# Public key in fully homomorphic encryption over the integers

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?

• What happens if you set it even? – xagawa Aug 4 '14 at 12:23
• 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. – JongHyun Kim Aug 4 '14 at 15:04
• @JongHyunKim, you could post that as an answer, so if someone else arrives here wondering about it, they can easily find it. – otus Aug 5 '14 at 7:46

(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.