**Non-Linear Reduced Round Attacks Against SHA-2 Hash family**

*Somitra Kumar Sanadhya and Palash Sarkar*

**Abstract: **Most of the attacks against (reduced) SHA-2 family in literature
have used local collisions which are valid for linearized version of SHA-2 hash functions. Recently, at FSE '08, an attack against reduced round SHA-256 was presented by Nikolić and Biryukov which used
a local collision which is valid for the actual SHA-256 function. It is a 9-step local collision which starts by introducing a modular difference of 1 in the two messages. It succeeds with probability roughly 1/3. We build on the work of Nikolić and Biryukov and provide a generalized nonlinear local collision which accepts an arbitrary initial message difference. This local collision succeeds with probability 1. Using this local collision we present attacks against 18-step SHA-256 and 18-step SHA-512 with arbitrary initial difference. Both of these attacks succeed with probability 1.
We then present special cases of our local collision and show two different differential paths for attacking 20-step SHA-256 and 20-step SHA-512. One of these paths is the same as presented by Nikolić and Biryukov while the other one is a new differential path. Messages following both these differential paths can be found with probability 1. This improves on the previous result where the success
probability of 20-step attack was 1/3. Finally, we present two differential paths for 21-step collisions for SHA-256 and SHA-512, one of which is a new path. The success probability of these paths for SHA-256 is roughly $2^{-15}$ and $2^{-17}$ which improves on the 21-step attack having probability $2^{-19}$ reported earlier.
We show examples of message pairs following all the presented differential paths for up to 21-step collisions in SHA-256. We also show first real examples of colliding message pairs for up to 20-step reduced SHA-512.

**Category / Keywords: **Cryptanalysis, SHA-256 Hash function, Reduced round attacks.

**Publication Info: **A shorter version of this paper will appear in the proceedings of ACISP 2008.

**Date: **received 16 Apr 2008

**Contact author: **somitra_r at isical ac in

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

**Version: **20080421:093515 (All versions of this report)

**Short URL: **ia.cr/2008/174

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