A local limit theorem for the distribution of a part of the spectrum of a random binary function

Денисов, Олег Викторович

doi:10.1515/dma.2000.10.1.87

Cited by 9 publications

(6 citation statements)

References 6 publications

(13 reference statements)

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“…Denisov published a "correction" in 2000 (see [13]), but it turns out that his original result was correct and the latter paper is incorrect, as was shown by Canfield et al [5]. For k = 1, one can get a simpler estimate…”

Section: Remark 4 Bierbrauer and Friedmanmentioning

confidence: 98%

A Parallel Approach in Computing Correlation Immunity up to Six Variables

Etherington

Anderson

Bach

et al. 2016

Int. J. Found. Comput. Sci.

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We show the use of a reconfigurable computer in computing the correlation immunity of Boolean functions of up to 6 variables. Boolean functions with high correlation immunity are desired in cryptographic systems because they are immune to correlation attacks. The SRC-6 reconfigurable computer was programmed in Verilog to compute the correlation immunity of functions. This computation is performed at a rate that is 190 times faster than a conventional computer.Our analysis of the correlation immunity is across all n-variable Boolean functions, for 2  n  6, thus obtaining, for the first time, a complete distribution of such functions. We also compare correlation immunity with two other cryptographic properties, nonlinearity and degree.Keywords. reconfigurable computing, cryptographic Boolean functions, correlation immunity, symmetric functions, rotation symmetric functions 1 Report Documentation PageForm Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. We show the use of a recon gurable computer in computing the correlation immunity of Boolean functions of up to 6 variables. Boolean functions with high correlation immunity are desired in cryptographic systems because they are immune to correlation attacks. The SRC-6 recon gurable computer was programmed in Verilog to compute the correlation immunity of functions. This computation is performed at a rate that is 190 times faster than a conventional computer. Our analysis of the correlation immunity is across all n-variable Boolean functions for 2 n 6, thus obtaining, for the first time, a complete distribution of such func-tions. We also compare correlation immunity with two other cryptographic properties nonlinearity and degree.

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Section: Remark 4 Bierbrauer and Friedmanmentioning

confidence: 98%

A Parallel Approach in Computing Correlation Immunity up to Six Variables

Etherington

Anderson

Bach

et al. 2016

Int. J. Found. Comput. Sci.

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“…An asymptotic number (as n → ∞) of resilient Boolean functions of order r = const was found in [3] and [4]. Methods from [3] and [4] are developed in [1] and [5] for the calculation of the asymptotic number of resilient Boolean functions of order r = O(n/ ln n). The resilient Boolean functions of order n − c, where c = const, are listed in [8].…”

Section: Introductionmentioning

confidence: 99%

“…Further we will investigate resilient functions. An asymptotic number (as n → ∞) of resilient Boolean functions of order r = const was found in [3] and [4]. Methods from [3] and [4] are developed in [1] and [5] for the calculation of the asymptotic number of resilient Boolean functions of order r = O(n/ ln n).…”

Section: Introductionmentioning

confidence: 99%

A Lower Bound on the Number of Boolean Functions with Median Correlation Immunity

Potapov

2019

2019 XVI International Symposium "Problems of Redundancy in Information and Control Systems" (REDUNDANCY)

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“…In a later paper [7], Denisov repudiated Theorem 1.1 and proposed a different value. However, we will show that Denisov's repudiation was a mistake, and Theorem 1.1 is correct.…”

Section: Introductionmentioning

confidence: 99%

“…However, we will show that Denisov's repudiation was a mistake, and Theorem 1.1 is correct. More discussion of [7] is given in Section 8.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic enumeration of correlation-immune boolean functions

et al. 2010

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A boolean function of n boolean variables is correlation-immune of order k if the function value is uncorrelated with the values of any k of the arguments. Such functions are of considerable interest due to their cryptographic properties, and are also related to the orthogonal arrays of statistics and the balanced hypercube colourings of combinatorics. The weight of a boolean function is the number of argument values that produce a function value of 1. If this is exactly half the argument values, that is, 2 n−1 values, a correlation-immune function is called resilient. An asymptotic estimate of the number N(n, k) of n-variable correlation-immune boolean functions of order k was obtained in 1992 by Denisov for constant k. Denisov repudiated that estimate in 2000, but we show that the repudiation was a mistake. The main contribution of this paper is an asymptotic estimate of N(n, k) which holds 112 Cryptogr. Commun. (2010) 2:111-126 if k increases with n within generous limits and specialises to functions with a given weight, including the resilient functions. In the case of k = 1, our estimates are valid for all weights.

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A local limit theorem for the distribution of a part of the spectrum of a random binary function

Cited by 9 publications

References 6 publications

A Parallel Approach in Computing Correlation Immunity up to Six Variables

A Parallel Approach in Computing Correlation Immunity up to Six Variables

A Lower Bound on the Number of Boolean Functions with Median Correlation Immunity

Asymptotic enumeration of correlation-immune boolean functions

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