A Lower Bound on the Second-Order Nonlinearity of the Generalized Maiorana-McFarland Boolean Functions
Summary :
Boolean functions used in stream ciphers and block ciphers should have high second-order nonlinearity to resist several known attacks and some potential attacks which may exist but are not yet efficient and might be improved in the future. The second-order nonlinearity of Boolean functions also plays an important role in coding theory, since its maximal value equals the covering radius of the second-order Reed-Muller code. But it is an extremely hard task to calculate and even to bound the second-order nonlinearity of Boolean functions. In this paper, we present a lower bound on the second-order nonlinearity of the generalized Maiorana-McFarland Boolean functions. As applications of our bound, we provide more simpler and direct proofs for two known lower bounds on the second-order nonlinearity of functions in the class of Maiorana-McFarland bent functions. We also derive a lower bound on the second-order nonlinearity of the functions which were conjectured bent by Canteaut and whose bentness was proved by Leander, by further employing our bound.
- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E101-A No.12 pp.2397-2401
- Publication Date
- 2018/12/01
- Publicized
- Online ISSN
- 1745-1337
- DOI
- 10.1587/transfun.E101.A.2397
- Type of Manuscript
- Special Section LETTER (Special Section on Signal Design and Its Applications in Communications)
- Category
- Cryptography and Information Security
Authors
Qi GAO
Southwest Jiaotong University,Guangxi Key Laboratory of Cryptography and Information Security
Deng TANG
Southwest Jiaotong University,Guangxi Key Laboratory of Cryptography and Information Security
Keyword
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Qi GAO, Deng TANG, "A Lower Bound on the Second-Order Nonlinearity of the Generalized Maiorana-McFarland Boolean Functions" in IEICE TRANSACTIONS on Fundamentals,
vol. E101-A, no. 12, pp. 2397-2401, December 2018, doi: 10.1587/transfun.E101.A.2397.
Abstract: Boolean functions used in stream ciphers and block ciphers should have high second-order nonlinearity to resist several known attacks and some potential attacks which may exist but are not yet efficient and might be improved in the future. The second-order nonlinearity of Boolean functions also plays an important role in coding theory, since its maximal value equals the covering radius of the second-order Reed-Muller code. But it is an extremely hard task to calculate and even to bound the second-order nonlinearity of Boolean functions. In this paper, we present a lower bound on the second-order nonlinearity of the generalized Maiorana-McFarland Boolean functions. As applications of our bound, we provide more simpler and direct proofs for two known lower bounds on the second-order nonlinearity of functions in the class of Maiorana-McFarland bent functions. We also derive a lower bound on the second-order nonlinearity of the functions which were conjectured bent by Canteaut and whose bentness was proved by Leander, by further employing our bound.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E101.A.2397/_p
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@ARTICLE{e101-a_12_2397,
author={Qi GAO, Deng TANG, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={A Lower Bound on the Second-Order Nonlinearity of the Generalized Maiorana-McFarland Boolean Functions},
year={2018},
volume={E101-A},
number={12},
pages={2397-2401},
abstract={Boolean functions used in stream ciphers and block ciphers should have high second-order nonlinearity to resist several known attacks and some potential attacks which may exist but are not yet efficient and might be improved in the future. The second-order nonlinearity of Boolean functions also plays an important role in coding theory, since its maximal value equals the covering radius of the second-order Reed-Muller code. But it is an extremely hard task to calculate and even to bound the second-order nonlinearity of Boolean functions. In this paper, we present a lower bound on the second-order nonlinearity of the generalized Maiorana-McFarland Boolean functions. As applications of our bound, we provide more simpler and direct proofs for two known lower bounds on the second-order nonlinearity of functions in the class of Maiorana-McFarland bent functions. We also derive a lower bound on the second-order nonlinearity of the functions which were conjectured bent by Canteaut and whose bentness was proved by Leander, by further employing our bound.},
keywords={},
doi={10.1587/transfun.E101.A.2397},
ISSN={1745-1337},
month={December},}
Copy
TY - JOUR
TI - A Lower Bound on the Second-Order Nonlinearity of the Generalized Maiorana-McFarland Boolean Functions
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 2397
EP - 2401
AU - Qi GAO
AU - Deng TANG
PY - 2018
DO - 10.1587/transfun.E101.A.2397
JO - IEICE TRANSACTIONS on Fundamentals
SN - 1745-1337
VL - E101-A
IS - 12
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - December 2018
AB - Boolean functions used in stream ciphers and block ciphers should have high second-order nonlinearity to resist several known attacks and some potential attacks which may exist but are not yet efficient and might be improved in the future. The second-order nonlinearity of Boolean functions also plays an important role in coding theory, since its maximal value equals the covering radius of the second-order Reed-Muller code. But it is an extremely hard task to calculate and even to bound the second-order nonlinearity of Boolean functions. In this paper, we present a lower bound on the second-order nonlinearity of the generalized Maiorana-McFarland Boolean functions. As applications of our bound, we provide more simpler and direct proofs for two known lower bounds on the second-order nonlinearity of functions in the class of Maiorana-McFarland bent functions. We also derive a lower bound on the second-order nonlinearity of the functions which were conjectured bent by Canteaut and whose bentness was proved by Leander, by further employing our bound.
ER -
English
