Five-valued Boolean functions play an important role in the design of symmetric cryptography.While the design and properties of single-output almost optimal five-valued spectra Boolean functions have been extensively ...Five-valued Boolean functions play an important role in the design of symmetric cryptography.While the design and properties of single-output almost optimal five-valued spectra Boolean functions have been extensively studied over the past few decades,there has been limited research on the construction of almost optimal five-valued spectra vectorial Boolean functions.In this paper,we present a construction method for even-variable 2-output almost optimal five-valued spectra balanced Boolean functions,whose Walsh spectra values belong to the set{0,±2^(n/2),±2^(n/2+1)},at the same time,we discuss the existence of sufficient conditions in the construction.Additionally,this paper presents a novel construction method for balanced single-output Boolean functions with even variables featuring a special five-valued spectral structure,whose Walsh spectra values are constrained to the set{0,±2^(n/2),±3·2^(n/2)}.These functions provide new canonical examples for the study of Boolean function spectral theory.展开更多
For an odd integer n ≥ 7, this paper presented a class of n-variable rotation symmetric Boolean functions （RSBFs） with optimum algebraic immunity. The nonlinearity of the constructed functions is determined.
The Boolean functions in an affine equivalence class are of the same algebraicdegree and nonlinearity, but may satisfy different order of correlation immunity and propa-gation criterion. A method is presented in this ...The Boolean functions in an affine equivalence class are of the same algebraicdegree and nonlinearity, but may satisfy different order of correlation immunity and propa-gation criterion. A method is presented in this paper to find Boolean functions with higherorder correlation immunity or satisfying higher order propagation criterion in an affine equiv-alence class. 8 AES s-box functions are not better Boolean functions in their affine equiva-lence class.展开更多
This paper proposes a practical algorithm for systematically generating strong Boolean functions (f:GF(2) n →GF(2)) with cryptographic meaning. This algorithm takes bent function as input and directly outputs the res...This paper proposes a practical algorithm for systematically generating strong Boolean functions (f:GF(2) n →GF(2)) with cryptographic meaning. This algorithm takes bent function as input and directly outputs the resulted Boolean function in terms of truth table sequence. This algorithm was used to develop two classes of balanced Boolean functions, one of which has very good cryptographic properties:nl(f)=2 2k?1?2k+2k?2 (n=2k), with the sum-of-squares avalanche characteristic off satisfying σf=24k+23k+2+23k-2 and the absolute avalanche characteristic off satisfying σf=24k+23k+2+23k-2. This is the best result up to now compared to existing ones. Instead of bent sequences, starting from random Boolean functions was also tested in the algorithm. Experimental results showed that starting from bent sequences is highly superior to starting from random Boolean functions. Key words Boolean functions - Bent sequences - Nonlinearity - GAC - PC - Balancedness Document code A CLC number TP301.6展开更多
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 ...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 one main result to find balanced rotation symmetric Boolean functions with maximum algebraic immunity. Through swapping the values of two orbits of rotation class of the majority function, a class of 4k＋l variable Boolean functions with maximum algebraic immu- nity is constructed. The function f（x） we construct always has terms of degree n-2 independence of what ever n is. And the nonlinearity off（x） is relatively good for large n.展开更多
To protect against algebraic attacks, a high algebraic immunity is now an important criterion for Boolean functions used in stream ciphers. In this paper, a new method based on a univariate polynomial representation o...To protect against algebraic attacks, a high algebraic immunity is now an important criterion for Boolean functions used in stream ciphers. In this paper, a new method based on a univariate polynomial representation of Boolean functions is proposed. The proposed method is used to constmct Boolean functions with an odd number of variables and with maximum algebraic immunity. We also discuss the nonlinearity of the constructed functions. Moreover, a lower bound is deter- mined for the number of Boolean functions with rmximum algebraic immunity.展开更多
The Walsh transform is an important tool to investigate cryptographic properties of Boolean functions.This paper is devoted to study the Walsh transform of a class of Boolean functions defined as g(x)=f(x)Tr^(n)_(1)(x...The Walsh transform is an important tool to investigate cryptographic properties of Boolean functions.This paper is devoted to study the Walsh transform of a class of Boolean functions defined as g(x)=f(x)Tr^(n)_(1)(x)+h(x)Tr^(n)_(1)(δx),by making use of the known conclusions of Walsh transform and the properties of trace function,and the conclusion is obtained by generalizing an existing result.展开更多
