C-DIFFERENTIALS AND GENERALIZED CRYPTOGRAPHIC PROPERTIES OF VECTORIAL BOOLEAN AND P-ARY FUNCTIONS
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Authors
Geary, Aaron C.
Subjects
cryptography
vectorial Boolean functions
p-ary functions
substitution boxes
nonlinearity
bent functions
avalanche characteristics
differential cryptanalysis
differential uniformity
higher order discrete derivatives
communications security
vectorial Boolean functions
p-ary functions
substitution boxes
nonlinearity
bent functions
avalanche characteristics
differential cryptanalysis
differential uniformity
higher order discrete derivatives
communications security
Advisors
Dinolt, George W.
Martinsen, Thor
Gera, Ralucca
Stanica, Pantelimon
Medina, Luis, University of Puerto Rico, Rio Piedras
Date of Issue
2022-06
Date
Publisher
Monterey, CA; Naval Postgraduate School
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Abstract
This dissertation investigates a newly defined cryptographic differential, called a c-differential, and its relevance to the nonlinear substitution boxes of modern symmetric block ciphers. We generalize the notions of perfect nonlinearity, bentness, and avalanche characteristics of vectorial Boolean and p-ary functions using the c-derivative and a new autocorrelation function, while capturing the original definitions as special cases (i.e., when c=1). We investigate the c-differential uniformity property of the inverse function over finite fields under several extended affine transformations. We demonstrate that c-differential properties do not hold in general across equivalence classes typically used in Boolean function analysis, and in some cases change significantly under slight perturbations. Thus, choosing certain affine equivalent functions that are easy to implement in hardware or software without checking their c-differential properties could potentially expose an encryption scheme to risk if a c-differential attack method is ever realized. We also extend the c-derivative and c-differential uniformity into higher order, investigate some of their properties, and analyze the behavior of the inverse function's second order c-differential uniformity. Finally, we analyze the substitution boxes of some recognizable ciphers along with certain extended affine equivalent variations and document their performance under c-differential uniformity.
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Thesis
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Distribution Statement
Approved for public release. Distribution is unlimited.
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This publication is a work of the U.S. Government as defined in Title 17, United States Code, Section 101. Copyright protection is not available for this work in the United States.