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A Note on the Algebraic Immunity of the Enhanced Boolean Functions
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In 2015, Carlet and Tang [Des. Codes Cryptogr. 76(3): 571-587, 2015] proposed a concept called enhanced Boolean functions and a class of such kind of functions on odd number of variables was constructed. They proved that the constructed functions in this class have optimal algebraic immunity if the numbers of variables are a power of 2 plus 1 and at least sub-optimal algebraic immunity otherwise. In addition, an open problem that if there are enhanced Boolean functions with optimal algebraic immunity and maximal algebraic degree n-1 on odd variables n≠2k+1 was proposed. In this letter, we give a negative answer to the open problem, that is, we prove that there is no enhanced Boolean function on odd n≠2k+1 variables with optimal algebraic immunity and maximal algebraic degree n-1.
- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E103-A No.1 pp.366-369
- Publication Date
- 2020/01/01
- Publicized
- Online ISSN
- 1745-1337
- DOI
- 10.1587/transfun.2019EAL2049
- Type of Manuscript
- LETTER
- Category
- Cryptography and Information Security
Authors
Deng TANG
Southwest Jiaotong University,Guangxi Key Laboratory of Cryptography and Information Security
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Deng TANG, "A Note on the Algebraic Immunity of the Enhanced Boolean Functions" in IEICE TRANSACTIONS on Fundamentals,
vol. E103-A, no. 1, pp. 366-369, January 2020, doi: 10.1587/transfun.2019EAL2049.
Abstract: In 2015, Carlet and Tang [Des. Codes Cryptogr. 76(3): 571-587, 2015] proposed a concept called enhanced Boolean functions and a class of such kind of functions on odd number of variables was constructed. They proved that the constructed functions in this class have optimal algebraic immunity if the numbers of variables are a power of 2 plus 1 and at least sub-optimal algebraic immunity otherwise. In addition, an open problem that if there are enhanced Boolean functions with optimal algebraic immunity and maximal algebraic degree n-1 on odd variables n≠2k+1 was proposed. In this letter, we give a negative answer to the open problem, that is, we prove that there is no enhanced Boolean function on odd n≠2k+1 variables with optimal algebraic immunity and maximal algebraic degree n-1.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.2019EAL2049/_p
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@ARTICLE{e103-a_1_366,
author={Deng TANG, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={A Note on the Algebraic Immunity of the Enhanced Boolean Functions},
year={2020},
volume={E103-A},
number={1},
pages={366-369},
abstract={In 2015, Carlet and Tang [Des. Codes Cryptogr. 76(3): 571-587, 2015] proposed a concept called enhanced Boolean functions and a class of such kind of functions on odd number of variables was constructed. They proved that the constructed functions in this class have optimal algebraic immunity if the numbers of variables are a power of 2 plus 1 and at least sub-optimal algebraic immunity otherwise. In addition, an open problem that if there are enhanced Boolean functions with optimal algebraic immunity and maximal algebraic degree n-1 on odd variables n≠2k+1 was proposed. In this letter, we give a negative answer to the open problem, that is, we prove that there is no enhanced Boolean function on odd n≠2k+1 variables with optimal algebraic immunity and maximal algebraic degree n-1.},
keywords={},
doi={10.1587/transfun.2019EAL2049},
ISSN={1745-1337},
month={January},}
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TY - JOUR
TI - A Note on the Algebraic Immunity of the Enhanced Boolean Functions
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 366
EP - 369
AU - Deng TANG
PY - 2020
DO - 10.1587/transfun.2019EAL2049
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E103-A
IS - 1
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - January 2020
AB - In 2015, Carlet and Tang [Des. Codes Cryptogr. 76(3): 571-587, 2015] proposed a concept called enhanced Boolean functions and a class of such kind of functions on odd number of variables was constructed. They proved that the constructed functions in this class have optimal algebraic immunity if the numbers of variables are a power of 2 plus 1 and at least sub-optimal algebraic immunity otherwise. In addition, an open problem that if there are enhanced Boolean functions with optimal algebraic immunity and maximal algebraic degree n-1 on odd variables n≠2k+1 was proposed. In this letter, we give a negative answer to the open problem, that is, we prove that there is no enhanced Boolean function on odd n≠2k+1 variables with optimal algebraic immunity and maximal algebraic degree n-1.
ER -
English
