## Cryptology ePrint Archive: Report 2013/415

**Short collision search in arbitrary SL2 homomorphic hash functions**

*Ciaran Mullan and Boaz Tsaban*

**Abstract: **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.

**Category / Keywords: **foundations / SL2 hash, homomorphic hash

**Date: **received 24 Jun 2013

**Contact author: **tsaban at math biu ac il

**Available format(s): **PDF | BibTeX Citation

**Version: **20130625:160614 (All versions of this report)

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