### Another Look at Key Randomisation Hypotheses

Subhabrata Samajder and Palash Sarkar

##### Abstract

In the context of linear cryptanalysis of block ciphers, let $p_0$ (resp. $p_1$) be the probability that a particular linear approximation holds for the right (resp. a wrong) key choice. The standard right key randomisation hypothesis states that $p_0$ is a constant $p\neq 1/2$ and the standard wrong key randomisation hypothesis states that $p_1=1/2$. Using these hypotheses, the success probability $P_S$ of the attack can be expressed in terms of the data complexity $N$. The resulting expression for $P_S$ is a monotone increasing function of $N$. Building on earlier work by Daemen and Rijmen (2007), Bogdanov and Tischhauser (2014) argued that $p_1$ should be considered to be a random variable. They postulated the adjusted wrong key randomisation hypothesis which states that $p_1$ follows a normal distribution. A non-intuitive consequence was that the resulting expression for $P_S$ is no longer a monotone increasing function of $N$. A later work by Blondeau and Nyberg (2017) argued that $p_0$ should also be considered to be a random variable and they postulated the adjusted right key randomisation hypothesis which states that $p_0$ follows a normal distribution. In this work, we revisit the key randomisation hypotheses. While the argument that $p_0$ and $p_1$ should be considered to be random variables is indeed valid, we consider the modelling of their distributions by normal to be inappropriate. Being probabilities, the support of the distributions of $p_0$ and $p_1$ should be subsets of $[0,1]$ which does not hold for normal distributions. We show that if $p_0$ and $p_1$ follow any distributions with supports which are subsets of $[0,1]$, and $E[p_0]=p$ and $E[p_1]=1/2$, then the expression for $P_S$ that is obtained is exactly the same as the one obtained using the standard key randomisation hypotheses. Consequently, $P_S$ is a monotone increasing function of $N$ even when $p_0$ and $p_1$ are considered to be random variables.

Available format(s)
Category
Secret-key cryptography
Publication info
Preprint. MINOR revision.
Keywords
linear cryptanalysiskey randomisation hypotheses
Contact author(s)
subhabrata samajder @ gmail com
palash @ isical ac in
History
2020-01-31: revised
See all versions
Short URL
https://ia.cr/2019/960

CC BY

BibTeX

@misc{cryptoeprint:2019/960,
author = {Subhabrata Samajder and Palash Sarkar},
title = {Another Look at Key Randomisation Hypotheses},
howpublished = {Cryptology ePrint Archive, Paper 2019/960},
year = {2019},
note = {\url{https://eprint.iacr.org/2019/960}},
url = {https://eprint.iacr.org/2019/960}
}

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