Improving the Lower Bound on the Higher Order Nonlinearity of Boolean Functions With Prescribed Algebraic Immunity

The recent algebraic attacks have received a lot of attention in cryptographic literature. The algebraic immunity of a Boolean function quantifies its resistance to the standard algebraic attacks of the pseudorandom generators using it as a nonlinear filtering or combining function. Very few results...

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Bibliographic Details
Published in:IEEE transactions on information theory Vol. 54; no. 8; pp. 3656 - 3662
Main Author: Mesnager, S.
Format: Journal Article
Language:English
Published: New York, NY IEEE 01.08.2008
Institute of Electrical and Electronics Engineers
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algebra
Algebraic attack
algebraic degree
algebraic immunity
annihilator
Applied sciences
block cipher
Blocking
Boolean function
Boolean functions
Codes
Cryptography
Data encryption
Detection, estimation, filtering, equalization, prediction
Exact sciences and technology
Filtering
Hamming weight
higher order nonlinearity
Immunity
Information theory
Information, signal and communications theory
Linear feedback shift registers
Lower bounds
Mathematics
Nonlinear equations
Nonlinear filters
Nonlinearity
Polynomials
Signal and communications theory
Signal, noise
stream cipher
Telecommunications and information theory
algebraic immunity
Lower bound
block cipher
Stream ciphering
Algebraic attack
stream cipher
algebraic degree
Boolean function
Non linear phenomenon
higher order nonlinearity
Non linear filtering
Private key
First order
Block ciphering
Security of data
annihilator
Cryptography
Non linear effect
ISSN:0018-9448, 1557-9654
Online Access:Get full text
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Summary:The recent algebraic attacks have received a lot of attention in cryptographic literature. The algebraic immunity of a Boolean function quantifies its resistance to the standard algebraic attacks of the pseudorandom generators using it as a nonlinear filtering or combining function. Very few results have been found concerning its relation with the other cryptographic parameters or with the r th-order nonlinearity. As recalled by Carlet at CRYPTO'06, many papers have illustrated the importance of the r th-order nonlinearity profile (which includes the first-order nonlinearity). The role of this parameter relatively to the currently known attacks has been also shown for block ciphers. Recently, two lower bounds involving the algebraic immunity on the r th-order nonlinearity have been shown by Carlet . None of them improves upon the other one in all situations. In this paper, we prove a new lower bound on the r th-order nonlinearity profile of Boolean functions, given their algebraic immunity, that improves significantly upon one of these lower bounds for all orders and upon the other one for low orders.
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2008.926360