**Is SHA-1 conceptually sound?**

*Charanjit S. Jutla and Anindya C. Patthak*

**Abstract: **We argue that if the message expansion code of SHA-1 is replaced
by a linear code with a better minimum distance, then the resulting
hash function is collision resistant. To support this argument,
we characterize the disturbance vectors which are used to build
local collision attacks as a linear code. This linear code is the xor-sum of two codes, the message expansion code and a linear code representing the underlying block cipher in SHA-1.

We also show that the following constraint satisfaction problem is $\np$-hard. The constraints are restricted to being XOR constraints, or Majority constraints on at most three variables each. The instances are further restricted by requiring that the constraints can be listed in a sequence C_1, C_2,...,C_m, such that for every constraint C_i, two of the variables in it occur only in constraints C_j, with |j-i|< 48. This problem is similar to the problem modeling the one-way function property of SHA-1.

**Category / Keywords: **Hash Functions, Linear Codes, minimum distance, CSPs, oneway functions

**Date: **received 3 Oct 2005, last revised 28 Nov 2005

**Contact author: **csjutla at us ibm com

**Available format(s): **Postscript (PS) | Compressed Postscript (PS.GZ) | BibTeX Citation

**Version: **20051128:230646 (All versions of this report)

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