Gottfried Herold

Polynomial Spaces: A New Framework for Composite-to-Prime-Order Transformations

At Eurocrypt 2010, Freeman presented a framework to convert cryptosystems based on composite-order groups into ones that use prime-order groups. Such a transformation is interesting not only from a conceptual point of view, but also since for relevant parameters, operations in prime-order groups are faster than composite-order operations by an order of magnitude. Since Freeman’s work, several other works have shown improvements, but also lower bounds on the efficiency of such conversions.

Polly Cracker, revisited

We formally treat cryptographic constructions based on the hardness of deciding ideal membership in multivariate polynomial rings. Of particular interest to us is a class of schemes known as “Polly Cracker.” We start by formalising and studying the relation between the ideal membership problem and the problem of computing a Gröbner basis. We show both positive and negative results. On the negative side, we define a symmetric Polly Cracker encryption scheme and prove that this scheme only achieves bounded

On the asymptotic complexity of solving LWE

\mathsf {CPA}

Improved Algorithms for the Approximate k-List Problem in Euclidean Norm

CPA security under the hardness of the ideal membership problem. Furthermore, we show that a large class of algebraic transformations cannot convert this scheme to a fully secure Polly Cracker-style scheme. On the positive side, we formalise noisy variants of the ideal-theoretic problems. These problems can be seen as natural generalisations of the learning with errors (

New attacks for knapsack based cryptosystems

\mathsf {LWE}

New Techniques for Structural Batch Verification in Bilinear Groups with Applications to Groth-Sahai Proofs

LWE) and the approximate GCD problems over polynomial rings. After formalising and justifying the hardness of the noisy assumptions, we show that noisy encoding of messages results in a fully