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Since we know that Homomorphic encryption allow computation on encrypted data. Let two n-bit integers $(x,y)$ are encrypted using some LWE public key cryptography. For example - if $x'= HEnc(x,pk)$ and $y'= HEnc(y,pk)$, Finding $Max(x',y')$ or $Min(x',y')$ or $Equal(x',y')$ the above comparison is necessary.

Now, we need to perform comparison (>=,==) operation homomorphically. How can we do this? Can anyone describe this with example?

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    $\begingroup$ So both operands are encrypted and the result is an encrypted bit? Is that the scenario? $\endgroup$ – mikeazo Jan 26 '16 at 12:50
  • $\begingroup$ Yes, this scenario is right. Homomorphic encryption allow computation on encrypted data. For example if x'=HEnc(x,pk) and y'=HEnc(y,pk), Finding Max(x',y') or Min(x',y') the above comparison is necessary. $\endgroup$ – Tushar Saha Jan 27 '16 at 4:21
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You'll have to write the function you are trying to calculate as a polynomial in the two inputs $x$ and $y$. If you are working over the field with $q$ elements as plaintexts, you have to calculate for equality the polynomial $(x-y)^{q-1}$. Greater-than-or-equal (however you define that for finite fields) will be even more complicated.

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Have a look at order revealing encryption (ORE) schemes:

https://eprint.iacr.org/2014/834.pdf

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  • $\begingroup$ Many thanks for suggesting this paper. The paper is about 'Order revealing encryption'. Is there any relation between 'Order revealing Encryption' and 'Homomorphic encryption'? I want to do this job homomorphically because I've some other job with this encryption. $\endgroup$ – Tushar Saha Jan 27 '16 at 7:18
  • $\begingroup$ No, ORE is rather unrelated to HOM. This "answer" is more of a comment. $\endgroup$ – Thomas M. DuBuisson Jan 27 '16 at 17:52

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