Lets assume that *n*=4*kn*_{2}+1 is a prime,
and that
is also a prime.
This condition is possible to satisfy, and these numbers are similar
in concept to *safe primes*.
With them we can represent *S*_{n}(*i*) and also
*S*_{n2}(*i*).
Now we will choose
*S*(*e*)=*p*^{4k}, where *p* is a primitive root (mod *n*).
By construction,
, so
for any choice of *S*(*e*).
This means that the exponent arithmetic can be done (mod *n*_{2}),
which implies that we can compute the signatures of the exponents
in a field.
This has many advantages, among others the ability to represent *i*and being free of any bad divisors.
The counterpart is that the signatures of *e*^{x} will have a smaller
set of possible values, *n*_{2} instead of *n*-1, or a factor of 4*k*.
Assuming that *k* is not too large (it could just be 1), we need to
choose *n* a factor of 4*k* larger than what we would have done.
This is a small price to pay for being able to compute exponent
signatures in a field.

As before, we will select *S*(*e*) so that it does not satisfy any
simple algebraic relationship.
In this case we want to select it also such that *S*(*e*)^{i} does not
satisfy any simple relation.
For example, if we choose *n*=4000133 and
*n*_{2}=1000033, which
satisfy the above constraints, *p*=1351050 is a primitive root (mod *n*)
which gives
*S*(*e*)=266466.
Neither *S*(*e*) nor *S*(*e*)^{i} satisfy any simple algebraic relation.