Practical Threshold Signatures

Question

In Practical Threshold Signature it says that there is a requirement in which the RSA public exponent $e$ must be more than $l$. $l$ in this case is the number of players in the threshold group.

My question is: Why is there a constraint that the public exponent must be larger than the number of players?

In my case, my public exponent must be 3 (due to limitation on another component), hence the number of player can only be 2.

Thanks.

Answer 1

I am not too familiar with this paper, but there is (at least) the following reason. They define $\Delta = l!$, then require (right after equation 6) that $1 = \mathsf{gcd}(e', e) = \mathsf{gcd}(4\Delta^2, e) = \mathsf{gcd}(4(l!)^2, e)$. This is not possible if $e \leq l$ (as then the GCD is $e\neq 1$).