Groups where the discrete logarithm problem is assumed to be intractable are central in the design of public-key cryptography. This was first pointed out by diffie and hellman in their seminal paper. The security of the diffie-hellman key-distribution system relies on the intractability of the discrete logarithm problem in the multiplicative group of finite fields. Such groups also allow one to construct encryption schemes, digital signature schemes, and many other cryptographic primitives and protocols. As an application, the authors present an improved, torus-based implementation of the ACJT group signature scheme.