  Since fully homomorphic encryption and homomorphically encrypted computing
preserve algebraic identities such as 2*2=2+2, a natural question is whether
this extremely utilitarian feature also sets up cryptographic attacks that use
the encrypted arithmetic operators to generate or identify the encryptions of
known constants. In particular, software or hardware might use encrypted
addition and multiplication to do encrypted division and deliver the encryption
of x/x=1. That can then be used to generate 1+1=2, etc, until a complete
codebook is obtained.
  This paper shows that there is no formula or computation using 32-bit
multiplication x*y and three-input addition x+y+z that yields a known constant
from unknown inputs. We characterise what operations are similarly `safe' alone
or in company, and show that 32-bit division is not safe in this sense, but
there are trivial modifications that make it so.
