An Open Question on the Uniqueness of (Encrypted) Arithmetic

We ask whether two or more images of arithmetic may inhabit the same space via different encodings. The answers have significance for a class of processor design that does all its computation in an encrypted form, without ever performing any decryption or encryption itself. Against the possibility of algebraic attacks against the arithmetic in a `crypto-processor' (KPU) we propose a defence called `ABC encryption' and show how this kind of encryption makes it impossible for observations of the arithmetic to be used by an attacker to discover the actual values. We also show how to construct such encrypted arithmetics.