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Lalley, Steven P.

doi:10.1023/a:1021636902356

Cited by 6 publications

(1 citation statement)

References 8 publications

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“…More recently, statistical tests for randomness, such as the FSU DIEHARD suite [28], were used to quantify the randomness of a permutation [29]. Additionally, entropy formulations, such as fiber entropy, were utilized to measure shuffle output randomness [30]. In our application, we determined the quality of the shuffle on the basis of Shannon entropy loss of the mapping process within the shuffle and simulating the shuffle algorithm using MATLAB with different PRNGs.…”

Section: Introductionmentioning

confidence: 99%

Shannon Entropy Loss in Mixed-Radix Conversions

Vennos

Michaels

2021

Entropy

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This paper models a translation for base-2 pseudorandom number generators (PRNGs) to mixed-radix uses such as card shuffling. In particular, we explore a shuffler algorithm that relies on a sequence of uniformly distributed random inputs from a mixed-radix domain to implement a Fisher–Yates shuffle that calls for inputs from a base-2 PRNG. Entropy is lost through this mixed-radix conversion, which is assumed to be surjective mapping from a relatively large domain of size 2J to a set of arbitrary size n. Previous research evaluated the Shannon entropy loss of a similar mapping process, but this previous bound ignored the mixed-radix component of the original formulation, focusing only on a fixed n value. In this paper, we calculate a more precise formula that takes into account a variable target domain radix, n, and further derives a tighter bound on the Shannon entropy loss of the surjective map, while demonstrating monotonicity in a decrease in entropy loss based on increased size J of the source domain 2J. Lastly, this formulation is used to specify the optimal parameters to simulate a card-shuffling algorithm with different test PRNGs, validating a concrete use case with quantifiable deviations from maximal entropy, making it suitable to low-power implementation in a casino.

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Section: Introductionmentioning

confidence: 99%

Shannon Entropy Loss in Mixed-Radix Conversions

Vennos

Michaels

2021

Entropy

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Applications of the Brauer Complex: Card Shuffling, Permutation Statistics, and Dynamical Systems

Fulman¹

2001

Journal of Algebra

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By algebraic group theory, there is a map from the semisimple conjugacy classes of a finite group of Lie type to the conjugacy classes of the Weyl group. Picking a semisimple class uniformly at random yields a probability measure on conjugacy classes of the Weyl group. Using the Brauer complex, it is proved that this measure agrees with a second measure on conjugacy classes of the Weyl group induced by a construction of Cellini using the affine Weyl group. Formulas for Cellini's measure in type A are found. This leads to new models of card shuffling and has interesting combinatorial and number-theoretic consequences. An analysis of type C gives another solution to a problem of Rogers in dynamical systems: the enumeration of unimodal permutations by cycle structure. The proof uses the factorization theory of palindromic polynomials over finite fields. Contact is made with symmetric function theory. 

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Random orderings of the integers and card shuffling

Jacka

Warren

2007

Stochastic Processes and their Applications

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In this paper we study random orderings of the integers with a certain invariance property. We describe all such orders in a simple way. We define and represent random shuffles of a countable set of labels and then give an interpretation of these orders in terms of a class of generalized riffle shuffles.2001 MSC Subject classifications: Primary: 60B15; Secondary: 60G09, 37A40, 37H99, 60J05, 03E10.

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Cited by 6 publications

References 8 publications

Shannon Entropy Loss in Mixed-Radix Conversions

Shannon Entropy Loss in Mixed-Radix Conversions

Applications of the Brauer Complex: Card Shuffling, Permutation Statistics, and Dynamical Systems

Random orderings of the integers and card shuffling

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