Short collision search in arbitrary SL2 homomorphic hash functions, by Ciaran Mullan and Boaz Tsaban
We study homomorphic hash functions into SL2(q), the 2x2 matrices with determinant 1 over the
field with q elements.
Modulo a well supported number theoretic hypothesis, which holds in particular for all concrete
homomorphisms proposed thus far, we prove that
a random homomorphism is at least as secure as any concrete homomorphism.
For a family of homomorphisms containing several concrete proposals in the literature,
we prove that collisions of length O(log q) can be found in running time O(sqrt q).
For general homomorphisms we offer an algorithm that, heuristically and according to experiments,
in running time O(sqrt q) finds collisions of length O(log q) for q even, and length O(log^2 q/loglog q) for arbitrary q.
For any conceivable practical scenario, our algorithms are substantially faster than all earlier algorithms
and produce much shorter collisions.
Computational Fuzzy Extractors, by Benjamin Fuller and Xianrui Meng and Leonid Reyzin
Fuzzy extractors derive strong keys from noisy sources. Their security is defined information- theoretically, which limits the length of the derived key, sometimes making it too short to be useful. We ask whether it is possible to obtain longer keys by considering computational security, and show the following.
-Negative Result: Noise tolerance in fuzzy extractors is usually achieved using an information reconciliation component called a \"secure sketch.\" The security of this component, which directly affects the length of the resulting key, is subject to lower bounds from coding theory. We show that, even when defined computationally, secure sketches are still subject to lower bounds from coding theory. Specifically, we consider two computational relaxations of the information-theoretic security requirement of secure sketches, using conditional HILL entropy and unpredictability entropy. For both cases we show that computational secure sketches cannot outperform the best information-theoretic secure sketches in the case of high-entropy Hamming metric sources.
-Positive Result: We show that the negative result can be overcome by analyzing computational fuzzy extractors directly. Namely, we show how to build a computational fuzzy extractor whose output key length equals the entropy of the source (this is impossible in the information-theoretic setting). Our construction is based on the hardness of the Learning with Errors (LWE) problem, and is secure when the noisy source is uniform or symbol-fixing (that is, each dimension is either uniform or fixed). As part of the security proof, we show a result of independent interest, namely that the decision version of LWE is secure even when a small number of dimensions has no error.
Unconditional Tightness Bounds for Generic Reductions: The Exact Security of Schnorr Signatures, Revisited, by Nils Fleischhacker and Tibor Jager and Dominique Schröder
A long line of research investigates the existence of tight security reductions for the Schnorr signature scheme. Most of these works presented lower tightness bounds, most recently Seurin (Eurocrypt 2012) showed that under certain assumptions the non-tight security proof for Schnorr signatures by Pointcheval and Stern (Eurocrypt 1996) is essentially optimal. All previous works in this direction share the same restrictions: The results hold only under the interactive one-more discrete logarithm assumption, they only consider algebraic reductions, and they only rule out tight reductions from the (one-more) discrete logarithm problem. The existence of a tight reduction from weaker computational problems, like CDH or DDH, remained open.
In this paper we introduce a new meta-reduction technique, which allows to prove lower bounds for the large and very natural class of generic reductions. A generic reduction is independent of a particular representation of group elements. Most reductions in state-of-the-art security proofs have this desirable property. This new approach allows to show unconditionally that there is no tight generic reduction from any natural computational problem \\Pi defined over algebraic groups (including even interactive problems) to breaking Schnorr signatures, unless solving \\Pi is easy.