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In this paper, we explicitly construct a large class of symmetric Boolean functions on 2*k* variables with algebraic immunity not less than *d*, where integer *k* is given arbitrarily and *d* is a given suffix of *k* in binary representation. If let *d* = *k*, our constructed functions achieve the maximum algebraic immunity. Remarkably, 2^{⌊ log2k ⌋ + 2} symmetric Boolean functions on 2*k* variables with maximum algebraic immunity are constructed, which are much more than the previous constructions. Based on our construction, a lower bound of symmetric Boolean functions with algebraic immunity not less than *d* is derived, which is 2^{⌊ log2d ⌋ + 2(k-d+1)}. As far as we know, this is the first lower bound of this kind.

- Publication
- IEICE TRANSACTIONS on Fundamentals Vol.E94-A No.1 pp.362-366

- Publication Date
- 2011/01/01

- Publicized

- Online ISSN
- 1745-1337

- DOI
- 10.1587/transfun.E94.A.362

- Type of Manuscript
- PAPER

- Category
- Cryptography and Information Security

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Yuan LI, Hui WANG, Haibin KAN, "Constructing Even-Variable Symmetric Boolean Functions with High Algebraic Immunity" in IEICE TRANSACTIONS on Fundamentals,
vol. E94-A, no. 1, pp. 362-366, January 2011, doi: 10.1587/transfun.E94.A.362.

Abstract: In this paper, we explicitly construct a large class of symmetric Boolean functions on 2*k* variables with algebraic immunity not less than *d*, where integer *k* is given arbitrarily and *d* is a given suffix of *k* in binary representation. If let *d* = *k*, our constructed functions achieve the maximum algebraic immunity. Remarkably, 2^{⌊ log2k ⌋ + 2} symmetric Boolean functions on 2*k* variables with maximum algebraic immunity are constructed, which are much more than the previous constructions. Based on our construction, a lower bound of symmetric Boolean functions with algebraic immunity not less than *d* is derived, which is 2^{⌊ log2d ⌋ + 2(k-d+1)}. As far as we know, this is the first lower bound of this kind.

URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/transfun.E94.A.362/_p

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@ARTICLE{e94-a_1_362,

author={Yuan LI, Hui WANG, Haibin KAN, },

journal={IEICE TRANSACTIONS on Fundamentals},

title={Constructing Even-Variable Symmetric Boolean Functions with High Algebraic Immunity},

year={2011},

volume={E94-A},

number={1},

pages={362-366},

abstract={In this paper, we explicitly construct a large class of symmetric Boolean functions on 2*k* variables with algebraic immunity not less than *d*, where integer *k* is given arbitrarily and *d* is a given suffix of *k* in binary representation. If let *d* = *k*, our constructed functions achieve the maximum algebraic immunity. Remarkably, 2^{⌊ log2k ⌋ + 2} symmetric Boolean functions on 2*k* variables with maximum algebraic immunity are constructed, which are much more than the previous constructions. Based on our construction, a lower bound of symmetric Boolean functions with algebraic immunity not less than *d* is derived, which is 2^{⌊ log2d ⌋ + 2(k-d+1)}. As far as we know, this is the first lower bound of this kind.},

keywords={},

doi={10.1587/transfun.E94.A.362},

ISSN={1745-1337},

month={January},}

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TY - JOUR

TI - Constructing Even-Variable Symmetric Boolean Functions with High Algebraic Immunity

T2 - IEICE TRANSACTIONS on Fundamentals

SP - 362

EP - 366

AU - Yuan LI

AU - Hui WANG

AU - Haibin KAN

PY - 2011

DO - 10.1587/transfun.E94.A.362

JO - IEICE TRANSACTIONS on Fundamentals

SN - 1745-1337

VL - E94-A

IS - 1

JA - IEICE TRANSACTIONS on Fundamentals

Y1 - January 2011

AB - In this paper, we explicitly construct a large class of symmetric Boolean functions on 2*k* variables with algebraic immunity not less than *d*, where integer *k* is given arbitrarily and *d* is a given suffix of *k* in binary representation. If let *d* = *k*, our constructed functions achieve the maximum algebraic immunity. Remarkably, 2^{⌊ log2k ⌋ + 2} symmetric Boolean functions on 2*k* variables with maximum algebraic immunity are constructed, which are much more than the previous constructions. Based on our construction, a lower bound of symmetric Boolean functions with algebraic immunity not less than *d* is derived, which is 2^{⌊ log2d ⌋ + 2(k-d+1)}. As far as we know, this is the first lower bound of this kind.

ER -