Modular Exponentiation in Java

问题

I need a way to calculate:

(g^u * y^v) mod p

in Java.

I've found this algorithm for calculating (g^u) mod p:

int modulo(int a,int b,int c) {
    long x=1
    long y=a;
    while(b > 0){
        if(b%2 == 1){
            x=(x*y)%c;
        }
        y = (y*y)%c; // squaring the base
        b /= 2;
    }
    return (int) x%c;
}

and it works great, but I can't seem to find a way to do this for

(g^u * y^v) mod p

as my math skills are lackluster.

To put it in context, it's for a java implementation of a "reduced" DSA - the verifying part requires this to be solved.

回答1:

Assuming that the two factors will not overflow, I believe you can simplify an expression like that in this way:

(x * y) mod p = ( (x mod p)*(y mod p) ) mod p. I'm sure you can figure it out from there.

回答2:

That fragment of code implements the well known "fast exponentiation" algorithm, also known as Exponentiation by squaring.

It also uses the fact that (a * b) mod p = ((a mod p) * (b mod p)) mod p. (Both addition and multiplications are preserved structures under taking a prime modulus -- it is a homomorphism). This way at every point in the algorithm it reduces to numbers smaller than p.

While you could try to calculate these in an interleaved fashion in a loop, there's no real benefit to doing so. Just calculate them separately, multiply them together, and take the mod one last time.

Be warned that you will get overflow if p^2 is greater than the largest representable int, and that this will cause you to have the wrong answer. For Java, switching to big integer might be prudent, or at least doing a runtime check on the size of p and throwing an exception.

Finally, if this is for cryptographic purposes, you should probably be using a library to do this, rather than implementing it yourself. It's very easy to do something slightly wrong that appears to work, but provides minimal to no security.

回答3:

Try

(Math.pow(q, u) * Math.pow(y, v)) % p

来源：https://stackoverflow.com/questions/4066952/modular-exponentiation-in-java