Algebraic immunity for cryptographically significant Boolean functions: analysis and construction

Carlet, Claude; Dalai, Deepak Kumar; Gupta, Kishan Chand; Maitra, Subhamoy

doi:10.1109/tit.2006.876253

Cited by 203 publications

(124 citation statements)

References 20 publications

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“…Also, we know that almost all Boolean functions have algebraic immunities close to this optimum; more precisely, for all a < 1, AI(f) is almost surely greater than n 2 − n 2 ln n a ln 2 when n tends to infinity: see [21]. Even when restricting ourselves to functions with optimum algebraic immunity, the algebraic attacks oblige to use now functions on at least 13 variables, see [4,8] (this number is a strict minimum and is in fact risky; a safer number of variables would better be near 20).…”

Section: An N-variable Boolean Function Let R Be a Positive Integer mentioning

confidence: 99%

“…This changes completely the situation with the nonlinearity profile. In this paper, we obtain a new bound which improves upon the bound of [8] for all values of AI(f ) when the number of variables is smaller than or equal to 12, and for most values of AI(f ) when the number of variables is smaller than or equal to 22 (which covers the practical situation of stream ciphers). It also improves asymptotically upon it.…”

Section: Introductionmentioning

confidence: 97%

“…In fact, until recently, almost nothing relevant was known on this criterion (see Section 2). Fortunately, it has been shown lately that, if the algebraic immunity AI(f ) of a function f is known, then we can deduce a lower bound on its r-th order nonlinearity for every r ≤ AI(f ) − 1: for the first order, see [17] and the improvement of [30]; for any order, see [8] (see Section 2 for a recall of these bounds). This changes completely the situation with the nonlinearity profile.…”

Section: Introductionmentioning

confidence: 99%

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On the Higher Order Nonlinearities of Algebraic Immune Functions

Carlet

2006

Lecture Notes in Computer Science

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Abstract. One of the most basic requirements concerning Boolean functions used in cryptosystems is that they must have high algebraic degrees. This simple criterion is not always well adapted to the concrete situation in which Boolean functions are used in symmetric cryptography, since changing one or several output bits of a Boolean function considerably changes its algebraic degree while it may not change its robustness. The proper characteristic is the r-th order nonlinearity profile (which includes the first-order nonlinearity). However, studying it is difficult and almost no paper, in the literature, has ever been able to give general effective results on it. The values of the nonlinearity profile are known for very few functions and these functions have little cryptographic interest. A recent paper has given a lower bound on the nonlinearity profile of functions, given their algebraic immunity. We improve upon it, and we deduce that it is enough, for a Boolean function, to have high algebraic immunity, for having non-weak low order nonlinearity profile (even when it cannot be evaluated), except maybe for the first order.

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Section: An N-variable Boolean Function Let R Be a Positive Integer mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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On the Higher Order Nonlinearities of Algebraic Immune Functions

Carlet

2006

Lecture Notes in Computer Science

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“…For instance, Maiorana-McFarland, i.e., M-M, together with its variations is a popular and favorable approach for a number of well-behaved functions. Being constructed by affine subfunctions, M-M construction, has an evident drawback against FAA [1] . It is an interesting fact that any randomly chosen balanced function on a large number of variables has good algebraic immunity with very high probability, whereas this is not so when considering a specific construction.…”

Section: Introductionmentioning

confidence: 99%

“…It seems that in [1], a class of 1-resilient and optimal algebraic immunity functions was first obtained through a doubly indexed recursive relation. But its low nonlinearity impedes the utilization in cryptographic models.…”

Section: Introductionmentioning

confidence: 99%

Construction of 1-Resilient Boolean Functions with Optimal Algebraic Immunity and Good Nonlinearity

Pan

Zhang

2011

J. Comput. Sci. Technol.

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This paper presents a construction for a class of 1-resilient functions with optimal algebraic immunity on an even number of variables. The construction is based on the concatenation of two balanced functions in associative classes. For some n, a part of 1-resilient functions with maximum algebraic immunity constructed in the paper can achieve almost optimal nonlinearity. Apart from their high nonlinearity, the functions reach Siegenthaler's upper bound of algebraic degree. Also a class of 1-resilient functions on any number n > 2 of variables with at least sub-optimal algebraic immunity is provided.

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Generating highly nonlinear resilient Boolean functions resistance against algebraic and fast algebraic attacks

Yang

Zhang

2014

Security Comm Networks

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Boolean functions play an important role in the design of stream ciphers. In this paper, a simulated annealing algorithm is designed to obtain Boolean functions satisfying all the needed criteria: high nonlinearity, 1-resiliency, optimal algebraic degree, optimal (or suboptimal) algebraic immunity, and good immunity to fast algebraic attacks. These functions provide a good trade-off among the criteria to resist the known cryptanalytic techniques.

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Algebraic immunity for cryptographically significant Boolean functions: analysis and construction

Cited by 203 publications

References 20 publications

On the Higher Order Nonlinearities of Algebraic Immune Functions

On the Higher Order Nonlinearities of Algebraic Immune Functions

Construction of 1-Resilient Boolean Functions with Optimal Algebraic Immunity and Good Nonlinearity

Generating highly nonlinear resilient Boolean functions resistance against algebraic and fast algebraic attacks

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