Compressed Modular Matrix Multiplication

Matrices of integers modulo a small prime can be compressed by storing several entries into a single machine word. Modular addition is performed by addition and possibly subtraction of a word containing several times the modulus. We show how modular multiplication can also be performed. In terms of arithmetic operations, the gain over classical matrix multiplication is equal to the number of integers that are stored inside a machine word. The gain in actual speed is also close to that number. First, modular dot product can be performed via an integer multiplication by the reverse integer. Modular multiplication by a word containing a single residue is also possible. We give bounds on the sizes of primes and matrices for which such a compression is possible. We also make explicit the details of the required compressed arithmetic routines and show some practical performance.