In 1994, the well-known Diffie-Hellman key exchange protocol was for the first time implemented in a non-group based setting. Here, the underlying key space was the set of reduced principal ideals of a real quadratic number field. This set does not possess a group structure, but instead exhibits a so-called infrastructure. More recently, the scheme was extended to real quadratic congruence function fields, whose set of reduced principal ideals has a similar infrastructure. As always, the security...

For $a$ generator of the multiplicative group of the field $GF(p,k)$, the discrete logarithm of an element $b$ of the field to the base $a$, $b\ne 0$ is that integer $z:1\le z\le p^k-1$, $b=a^z$. The $p$-ary digits which represent $z$ can be described with extremely simple polynomial forms.