  The BMR16 circuit garbling scheme introduces gadgets that allow for
ciphertext-free modular addition, while the multiplication of private inputs
modulo a prime p can be done with 2(p - 1) ciphertexts as described in Malkin,
Pastro, and Shelat's An algebraic approach to garbling. By using a residue
number system (RNS), we can construct a circuit to handle the squaring and
multiplication of inputs modulo a large N via the methods described in Hollman
and Gorissen's multi-layer residue number system. We expand on the existing
techniques for arithmetic modulo p to develop methods to handle arithmetic in a
positional, base-p number system. We evaluate the ciphertext cost of both of
these methods and compare their performance for squaring in various large
moduli.
