Abstract. This paper compares different approaches for computing power products � 1≤i≤k ge i i in commutative groups. We look at the conventional simultaneous exponentiation approach and present an alternative strategy, interleaving exponentiation. Our comparison shows that in general groups, sometimes the conventional method and sometimes interleaving exponentiation is more efficient. In groups where inverting elements is easy (e.g. elliptic curves), interleaving exponentiation with signed exponent recoding usually wins over the conventional method. 1

We explain a variant of the Fiat-Shamir identification and signature protocol that is based on the intractability of computing generators of principal ideals in algebraic number fields. We also

This paper gives a survey on cryptographic primitives based on class groups of imaginary quadratic orders (IQ cryptography, IQC). We present IQC versions of several well known cryptographic primitives, and we explain, why these primitives are secure if one assumes the hardness of the underlying problems. We give advice on the selection of the cryptographic parameters and show the impact of this advice on the eciency of some IQ cryptosystems.

This paper gives an overview of the state of art in cryptography based on quadratic orders. It discusses the intractable problems in class groups and the infrastructure of quadratic orders, approaches to the solution to these problems, and the crypto-systems employing them as underlying problems.