Construction and enumeration of Boolean functions with maximum algebraic immunity
Algebraic immunity is a new cryptographic criterion proposed against algebraic attacks. In order to resist algebraic attacks, Boolean functions used in many stream ciphers should possess high algebraic immunity. This paper presents two main results to find balanced Boolean functions with maximum alg...
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| Published in: | Science China. Information sciences Vol. 52; no. 1; pp. 32 - 40 |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Beijing
Science in China Press
2009
Springer Nature B.V |
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| ISSN: | 1009-2757, 1674-733X, 1862-2836, 1869-1919 |
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| Abstract | Algebraic immunity is a new cryptographic criterion proposed against algebraic attacks. In order to resist algebraic attacks, Boolean functions used in many stream ciphers should possess high algebraic immunity. This paper presents two main results to find balanced Boolean functions with maximum algebraic immunity. Through swapping the values of two bits, and then generalizing the result to swap some pairs of bits of the symmetric Boolean function constructed by Dalai, a new class of Boolean functions with maximum algebraic immunity are constructed. Enumeration of such functions is also n given. For a given function p（x） with deg（p（x）） 〈 [n/2], we give a method to construct functions in the form p（x）＋q（x） which achieve the maximum algebraic immunity, where every term with nonzero coefficient in the ANF of q（x） has degree no less than [n/2]. |
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| AbstractList | Algebraic immunity is a new cryptographic criterion proposed against algebraic attacks. In order to resist algebraic attacks, Boolean functions used in many stream ciphers should possess high algebraic immunity. This paper presents two main results to find balanced Boolean functions with maximum algebraic immunity. Through swapping the values of two bits, and then generalizing the result to swap some pairs of bits of the symmetric Boolean function constructed by Dalai, a new class of Boolean functions with maximum algebraic immunity are constructed. Enumeration of such functions is also n given. For a given function p（x） with deg（p（x）） 〈 [n/2], we give a method to construct functions in the form p（x）＋q（x） which achieve the maximum algebraic immunity, where every term with nonzero coefficient in the ANF of q（x） has degree no less than [n/2]. Algebraic immunity is a new cryptographic criterion proposed against algebraic attacks. In order to resist algebraic attacks, Boolean functions used in many stream ciphers should possess high algebraic immunity. This paper presents two main results to find balanced Boolean functions with maximum algebraic immunity. Through swapping the values of two bits, and then generalizing the result to swap some pairs of bits of the symmetric Boolean function constructed by Dalai, a new class of Boolean functions with maximum algebraic immunity are constructed. Enumeration of such functions is also given. For a given function p(x) with deg(p(x)) < , we give a method to construct functions in the form p(x)+q(x) which achieve the maximum algebraic immunity, where every term with nonzero coefficient in the ANF of q(x) has degree no less than . Algebraic immunity is a new cryptographic criterion proposed against algebraic attacks. In order to resist algebraic attacks, Boolean functions used in many stream ciphers should possess high algebraic immunity. This paper presents two main results to find balanced Boolean functions with maximum algebraic immunity. Through swapping the values of two bits, and then generalizing the result to swap some pairs of bits of the symmetric Boolean function constructed by Dalai, a new class of Boolean functions with maximum algebraic immunity are constructed. Enumeration of such functions is also given. For a given function p(x) with deg(p(x)) < {expression}, we give a method to construct functions in the form p(x)+q(x) which achieve the maximum algebraic immunity, where every term with nonzero coefficient in the ANF of q(x) has degree no less than {expression}. Algebraic immunity is a new cryptographic criterion proposed against algebraic attacks. In order to resist algebraic attacks, Boolean functions used in many stream ciphers should possess high algebraic immunity. This paper presents two main results to find balanced Boolean functions with maximum algebraic immunity. Through swapping the values of two bits, and then generalizing the result to swap some pairs of bits of the symmetric Boolean function constructed by Dalai, a new class of Boolean functions with maximum algebraic immunity are constructed. Enumeration of such functions is also given. For a given function p ( x ) with deg( p ( x )) < , we give a method to construct functions in the form p ( x )+ q ( x ) which achieve the maximum algebraic immunity, where every term with nonzero coefficient in the ANF of q ( x ) has degree no less than . |
| Author | ZHANG WenYing WU ChuanKun LIU XiangZhong |
