Torus-Based Cryptography

Synonyms

Subgroup cryptosystems

Related Concepts

Discrete Logarithm Problem; Finite Field; Primitive Element

Definition

Torus-based cryptography aims at representing certain field elements in a compact form, while keeping the difficulty of the Discrete Logarithm Problem unchanged. The main ideas come from arithmetic, in particular, algebraic tori.

Background

Torus-based cryptography is a recent development of Subgroup Cryptosystems. Classically, the multiplicative group ${\mathbb{F}}_{{p}^{n}}^{{_\ast}}$ of a finite field ${\mathbb{F}}_{{p}^{n}}$ is used in the context of public-key cryptography (e.g., the Diffie–Hellmann Key Agreement). In [6], Schnorr proposes to use a proper subgroup $\langle g\rangle$ of ${\mathbb{F}}_{{p}^{n}}^{{_\ast}}$ and argues that this approach presents advantages, as long as the discrete logarithm problem in $\langle g\rangle$ is as hard as in ${\mathbb{F}}_{{p}^{n}}^{{_\ast}}$. A necessary requirement is then that gdoes not belong to any...