# Are there any noisy homomorphic encryption schemes?

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

• Presumably you want to avoid assuming the existence of standard homomorphic encryption schemes, since otherwise there's a "Just Do It" construction. $\;$ – user991 Apr 8 '15 at 4:24
• @RickyDemer sorry i did not get you, can u be more clear ? what is a "just do it" construction ? – sashank Apr 8 '15 at 4:27
• If there is a standard homomorphic encryption schemes then there is a noisy homomorphic encryption scheme. $\:$ That can be proven by showing how one can directly turn a standard homomorphic encryption scheme into a noisy homomorphic encryption scheme. $\:$ (I have given two other "Just Do It" constructions on this site.) $\;\;\;\;$ – user991 Apr 8 '15 at 4:32
• @RickyDemer got it ! now the question left is whether there are natively noisy algorithms – sashank Apr 8 '15 at 5:13
• doesn't Regev scheme based on LWE verify this if you set the right parameters for the error distribution? – Florian Bourse Apr 8 '15 at 8:38

That is, if $$c$$ is an encryption of $$m$$, then $$Dec(c) = m + e$$, where $$e$$ is some small error.
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...).