Time-Memory-Data Trade-Off Attack on Stream Ciphers Based on Maiorana-McFarland Functions

Abstract

In this paper, we present the time-memory-data (TMD) trade-off attack on stream ciphers filter function generators and filter cominers based on Maiorana-McFarland functions. This can be considered as a generalization of the time-memory-data trade-off attack of Mihaljevic and Imai on Toyocrypt. First, we substitute the filter function in Toyocrypt (which has the same size as the LFSR) with a general Maiorana-McFarland function. This allows us to apply the attack to a wider class of stream ciphers. Second, we highlight how the choice of different Maiorana-McFarland functions can affect the effectiveness of our attack. Third, we show that the attack can be modified to apply on filter functions which are smaller than the LFSR and on filter-combiner stream ciphers. This allows us to cryptanalyze other configurations commonly found in practice. Finally, filter functions with vector output are sometimes used in stream ciphers to improve the throughput. Therefore the case when the Maiorana-McFarland functions have vector output is investigated. We found that the extra speed comes at the price of additional weaknesses which make the attacks easier.

On the approximation of S‐boxes via Maiorana–McFarland functions

“…1. In this case, our indicator suggests that either standard algebraic attacks or Time-Memory-Data (TMD) trade-off attacks [8] can also be applied in this scenario. In the extreme case when EN (r) F = 0 linear equations in secret state bits are obtained, as demonstrated in Example 1 below.…”

“…In the first place this class has a particular algebraic structure, which is characterised by the property that when a fixed subset of input variables is kept fixed (restriction on say x 1 , …, x k ) the function becomes linear or in general affine. Certain cryptographic weaknesses, especially in a straightforward applications such as a filtering function, are somehow immanent to this class and consequently many cryptanalytic attacks perform better if such a function is employed [8][9][10]. Nevertheless, Maiorana-McFarland functions have received a lot of attention as a construction method both in the design of Boolean functions and resilient S-boxes.…”