Java Pow BigInteger实现

Question

I am working on a cryptography implementation and part of the design includes the following:

( (y^a)^b / (y^c)^b ) mod p

I have the following snippet:

BigInteger yab = y.pow(ab.intValue());
BigInteger ycb = y.pow(cb.intValue());

BigInteger ans = (yab.divide(ycb)).mod(p);

It works fine for small integer. Once I replaced it with generated keys, the exponent grew so huge and I will hit the "BigInteger out of int range" error. I have tried the modPow function but the result is different.

I understand that casting it to int has its limitation. Does that means my implementation is infeasible?

Answer 1

It seems like you're doing modular arithmetic in group where n is a prime (in your case is n = p ). This means that

x / y

is not a division but a multiplication of x with the y ^-1 (modular inverse of y ).

Good thing is that the BigInteger class provides such a method:

BigInteger ans = yab.multiply(ycb.modInverse(p)).mod(p);

where yab and ycb can be efficiently computed without overflow (assuming ab is the product of a and b ):

BigInteger yab = y.modPow(ab, p);
BigInteger ycb = y.modPow(cb, p);

Answer 2

You can simplify the code and this will also make it faster

x^y / x^z = x^(y - z)

so

BigInteger yab = y.pow(ab.intValue());
BigInteger ycb = y.pow(cb.intValue());

BigInteger ans = (yab.divide(ycb)).mod(p);

can be simplified to

BigInteger yabc = y.pow((int) (ab.longValue() - cb.longValue()));
BigInteger ans = yabc.mod(p);

or

BigInteger and = y.modPow(ab.minus(cb), p);