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Paper 2017/800

Collisions and Semi-Free-Start Collisions for Round-Reduced RIPEMD-160

Fukang Liu, Florian Mendel, and Gaoli Wang

Abstract

In this paper, we propose an improved cryptanalysis of the double-branch hash function RIPEMD-160 standardized by ISO/IEC. Firstly, we show how to theoretically calculate the step differential probability of RIPEMD-160, which was stated as an open problem by Mendel $et$ $al.$ at ASIACRYPT 2013. Secondly, based on the method proposed by Mendel $et$ $al.$ to automatically find a differential path of RIPEMD-160, we construct a 30-step differential path where the left branch is sparse and the right branch is controlled as sparse as possible. To ensure the message modification techniques can be applied to RIPEMD-160, some extra bit conditions should be pre-deduced and well controlled. These extra bit conditions are used to ensure that the modular difference can be correctly propagated. This way, we can find a collision of 30-step RIPEMD-160 with complexity $2^{70}$. This is the first collision attack on round-reduced RIPEMD-160. Moreover, by a different choice of the message words to merge two branches and adding some conditions to the starting point, the semi-free-start collision attack on the first 36-step RIPEMD-160 from ASIACRYPT 2013 can be improved. However, the previous way to pre-compute the equation $T^{\lll S_0}\boxplus C_0=(T\boxplus C_1)^{\lll S_1}$ costs too much. To overcome this obstacle, we are inspired by Daum's $et~al$. work on MD5 and describe a method to reduce the time complexity and memory complexity to pre-compute that equation. Combining all these techniques, the time complexity of the semi-free-start collision attack on the first 36-step RIPEMD-160 can be reduced by a factor of $2^{15.3}$ to $2^{55.1}$.

Note: 1. We negelect three uncontrolled bit conditions on the right branch when mounting collision attack by mistake. Therefore, the time complexity shoule become $2^{70}$. 2. We also correct the probability that $\Delta X_4$ = 0 for the semi-free-start collision attack. But this will not influence the original cpmlexity or probability of the 36-step semi-free-start collision attack.