We use evolutionaly computing to synthesize Boolean functions randomly Byusing specific crossover and mutation operator,in evolving process and modifying search space andfitness function,we get some high non-linearity...We use evolutionaly computing to synthesize Boolean functions randomly Byusing specific crossover and mutation operator,in evolving process and modifying search space andfitness function,we get some high non-linearity functions which have other good cryptographycharacteristics such as autocorrelation etc Comparing to other heuristic search techniques,evolutionary computing approach is more effective because of global search strategy and implicitparallelism.展开更多
Boolean or switching functions can be associated to finite aligned spaces in a way similar to the way they can be associated to finite topological spaces. We prove a characterization of switching functions associated ...Boolean or switching functions can be associated to finite aligned spaces in a way similar to the way they can be associated to finite topological spaces. We prove a characterization of switching functions associated to aligned spaces which is similar to the one we have given for switching functions associated to finite topological spaces.展开更多
In this paper, we survey a number of studies in the literature on improving lightweight systems in the Internet of Things (IoT). The paper illustrates recent development of Boolean cryptographic function Application a...In this paper, we survey a number of studies in the literature on improving lightweight systems in the Internet of Things (IoT). The paper illustrates recent development of Boolean cryptographic function Application and how it assists in using hardware such as the internet of things. For a long time there seems to be little progress in applying pure mathematics in providing security since the wide progress made by George Boole and Shannon. We discuss cryptanalysis of Boolean functions to avoid trapdoors and vulnerabilities in the development of block ciphers. It appears that there is significant progress. A comparative analysis of lightweight cryptographic schemes is reported in terms of execution time, code size and throughput. Depending on the schemes and the structure of the algorithms, these parameters change but remain within reasonable values making them suited for Internet of things applications. The driving force of lightweight cryptography (LWC) stems mainly from its direct applications in the real world since it provides solutions to actual problems faced by designers of IoT systems. Broadly speaking, lightweight cryptographic algorithms are designed to achieve two main goals. The first goal of a cryptographic algorithm is to withstand all known cryptanalytic attacks and thus to be secure in the black-box model. The second goal is to build the cryptographic primitive in such a way that its implementations satisfy a clearly specified set of constraints that depend on a case-by-case basis.展开更多
In order to measure the correlation propeties of two Boolean functions,the global avalanche characteristics of Boolean functions constructed by concatenation are discussed,i.e.,f_1‖f_2and f_1‖f_2‖f_3‖f_4.Firstly,f...In order to measure the correlation propeties of two Boolean functions,the global avalanche characteristics of Boolean functions constructed by concatenation are discussed,i.e.,f_1‖f_2and f_1‖f_2‖f_3‖f_4.Firstly,for the function f = f_1‖f_2,the cross-correlation function of f_1,f_2 in the special condition are studied.In this case,f,f_1,f_2 must be in desired form.By computing their sum-of-squares indicators,the crosscorrelation function between f_1,f_2 is obtained.Secondly,for the function g = f_1‖f_2‖f_3‖f_4,by analyzing the relation among their auto-correlation functions,their sum-of-squares indicators are investigated.Based on them,the sum-of-squares indicators of functions obtained by Canteaut et al.are investigated.The results show that the correlation property of g is good when the correlation properties of Boolean functions f_1,f_2,f_3,f_4 are good.展开更多
A large class of linear structural functions(LSF) satisfying the condition of correlational immunity of order one are constructed by studying the linear structural Boolean functions. With these new founded functions, ...A large class of linear structural functions(LSF) satisfying the condition of correlational immunity of order one are constructed by studying the linear structural Boolean functions. With these new founded functions, the known enumeration bounds of correlation-immune functions of order one are greatly improved. In fact, the best, up to now, lower bound is found.展开更多
