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Covering Radius of the (n-3)-rd Order Reed-Muller Code in the Set of Resilient Functions
Authors: Yuri Borissov, An Braeken, Svetla NikovaAbstract:
In this paper, we continue the study of the covering
radius in the set of resilient functions, which has been defined by Kurosawa. This new concept is meaningful to cryptography especially in the context of the new class of algebraic attacks on stream ciphers proposed by Courtois and Meier at Eurocrypt 2003 and Courtois at Crypto 2003. In order to resist such attacks the combining Boolean function should be at high distance from lower degree functions.
Using a result from coding theory on the covering radius of
(n-3)-rd Reed-Muller codes, we establish exact values of the
the covering radius of RM(n-3,n) in the set of 1-resilient Boolean
functions of n variables, when \lfloor n/2 \rfloor = 1 \mod\;2. We also improve the lower bounds for covering radius of the Reed-Muller
codes RM(r,n) in the set of t-resilient functions, where
\lceil r/2 \rceil = 0 \mod\;2, t \leq n-r-2 and n\geq r+3.
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