where

Over rational expressions of polynomials, the only relations that
*S*(*e*^{a}) has to follow are:
*S*(*e*^{0}) = 1 and
*S*(*e*^{a+b}) = *S*(*e*^{a}) *S*(*e*^{b}).
These are satisfied with the above definition.

*S*(*e*^{a}) should also satisfy a negative condition, which is
that there is no relation
for any finite-degree polynomials
*P*(*x*) and *Q*(*x*).
This can be proved over the reals by showing that the
expansion of *Q*(*x*)*e*^{x} will have a term with degree higher than
the degree of *P*(*x*), and hence for sufficiently large *x*,
|*Q*(*x*)*e*^{x}| > |*P*(*x*)|.

This property should also be maintained over (mod *n*).
To prove this we make use of the operator
.
Now, each time that we apply to a polynomial, we reduce
its degree by one.
So

where

or

Suppose that

Choosing *S*(*e*) to be a primitive root guarantees that the
range of the signatures *S*(*e*^{x}) is as large as possible,
i.e. there will be different possible values for
the signature.
An extremely small range would force us to reconsider the
probability bounds, in particular if we were computing
with polynomials on exponentials.