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Frequently, I have seen people use the term "order" in cryptography (the group-theoretic one). I have a mathematical background and "order" (say, for prime modulus $p$) is defined as the smallest integer, such that:

$$a^r \equiv 1 \pmod p$$

So, a generator (something which has the max possible order; i.e., order is $\phi(p)=p-1$) will have order $p-1$. In case of a composite group, the generator should have order $\phi(N)$.

How is it that, frequently, I read in crypto literature that people say generator has order $N$?

For example, the first answer at: When do we need composite order groups for bilinear maps and when prime order?

Am I missing some details or is my understanding incorrect?

Alin Tomescu
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CutestPenguin
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1 Answers1

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You probably know that cyclic groups of any order exist, so it is perfectly possible to have a cyclic group of order $pq$, and to consider a generator thereof.

fkraiem
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