The nonlinear filter Boolean function of LILI-128 stream cipher generator is studied in this paper. First we measure the complexity of the stream ciphers sequence of LILI-128 stream cipher generator and obtain the sho...The nonlinear filter Boolean function of LILI-128 stream cipher generator is studied in this paper. First we measure the complexity of the stream ciphers sequence of LILI-128 stream cipher generator and obtain the shortest bit stream sequence reconstructed Boolean function of nonlinear filter in LILI-128 stream cipher generator. Then the least nonlinear Boolean function of generating stream cipher sequence is reconstructed by clusterig, nonlinear predictive and nonlinear synchronization from shortest bit stream sequence. We have verified the correctness of our reconstruction result by simulating the block diagram of Lili-128 keystream generator using our getting Boolean function and implement designers’ reference module of Lili-128 stream cipher public online, and two methods produce the same synchronous keystream sequence under same initial state, so that our research work proves that the nonlinear Boolean function of LILI-128 stream cipher generator is successfully broken.展开更多
From the motivation of algebraic attacks on stream and block ciphers,the concept of algebraic immunity（AI） of a Boolean function was introduced and studied extensively.High algebraic immunity is a necessary conditio...From the motivation of algebraic attacks on stream and block ciphers,the concept of algebraic immunity（AI） of a Boolean function was introduced and studied extensively.High algebraic immunity is a necessary condition for resisting algebraic attacks.In this paper,we give some lower bounds on the algebraic immunity of Boolean functions.The results are applied to give lower bounds on the AI of symmetric Boolean functions and rotation symmetric Boolean functions.Some balanced rotation symmetric Boolean functions with their AI near the maximum possible value「n/2」are constructed.展开更多
A Boolean function /(x1,x2,…,xn) is said to be elusive, if every decision tree algorithm computing / must examine all n variables in the worst case. In 1988, A.C.C. Yao introduced a question: Is any nontrivial monoto...A Boolean function /(x1,x2,…,xn) is said to be elusive, if every decision tree algorithm computing / must examine all n variables in the worst case. In 1988, A.C.C. Yao introduced a question: Is any nontrivial monotone Boolean function that is invariant under the transitive act of group Cm × Cn elusive? The positive answer to this question supports the famous Rivest-Vuillemin conjecture on decision tree complexity. In this paper, we shall partly answer this question.展开更多
This paper shows that monotone self-dual Boolean functions in irredundant disjuntive normal form （IDNF） do not have more variables than disjuncts. Monotone self-dual Boolean functions in IDNF with the same number of...This paper shows that monotone self-dual Boolean functions in irredundant disjuntive normal form （IDNF） do not have more variables than disjuncts. Monotone self-dual Boolean functions in IDNF with the same number of variables and disjuncts are examined. An algorithm is proposed to test whether a monotone Boolean function in IDNF with n variables and n disjuncts is self-dual. The runtime of the algorithm is O（n3）.展开更多
This paper presents a construction for a class of 1-resilient functions with optimal algebraic immunity on an even number of variables. The construction is based on the concatenation of two balanced functions in assoc...This paper presents a construction for a class of 1-resilient functions with optimal algebraic immunity on an even number of variables. The construction is based on the concatenation of two balanced functions in associative classes. For some n, a part of 1-resilient functions with maximum algebraic immunity constructed in the paper can achieve almost optimal nonlinearity. Apart from their high nonlinearity, the functions reach Siegenthaler＇s upper bound of algebraic degree. Also a class of l-resilient functions on any number n 〉 2 of variables with at least sub-optimal algebraic immunity is provided.展开更多
The global avalanche characteristics （the sum- of-squares indicator and the absolute indicator） measure the overall avalanche characteristics of a cryptographic Boolean function. Sung et al. （1999） gave the lower ...The global avalanche characteristics （the sum- of-squares indicator and the absolute indicator） measure the overall avalanche characteristics of a cryptographic Boolean function. Sung et al. （1999） gave the lower bound on the sum- of-squares indicator for a balanced Boolean function satisfy- ing the propagation criterion with respect to some vectors. In this paper, if balanced Boolean functions satisfy the propaga- tion criterion with respect to some vectors, we give three nec- essary and sufficient conditions on the auto-correlation distri- bution of these functions reaching the minimum the bound on the sum-of-squares indicator. And we also find all Boolean functions with 3-variable, 4-variable, and 5-variable reaching the minimum the bound on the sum-of-squares indicator.展开更多
