$N$ is a big integer value, with only two non trivial factors.
Value $k$ will be Symmetric mod if and only if, all the possible nontrivial factors of $N$ will be equal mod k.
For example for N = $557*677=377089$ the values 3,4 and 6 are Symmetric mod
- $557 \mod 3 = 677 \mod 3 = 2$
- $557 \mod 4 = 677 \mod 4 = 1$
- $557 \mod 6 = 677 \mod 6 = 5$
I thought that in order to check if value is Symmetric mod, you do not need to factor $N$, you just need to solve:
$$(a\mod k)(b\mod k) = N \mod k$$
How ever for $k = 5$, the possible options are:
- $a \equiv b \equiv 2(\mod 5)$
- $a \equiv b \equiv 3(\mod 5)$
- $a \equiv 1(\mod 5), b \equiv 4(\mod 5)$
And only after factoring you come to conclusion that $5$ is indeed Symmetric mod
- $557 \mod 5 = 677 \mod 5 = 2$
How to determinate if $k$ is Symmetric mod without factoring $N$?