Maximal values of generalized algebraic immunity

Authors:
Keqin Feng;Qunying Liao;Jing Yang
Affiliations:
Department of Mathematical Sciences, Tsinghua University, Beijing, China 100084;Department of Mathematical Sciences, Tsinghua University, Beijing, China 100084 and College of Mathematics and Software Science, Sichuan Normal University, Chengdu, China 610066;Department of Mathematical Sciences, Tsinghua University, Beijing, China 100084
Venue:
Designs, Codes and Cryptography
Year:
2009

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Abstract

The notion of algebraic immunity of Boolean functions has been generalized in several ways to vector-valued functions and/or over arbitrary finite fields and reasonable upper bounds for such generalized algebraic immunities has been proved in Armknecht and Krause (Proceedings of ICALP 2006, LNCS, vol. 4052, pp 180---191, 2006), Ars and Faugere (Algebraic immunity of functions over finite fields, INRIA, No report 5532, 2005) and Batten (Canteaut, Viswanathan (eds.) Progress in Cryptology--INDOCRYPT 2004, LNCS, vol. 3348, pp 84---91, 2004). In this paper we show that the upper bounds can be reached as the maximal values of algebraic immunities for most of generalizations by using properties of Reed---Muller codes.