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I'm reading “Fully Homomorphic Encryption over the Integers” by van Dijk et al.

I wonder why $x_0$, which is a component of the public key, should be an odd number?

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    $\begingroup$ What happens if you set it even? $\endgroup$ – xagawa Aug 4 '14 at 12:23
  • $\begingroup$ Thanks for giving me a hint. (even number mod even number) is always even number and (odd number mod even number) is always odd number. If m=0, the result of modular $x_0$ will be always even number. If m=1, the result will be odd number. So $x_0$ should be a odd number. $\endgroup$ – JongHyun Kim Aug 4 '14 at 15:04
  • $\begingroup$ @JongHyunKim, you could post that as an answer, so if someone else arrives here wondering about it, they can easily find it. $\endgroup$ – otus Aug 5 '14 at 7:46
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(even number mod even number) is always even number and (odd number mod even number) is always odd number. If $m=0$, the result of modular $x_0$ will be always even number. If $m=1$, the result will be odd number. So $x_0$ should be a odd number.

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