The properties of the 2m-variable symmetric Boolean functions with maximum al- gebraic immunity are studied in this paper. Their value vectors, algebraic normal forms, and algebraic degrees and weights are all obtaine...The properties of the 2m-variable symmetric Boolean functions with maximum al- gebraic immunity are studied in this paper. Their value vectors, algebraic normal forms, and algebraic degrees and weights are all obtained. At last, some necessary conditions for a symmetric Boolean function on even number variables to have maximum algebraic immunity are introduced.展开更多
By some basic transforms and invariant theory, we give two results： 1） an algorithm, which can be used to judge if two Boolean functions are affinely equivalent and to obtain the equivalence relationship if they are...By some basic transforms and invariant theory, we give two results： 1） an algorithm, which can be used to judge if two Boolean functions are affinely equivalent and to obtain the equivalence relationship if they are equivalent. This is useful in studying Boolean functions and in engineering. For example, we classify all 8-variable homogeneous bent functions of degree 3 into two classes; 2） Reed-Muller codes R（4,6）/R（1,6）, R（3,7）/R（1,7） are classified efficiently.展开更多
基金National Natural Science Foundation of China(62272360)。
文摘Five-valued Boolean functions play an important role in the design of symmetric cryptography.While the design and properties of single-output almost optimal five-valued spectra Boolean functions have been extensively studied over the past few decades,there has been limited research on the construction of almost optimal five-valued spectra vectorial Boolean functions.In this paper,we present a construction method for even-variable 2-output almost optimal five-valued spectra balanced Boolean functions,whose Walsh spectra values belong to the set{0,±2^(n/2),±2^(n/2+1)},at the same time,we discuss the existence of sufficient conditions in the construction.Additionally,this paper presents a novel construction method for balanced single-output Boolean functions with even variables featuring a special five-valued spectral structure,whose Walsh spectra values are constrained to the set{0,±2^(n/2),±3·2^(n/2)}.These functions provide new canonical examples for the study of Boolean function spectral theory.
基金Supported by the National Natural Science Foundation of China ( 60603012)the Foundation of Hubei Provincial Department of Education, China (D200610004)
文摘For an odd integer n ≥ 7, this paper presented a class of n-variable rotation symmetric Boolean functions （RSBFs） with optimum algebraic immunity. The nonlinearity of the constructed functions is determined.
文摘The Boolean functions in an affine equivalence class are of the same algebraicdegree and nonlinearity, but may satisfy different order of correlation immunity and propa-gation criterion. A method is presented in this paper to find Boolean functions with higherorder correlation immunity or satisfying higher order propagation criterion in an affine equiv-alence class. 8 AES s-box functions are not better Boolean functions in their affine equiva-lence class.
文摘This paper proposes a practical algorithm for systematically generating strong Boolean functions (f:GF(2) n →GF(2)) with cryptographic meaning. This algorithm takes bent function as input and directly outputs the resulted Boolean function in terms of truth table sequence. This algorithm was used to develop two classes of balanced Boolean functions, one of which has very good cryptographic properties:nl(f)=2 2k?1?2k+2k?2 (n=2k), with the sum-of-squares avalanche characteristic off satisfying σf=24k+23k+2+23k-2 and the absolute avalanche characteristic off satisfying σf=24k+23k+2+23k-2. This is the best result up to now compared to existing ones. Instead of bent sequences, starting from random Boolean functions was also tested in the algorithm. Experimental results showed that starting from bent sequences is highly superior to starting from random Boolean functions. Key words Boolean functions - Bent sequences - Nonlinearity - GAC - PC - Balancedness Document code A CLC number TP301.6
基金Supported by the National Natural Science Foundation of China(61272434)the Natural Science Foundation of Shandong Province(ZR 2012FM004,ZR2013FQ021)the Foundation of Science and Technology on Information Assume Laboratory(KJ-13-004)
文摘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 one main result to find balanced rotation symmetric Boolean functions with maximum algebraic immunity. Through swapping the values of two orbits of rotation class of the majority function, a class of 4k＋l variable Boolean functions with maximum algebraic immu- nity is constructed. The function f（x） we construct always has terms of degree n-2 independence of what ever n is. And the nonlinearity off（x） is relatively good for large n.
