**SL2 homomorphic hash functions: Worst case to average case reduction and short collision search**

*Ciaran Mullan and Boaz Tsaban*

**Abstract: **We study homomorphic hash functions into SL(2,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 concrete
homomorphisms proposed thus far, we provide a worst case to average case reduction for these hash functions:
upto a logarithmic factor, a random homomorphism is 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.
While exponetial time, our algorithms are faster in practice than all earlier generic algorithms,
and produce much shorter collisions.

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

**Date: **received 24 Jun 2013, last revised 2 Nov 2015

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

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

**Note: **To appear in Designs Codes and Cryptography

**Version: **20151102:224852 (All versions of this report)

**Short URL: **ia.cr/2013/415

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