# American Institute of Mathematical Sciences

February  2018, 12(1): 115-122. doi: 10.3934/amc.2018007

## Generalized nonlinearity of $S$-boxes

 1 Department of Computer Science and Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, India 2 Cryptology and Security Research Unit, R. C. Bose Centre for Cryptology and Security, Indian Statistical Institute, Kolkata 700108, India 3 Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA 93943-5216, USA

* Corresponding author: Goutam Paul

Nishant Sinha thanks IIT Roorkee for supporting his research

Received  September 2016 Published  March 2018

While analyzing $S$-boxes, or vectorial Boolean functions, it is of interest to approximate its component functions by affine functions. In the usual attack models, it is assumed that all input vectors to an $S$-box are equiprobable. The nonlinearity of an $S$-box is defined, subject to this assumption. In this paper, we explore the possibility of linear cryptanalysis of an $S$-box by introducing biased inputs and thus propose a generalized notion of nonlinearity along with a generalization of the Walsh-Hadamard spectrum of an $S$-box.

Citation: Sugata Gangopadhyay, Goutam Paul, Nishant Sinha, Pantelimon Stǎnicǎ. Generalized nonlinearity of $S$-boxes. Advances in Mathematics of Communications, 2018, 12 (1) : 115-122. doi: 10.3934/amc.2018007
##### References:

show all references

##### References:
Maximum bias without and with biased inputs for all DES S-boxes.
 $F$ $S_1$ $S_2$ $S_3$ $S_4$ $S_5$ $S_6$ $S_7$ $S_8$ $\displaystyle \max_{{\bf{u}} \in {\mathbb{F}}_2^n}\epsilon({\bf{u}}, {\bf{v}} \cdot F)$ 0.219 0.219 0.219 0.156 0.219 0.188 0.281 0.188 $\displaystyle \max_{{\bf{u}} \in {\mathbb{F}}_2^n} \epsilon^{(p)}_{{\mathcal{S}}}({\bf{u}}, {\bf{v}}\cdot F)$ 0.494 0.494 0.497 0.489 0.494 0.491 0.494 0.494
 $F$ $S_1$ $S_2$ $S_3$ $S_4$ $S_5$ $S_6$ $S_7$ $S_8$ $\displaystyle \max_{{\bf{u}} \in {\mathbb{F}}_2^n}\epsilon({\bf{u}}, {\bf{v}} \cdot F)$ 0.219 0.219 0.219 0.156 0.219 0.188 0.281 0.188 $\displaystyle \max_{{\bf{u}} \in {\mathbb{F}}_2^n} \epsilon^{(p)}_{{\mathcal{S}}}({\bf{u}}, {\bf{v}}\cdot F)$ 0.494 0.494 0.497 0.489 0.494 0.491 0.494 0.494
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