Most optimized way to calculate modulus in C

Question

I have minimize cost of calculating modulus in C. say I have a number x and n is the number which will divide x

when n == 65536 (which happens to be 2^16):

m

Answer 1

rIf you don't have to fully reduce your integers modulo 65521, then you can use the fact that 65521 is close to 2**16. I.e. if x is an unsigned int you want to reduce then you can do the following:

unsigned int low = x &0xffff;
unsigned int hi = (x >> 16);
x = low + 15 * hi;

This uses that 2**16 % 65521 == 15. Note that this is not a full reduction. I.e. starting with a 32-bit input, you only are guaranteed that the result is at most 20 bits and that it is of course congruent to the input modulo 65521.

This trick can be used in applications where there are many operations that have to be reduced modulo the same constant, and where intermediary results do not have to be the smallest element in its residue class.

E.g. one application is the implementation of Adler-32, which uses the modulus 65521. This hash function does a lot of operations modulo 65521. To implement it efficiently one would only do modular reductions after a carefully computed number of additions. A reduction shown as above is enough and only the computation of the hash will need a full modulo operation.