基金This work was supported by the National Natural Science Foundation of China under Grants No. 61103191, No. 61070215 the Funds of Key Lab of Fujian Province University Network Security and Cryptology under Crant No. 2011003 and the Open Research Fund of State Key Laboratory of Inforrmtion Security.
文摘To protect against algebraic attacks, a high algebraic immunity is now an important criterion for Boolean functions used in stream ciphers. In this paper, a new method based on a univariate polynomial representation of Boolean functions is proposed. The proposed method is used to constmct Boolean functions with an odd number of variables and with maximum algebraic immunity. We also discuss the nonlinearity of the constructed functions. Moreover, a lower bound is deter- mined for the number of Boolean functions with rmximum algebraic immunity.
基金Supported by the Natural Science Foundation of Anhui Higher Education Institutions of China(KJ2020ZD008)Key Research and Development Projects in Anhui Province(202004a05020043)the Graduate Innovation Fund of Huaibei Normal University(yx2021022)。
文摘The Walsh transform is an important tool to investigate cryptographic properties of Boolean functions.This paper is devoted to study the Walsh transform of a class of Boolean functions defined as g(x)=f(x)Tr^(n)_(1)(x)+h(x)Tr^(n)_(1)(δx),by making use of the known conclusions of Walsh transform and the properties of trace function,and the conclusion is obtained by generalizing an existing result.
基金Supported by the National Natural Science Foun-dation of China(69973034,60373087,60473023)the Ph.D Pro-grams Foundation of Ministry of Education of China(20020486046)
文摘We use evolutionaly computing to synthesize Boolean functions randomly Byusing specific crossover and mutation operator,in evolving process and modifying search space andfitness function,we get some high non-linearity functions which have other good cryptographycharacteristics such as autocorrelation etc Comparing to other heuristic search techniques,evolutionary computing approach is more effective because of global search strategy and implicitparallelism.
文摘Boolean or switching functions can be associated to finite aligned spaces in a way similar to the way they can be associated to finite topological spaces. We prove a characterization of switching functions associated to aligned spaces which is similar to the one we have given for switching functions associated to finite topological spaces.
文摘In this paper, we survey a number of studies in the literature on improving lightweight systems in the Internet of Things (IoT). The paper illustrates recent development of Boolean cryptographic function Application and how it assists in using hardware such as the internet of things. For a long time there seems to be little progress in applying pure mathematics in providing security since the wide progress made by George Boole and Shannon. We discuss cryptanalysis of Boolean functions to avoid trapdoors and vulnerabilities in the development of block ciphers. It appears that there is significant progress. A comparative analysis of lightweight cryptographic schemes is reported in terms of execution time, code size and throughput. Depending on the schemes and the structure of the algorithms, these parameters change but remain within reasonable values making them suited for Internet of things applications. The driving force of lightweight cryptography (LWC) stems mainly from its direct applications in the real world since it provides solutions to actual problems faced by designers of IoT systems. Broadly speaking, lightweight cryptographic algorithms are designed to achieve two main goals. The first goal of a cryptographic algorithm is to withstand all known cryptanalytic attacks and thus to be secure in the black-box model. The second goal is to build the cryptographic primitive in such a way that its implementations satisfy a clearly specified set of constraints that depend on a case-by-case basis.
