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Algorithm Implementation/Mathematics/Modular Exponentiation
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Modular exponentiation
Algorithm
Here we show the modular exponentiation algorithm for integers - a way to efficiently compute ae (mod n).
This general algorithm may also be used for other algebraic structures which have multiplication and exponentiation and is
efficient when the size of values has an upper bound - the modulus. Other structures which can use this basic algorithm
include matrix exponentiation with floating point coefficients and elliptic curve computations over finite fields (although in
that case the operation is called multiplication, not exponentiation).
Low to High Algorithm
This is the most intuitive algorithm. The algorithm scans the exponent, a bit at a time, and if the nth bit has the value 1,
multiplies the accumulator by a2^n, which is computed by successive squaring as the exponent is traversed.
A pseudocode description is:
function powmod(a, e, n)
accum := 1
x = e // Scan the bits of the exponents
apow := a // Successive powers, a^(2^n)
while (x ≠ 0)
if (x is odd)
accum := accum × apow (mod n)
Divide x by 2
apow : = apow^2 (mod n)
return accum
High to Low Algorithm
This algorithm uses slightly less memory - not requiring separate storage of the powers of a, and is
also more efficient if the base a has a special structure (such as being small), making multiplication
by it easy. The algorithm scans the exponent, a bit at a time, from the bit immediately following the top bit to the low order bit.
A pseudocode description is:
function powmod(a, e, n)
accum := a
if (e = 0) return 1 // A special case
for (bitptr := highbit(e) - 1; bitptr >= 0; bitptr--)
accum := accum^2 (mod n)
if (e[bitptr] = 1)
accum := accum × a (mod n)
return accum
C
Low-to-High algorithm
long powmod(long a, long e, long n) {
long accum = 1, apow, x;
apow = a; x = e;
while (x != 0) {
if (x & 0x01)
accum = (accum*apow) % n;
x >>= 1;
apow = (apow * apow) % n;
};
return accum;
}
Low-to-High algorithm with GMP
GMP allows for calculating with arbitrary precision. The GMP library has a mpz_powm(result,a,e,n) function, so this is not needed,
but is here strictly for expository value.
// High to Low powmod algorithm
void powmod(mpz_t result, mpz_t a, mpz_t e, mpz_t n) {
// Use result as accum (temp variable)
if (mpz_cmp_si(e, 0) == 0) { // If exponent is zero
mpz_set_ui(result, 1); // Set result to 1
};
mpz_set(result, a); // Set value of accum to a
int bitptr = mpz_sizeinbase(e,2) - 1; // Find top bit in exponent
for (bitptr--; bitptr >= 0; bitptr--) {
mpz_mul(result, result, result); // result <-- result^2
mpz_fdiv_r(result, result, n); // result <-- result (mod n)
if (mpz_tstbit(e,bitptr)) { // Is bit e[bitptr] == 1?
mpz_mul(result, result, a); // result <-- result*a
mpz_fdiv_r(result, result, n); // result <-- result (mod n)
};
};
}
Java
Low-to-High algorithm
public static long powmod(long a, long e, long n){
long accum = 1;
long x = e;
long apow = a;
while (x != 0){
if ((x & 0x01) == 0x01){
accum = (accum * apow) % n;
};
x >>= 1;
apow = (apow * apow) % n;
};
return accum;
}
Low-to-High algorithm with BigInteger
The Java BigInteger class has a modPow(e, n) method, so this is not needed,
but is here strictly for expository value.
// High to low powmod algorithm
public static BigInteger POWMOD(BigInteger a, BigInteger e, BigInteger n) {
BigInteger accum = a;
if (e.equals(BigInteger.ZERO))
return BigInteger.ONE;
int bitptr = e.bitLength()-1;
for (bitptr--; bitptr >= 0; bitptr--) {
accum = accum.multiply(accum).mod(n);
if (e.testBit(bitptr)) {
accum = accum.multiply(a).mod(n);
};
};
return accum;
}
Python
Low-to-High algorithm
Python has a pow(a, e, n) function, so this is not needed,
but is here strictly for expository value.
def powmod(a: int, e: int, n: int) -> int:
accum: int = 1
apow2: int = a
while e > 0:
if e & 1:
accum = (accum * apow2) % n
apow2 = (apow2 * apow2) % n
e = e >> 1
return accum