TITLE: Torus based cryptography
QUESTION [5 upvotes]: In cryptography one needs finite groups $G$ in which the discrete logarithm problem is infeasible. Often they use the multiplicative group $\mathbb{G}_m(\mathbb{F}_p)$ where $p$ is a prime number of bit length $500$, say.
Rubin and Silverberg suggested (cf. [1]) to use certain tori instead, if the goal is to minimize the key size. In the easiest case, this comes down to using the group
$$T_2(p)=ker(Norm: \mathbb{F}_{p^2}^\times\to \mathbb{F}_p^\times).$$
If I understood correctly, then the underlying philosopy seems to be: The group $T_2(p)$ should be as secure as $\mathbb{F}_{p^2}^\times$, but its size is only $p+1$. (So, if you use groups of type $T_2(p)$ instead of groups of type $\mathbb{G}_m(\mathbb{F}_p)$, then you can achive the same security with half the key size.)
Question. What are the reasons, be they heurisical or strictly provable, to believe
in this philosopy?
Denote by $\mathbb{G}'_m$ the quadratic twist of the algebraic group $\mathbb{G}_m$.
It is easy to see that $T_2(p)$ is isomorphic to $\mathbb{G}'_m(\mathbb{F}_p)$.
(This isomorphism is easy to compute). The philosophy predicts: The quadratic twist of the multiplicative group should be better than the multiplicative group itself.
(Compare with elliptic curves: If $E/\mathbb{F}_p$ is an elliptic curve, then I would certainly not expect its quadratic twist to be better than $E$ itself.)
Remark: I concentrated on the simplest case above. One also considers certain groups $T_n(p)$ which are expected to be as secure as $\mathbb{F}_{p^n}^\times$, while their size is only $\approx\varphi(n)p$. Lemma 7 in [1] is meant to explain this. However, I would be keen on a more detailed explanation. 
[1] Lect. Notes in Comp. Sci. 2729 (2003) 349-365. (available at http://math.stanford.edu/~rubin/)

REPLY [6 votes]: Can I refer you to my paper `On the Discrete Logarithm Problem on Algebraic Tori', Advances in Cryptology – CRYPTO 2005, Lecture Notes in Computer Science, 2005, Volume 3621/2005, 66-85, in which myself and Frederik Vercauteren studied this very problem.
In particular, we showed that the compression mechanism afforded by the birationality of some algebraic tori may be exploited to obtain a faster discrete logarithm algorithm for some cryptographically practical field sizes. In these instances, attacking the discrete logarithm in $\mathbb{F}_{p^n}^{\times}$ via its decomposition 
$\prod_{d \mid n} T_d(\mathbb{F}_p)$
is faster than using L[1/3] index calculus techniques.
Since then, other work has improved the L[1/3] index calculus techniques. However, our work demonstrates that it is naive to argue that the DLP in algebraic tori must be 
hard purely because the DLP in the multiplicative group of the extension field is hard, precisely because an attack on the former provides an attack on the latter.