The covering radius of the&amp;lt;tex&amp;gt;(2^{15}, 16)&amp;lt;/tex&amp;gt;Reed-Muller code is at least 16276

Patterson, Nicholas J.; Wiedemann, Doug

doi:10.1109/tit.1983.1056679

Cited by 134 publications

(36 citation statements)

References 4 publications

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“…This lower bound can only be achieved for even values of n. When n is odd, the lowest achievable value of L(f) is unknown in the general case: there always exist some functions with L(f) = 2 (n+1)/2 and this value corresponds to the minimum possible nonlinearity for any n ≤ 7. On the other hand some functions with L(f) = 27 32 2 (n+1)/2 are known for any odd n ≥ 15 [20,21]. From now on, we will focus on highly nonlinear Boolean functions in the following sense: Definition 3.…”

Section: Definition 2 the Nonlinearity Of A Boolean Function F With mentioning

confidence: 99%

Propagation Characteristics and Correlation-Immunity of Highly Nonlinear Boolean Functions

Canteaut

Carlet

Charpin

et al. 2000

Advances in Cryptology — EUROCRYPT 2000

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Abstract. We investigate the link between the nonlinearity of a Boolean function and its propagation characteristics. We prove that highly nonlinear functions usually have good propagation properties regarding different criteria. Conversely, any Boolean function satisfying the propagation criterion with respect to a linear subspace of codimension 1 or 2 has a high nonlinearity. We also point out that most highly nonlinear functions with a three-valued Walsh spectrum can be transformed into 1-resilient functions.

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Section: Definition 2 the Nonlinearity Of A Boolean Function F With mentioning

confidence: 99%

Propagation Characteristics and Correlation-Immunity of Highly Nonlinear Boolean Functions

Canteaut

Carlet

Charpin

et al. 2000

Advances in Cryptology — EUROCRYPT 2000

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“…In the general case, we know [11] where the lower bound, called the quadratic bound, is notably tight for m 2 f3; 5; 7g [4], [12], [13]. On the other hand, it is known that (1; m) 2 m01 0 (27=32)2 when m 15 [14], [15].…”

Section: Preliminary Resultsmentioning

confidence: 99%

On the weight distributions of optimal cosets of the first-order Reed-Muller codes

Canteaut¹

2001

IEEE Trans. Inform. Theory

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Abstract-We study the weight distributions of cosets of the first-order Reed-Muller code (1 ) for odd , whose minimum weight is greater than or equal to the so-called quadratic bound. Some general restrictions on the weight distribution of a coset of (1 ) are obtained by partitioning its words according to their weight divisibility. Most notably, we show that there are exactly five weight distributions for optimal cosets of ( 1 7) in (5 7) and that these distributions are related to the degree of the function generating the coset. Moreover, for any odd 9, we exhibit optimal cubic cosets of (1 ) whose weights take on exactly five values.

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“…The nonlinearity is 2 6 − 2 3 = 56 which is optimal among 7-bit Boolean functions, see [18]. The additve autocorrelation is optimal, given by ∆ f = 2 (7+1)/2 = 16.…”

Section: Lemma 2 (Kasami-dillonmentioning

confidence: 99%

Additive Autocorrelation of Resilient Boolean Functions

Gong

Khoo

2004

Selected Areas in Cryptography

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Abstract. In this paper, we introduce a new notion called the dual function for studying Boolean functions. First, we discuss general properties of the dual function that are related to resiliency and additive autocorrelation. Second, we look at preferred functions which are Boolean functions with the lowest 3-valued spectrum. We prove that if a balanced preferred function has a dual function which is also preferred, then it is resilient, has high nonlinearity and optimal additive autocorrelation. We demonstrate four such constructions of optimal Boolean functions using the Kasami, Dillon-Dobbertin, Segre hyperoval and Welch-Gong Transformation functions. Third, we compute the additive autocorrelation of some known resilient preferred functions in the literature by using the dual function. We conclude that our construction yields highly nonlinear resilient functions with better additive autocorrelation than the Maiorana-McFarland functions. We also analysed the saturated functions, which are resilient functions with optimized algebraic degree and nonlinearity. We show that their additive autocorrelation have high peak values, and they become linear when we fix very few bits. These potential weaknesses have to be considered before we deploy them in applications.

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The covering radius of the<tex>(2^{15}, 16)</tex>Reed-Muller code is at least 16276

Cited by 134 publications

References 4 publications

Propagation Characteristics and Correlation-Immunity of Highly Nonlinear Boolean Functions

Propagation Characteristics and Correlation-Immunity of Highly Nonlinear Boolean Functions

On the weight distributions of optimal cosets of the first-order Reed-Muller codes

Additive Autocorrelation of Resilient Boolean Functions

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