign = 1;
while (v.signum() > 0)
{
// INV v >= 0, u >= 0, v odd, u odd (except last iteration when v is even (0))
while (u.compareTo(v) > BigInteger.EQUALS)
{
Elementary.inplaceSubtract(u, v);
toShift = u.getLowestSetBit();
BitLevel.inplaceShiftRight(u, toShift);
Elementary.inplaceAdd(r, s);
BitLevel.inplaceShiftLeft(s, toShift);
k += toShift;
}
while (u.compareTo(v) <= BigInteger.EQUALS)
{
Elementary.inplaceSubtract(v, u);
if (v.signum() == 0)
{
break;
}
toShift = v.getLowestSetBit();
BitLevel.inplaceShiftRight(v, toShift);
Elementary.inplaceAdd(s, r);
BitLevel.inplaceShiftLeft(r, toShift);
k += toShift;
}
}
if (!u.isOne())
{
// in u is stored the gcd
throw new ArithmeticException("BigInteger not invertible.");
}
if (r.compareTo(p) >= BigInteger.EQUALS)
{
Elementary.inplaceSubtract(r, p);
}
r = p.subtract(r);
// Have pair: ((BigInteger)r, (Integer)k) where r == a^(-1) * 2^k mod (module)
int n1 = calcN(p);
if (k > m)
{
r = monPro(r, BigInteger.ONE, p, n1);
k = k - m;
}
r = monPro(r, BigInteger.getPowerOfTwo(m - k), p, n1);
return(r);
}
