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popular mixing color approach

https://cse.sc.edu/~farkas/csce522/lectures/test1/With%20Math%20-%20Diffie-Hellman%20Rick.ppt

probably need to add it into your notes as well.

Hello,

In the lattice section, the coefficient modulus (q) is always stated to be prime (I guess it was taken from the shell implementation), but it does not need to be prime. It can in fact be any integer as long as it is smaller than the maximum bitsize for a given ring degree to ensure security.

Being prime has no effect on the security at all (although from recent research the structure of Q might lead to very efficient quantum attacks). It can however enable more efficient implementations (like RNS representation and the usage of the NTT) and sometime you do not want it to be prime, for example the official HEAAN implementation use q = 2^s (a power of two).

"The smallest number n for which nX = 0 is called the order of X."

If n is the identity (for example 0) then nX also equals 0 but the order of the group is not 0.

Hi Quan,

In section 9 (page 14), you wrote:

The pairing that we use has a few nice properties such as: e(P + Q, R) = e(P, R) + e(Q, R) and e(a.P, b.Q) = e(P, Q)^(a*b)

Maybe you meant:

• e(P + Q, R) = e(P, R) * e(Q, R) and e(a.P, b.Q) = e(P, Q)^(a*b), as ^ denotes repeated *.
• or e(P + Q, R) = e(P, R) + e(Q, R) and e(a.P, b.Q) = e(P, Q).(a*b), as . denotes repeated +.

Hey Quan, great note! It is always better to understand things under intuitive approach. I'm having a feedback while reading it.

In the Field section, you may miss the definition of multiplication. Without it, we don't have any clues how to do P1(x) ∗ P2(x) in 2.4 Polynomial modulus section. Or without definition, how do we understand an element multiply with another element?
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Do I miss it somewhere?