| AuthorAffiliation | School of Information Science and Engineering, Shandong Normal University, Jinan 250014, China State Key Lab of Information Security, Institute of Software, Chinese Academy of Sciences, Beijing 100080, China No. 2 Middle School Attached to Shandong Normal University, Jinan 250014, China |
| Author_xml | – sequence: 1 givenname: WenYing surname: Zhang fullname: Zhang, WenYing email: wenyingzh@is.iscas.ac.cn organization: School of Information Science and Engineering, Shandong Normal University, State Key Lab of Information Security, Institute of Software, Chinese Academy of Sciences – sequence: 2 givenname: ChuanKun surname: Wu fullname: Wu, ChuanKun organization: State Key Lab of Information Security, Institute of Software, Chinese Academy of Sciences – sequence: 3 givenname: XiangZhong surname: Liu fullname: Liu, XiangZhong organization: No. 2 Middle School Attached to Shandong Normal University |
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| Cites_doi | 10.1007/978-3-540-24676-3_28 10.1007/s10623-005-6300-x 10.1109/TIT.2006.872977 10.1007/11596219_4 10.1007/3-540-54973-0 10.1109/TIT.2006.876253 10.1007/11935230_6 10.1007/11779360_10 |
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| Copyright | Science China Press and Springer-Verlag Berlin Heidelberg 2009 Science China Press and Springer-Verlag Berlin Heidelberg 2009. |
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| Keywords | algebraic immunity cryptography Boolean function algebraic attack |
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| References | Courtois, Meier (CR1) 2003 Braeken, Praneel (CR5) 2005 CR3 Meier, Pasalic, Carlet (CR4) 2004 CR8 Li, Qi (CR7) 2006; 52 CR9 CR12 CR11 Ding, Xiao, Shan (CR13) 1991 Carlet, Dalai, Gupta (CR10) 2006; 52 Mihaljevie, Imai (CR2) 2002; E85-A Dalai, Maitra, Sarkar (CR6) 2006; 40 A. Braeken (7_CR5) 2005 D. Dalai (7_CR6) 2006; 40 7_CR11 7_CR12 7_CR3 C. Carlet (7_CR10) 2006; 52 N. Courtois (7_CR1) 2003 7_CR9 7_CR8 W. Meier (7_CR4) 2004 N. Li (7_CR7) 2006; 52 C. Ding (7_CR13) 1991 M. Mihaljevie (7_CR2) 2002; E85-A |
| References_xml | – start-page: 346 year: 2003 end-page: 359 ident: CR1 article-title: Algebraic attacks on stream ciphers with linear feedback publication-title: Advances in Cryptology-EUROCRYPT 2003 – ident: CR3 – ident: CR12 – ident: CR11 – start-page: 474 year: 2004 end-page: 491 ident: CR4 article-title: Algebraic attacks and decomposition of Boolean functions publication-title: Advances in Cryptology-EUROCRYPT 2004 doi: 10.1007/978-3-540-24676-3_28 – ident: CR9 – volume: 40 start-page: 41 issue: 1 year: 2006 end-page: 58 ident: CR6 article-title: Basic theory in construction of Boolean functions with maximum possible annihilator immunity publication-title: Design, Codes and Cryptography doi: 10.1007/s10623-005-6300-x – volume: 52 start-page: 2271 issue: 5 year: 2006 end-page: 2273 ident: CR7 article-title: Symmetric Boolean functions depending on an odd number of variables with maximum algebraic immunity publication-title: IEEE Trans Inf Theory doi: 10.1109/TIT.2006.872977 – ident: CR8 – start-page: 35 year: 2005 end-page: 48 ident: CR5 article-title: On the algebraic immunity of symmetric Boolean functions publication-title: Advances in Indocrypt 2005 doi: 10.1007/11596219_4 – year: 1991 ident: CR13 publication-title: The Stability Theory of Stream Ciphers doi: 10.1007/3-540-54973-0 – volume: 52 start-page: 3105 issue: 7 year: 2006 end-page: 3121 ident: CR10 article-title: Algebraic immunity for cryptographically significant Boolean functions: Analysis and construction publication-title: IEEE Trans Inf Theory doi: 10.1109/TIT.2006.876253 – volume: E85-A start-page: 66 year: 2002 end-page: 73 ident: CR2 article-title: Cryptanalysis of toyocrypt-HSI stream cipher publication-title: IEICE Trans Fundamentals – ident: 7_CR11 – volume: 52 start-page: 3105 issue: 7 year: 2006 ident: 7_CR10 publication-title: IEEE Trans Inf Theory doi: 10.1109/TIT.2006.876253 – ident: 7_CR8 doi: 10.1007/11935230_6 – volume: E85-A start-page: 66 year: 2002 ident: 7_CR2 publication-title: IEICE Trans Fundamentals – start-page: 474 volume-title: Advances in Cryptology-EUROCRYPT 2004 year: 2004 ident: 7_CR4 doi: 10.1007/978-3-540-24676-3_28 – ident: 7_CR12 doi: 10.1007/11779360_10 – volume-title: The Stability Theory of Stream Ciphers year: 1991 ident: 7_CR13 doi: 10.1007/3-540-54973-0 – ident: 7_CR3 – volume: 40 start-page: 41 issue: 1 year: 2006 ident: 7_CR6 publication-title: Design, Codes and Cryptography doi: 10.1007/s10623-005-6300-x – start-page: 35 volume-title: Advances in Indocrypt 2005 year: 2005 ident: 7_CR5 doi: 10.1007/11596219_4 – ident: 7_CR9 – start-page: 346 volume-title: Advances in Cryptology-EUROCRYPT 2003 year: 2003 ident: 7_CR1 – volume: 52 start-page: 2271 issue: 5 year: 2006 ident: 7_CR7 publication-title: IEEE Trans Inf Theory doi: 10.1109/TIT.2006.872977 |
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| SubjectTerms | Algebra Algorithms Boolean functions Computer Science Cryptography Encryption Enumeration Information Systems and Communication Service Mathematical analysis 代数攻击 加解密算法 密码学 密钥值 源代码 |
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| Title | Construction and enumeration of Boolean functions with maximum algebraic immunity |
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