Are there any Homomorphic Encryption(HE) schemes that result in noisy answers ? By noisy i mean , the answers could be approximately near the actual answers by noise factor $\epsilon$.
For example , we could get $dec_{sk}(eval_{pk}({enc_{pk}(5) + enc_{pk}(10))) = 16}$ where as the real answer is 15. Here $enc,eval,dec$ are functions of an HE scheme and $sk,pk$ are private and public keys. The scheme could be Fully HE or Semi HE or Partial HE
Yes, on the paper Homomorphic Encryption for Arithmetic of Approximate Numbers, Cheon et al. propose a scheme in which the error is not completely eliminated during decryption.
That is, if $c$ is an encryption of $m$, then $Dec(c) = m + e$, where $e$ is some small error.
Roughly speaking, that scheme simulates floating-point arithmetic, since this type of arithmetic also accumulate errors.
With that, the authors manage to perform operations that are hard to do with exact homomorphic encryption schemes. For example, in Algorithm 2 (page 15), they show how to compute homomorphically $\frac 1 m$ with $r$ bits of precision (with other schemes, that is only possible using bootstrapping...).