基金Sponsored by the National Natural Science Foundations of Anhui Higher Education Institutions of China(Grant No.KJ2014A220,KJ2014A231)the Anhui Provincial Natural Science Foundation(Grant No.1608085MF143)the Key Program in the Youth Elite Support Plan in Universities of Anhui Province(Grant No.gxyq ZD2016112)
文摘In order to measure the correlation propeties of two Boolean functions,the global avalanche characteristics of Boolean functions constructed by concatenation are discussed,i.e.,f_1‖f_2and f_1‖f_2‖f_3‖f_4.Firstly,for the function f = f_1‖f_2,the cross-correlation function of f_1,f_2 in the special condition are studied.In this case,f,f_1,f_2 must be in desired form.By computing their sum-of-squares indicators,the crosscorrelation function between f_1,f_2 is obtained.Secondly,for the function g = f_1‖f_2‖f_3‖f_4,by analyzing the relation among their auto-correlation functions,their sum-of-squares indicators are investigated.Based on them,the sum-of-squares indicators of functions obtained by Canteaut et al.are investigated.The results show that the correlation property of g is good when the correlation properties of Boolean functions f_1,f_2,f_3,f_4 are good.
文摘A large class of linear structural functions(LSF) satisfying the condition of correlational immunity of order one are constructed by studying the linear structural Boolean functions. With these new founded functions, the known enumeration bounds of correlation-immune functions of order one are greatly improved. In fact, the best, up to now, lower bound is found.
文摘The nonlinear filter Boolean function of LILI-128 stream cipher generator is studied in this paper. First we measure the complexity of the stream ciphers sequence of LILI-128 stream cipher generator and obtain the shortest bit stream sequence reconstructed Boolean function of nonlinear filter in LILI-128 stream cipher generator. Then the least nonlinear Boolean function of generating stream cipher sequence is reconstructed by clusterig, nonlinear predictive and nonlinear synchronization from shortest bit stream sequence. We have verified the correctness of our reconstruction result by simulating the block diagram of Lili-128 keystream generator using our getting Boolean function and implement designers’ reference module of Lili-128 stream cipher public online, and two methods produce the same synchronous keystream sequence under same initial state, so that our research work proves that the nonlinear Boolean function of LILI-128 stream cipher generator is successfully broken.
基金supported by the National Natural Science Foundation of China (10871068,61021004)DNRF-NSFC Joint (11061130539)
文摘From the motivation of algebraic attacks on stream and block ciphers,the concept of algebraic immunity（AI） of a Boolean function was introduced and studied extensively.High algebraic immunity is a necessary condition for resisting algebraic attacks.In this paper,we give some lower bounds on the algebraic immunity of Boolean functions.The results are applied to give lower bounds on the AI of symmetric Boolean functions and rotation symmetric Boolean functions.Some balanced rotation symmetric Boolean functions with their AI near the maximum possible value「n/2」are constructed.
基金Supported by the National Natural Science Foundation of China (10171095) Foundation of Chinese Academy of Science (J2001), and Foundation of GSCAS
文摘A Boolean function /(x1,x2,…,xn) is said to be elusive, if every decision tree algorithm computing / must examine all n variables in the worst case. In 1988, A.C.C. Yao introduced a question: Is any nontrivial monotone Boolean function that is invariant under the transitive act of group Cm × Cn elusive? The positive answer to this question supports the famous Rivest-Vuillemin conjecture on decision tree complexity. In this paper, we shall partly answer this question.
文摘This paper shows that monotone self-dual Boolean functions in irredundant disjuntive normal form （IDNF） do not have more variables than disjuncts. Monotone self-dual Boolean functions in IDNF with the same number of variables and disjuncts are examined. An algorithm is proposed to test whether a monotone Boolean function in IDNF with n variables and n disjuncts is self-dual. The runtime of the algorithm is O（n3）.
基金supported by the National Natural Science Foundations of China under Grant Nos. 60903200,61003299
文摘This paper presents a construction for a class of 1-resilient functions with optimal algebraic immunity on an even number of variables. The construction is based on the concatenation of two balanced functions in associative classes. For some n, a part of 1-resilient functions with maximum algebraic immunity constructed in the paper can achieve almost optimal nonlinearity. Apart from their high nonlinearity, the functions reach Siegenthaler＇s upper bound of algebraic degree. Also a class of l-resilient functions on any number n 〉 2 of variables with at least sub-optimal algebraic immunity is provided.
基金This work was supported by Sichuan Provincial Youth Science Fund, the Science and Technology on Communication Security Laboratory Project （9140C110201110C1102）, the National Natural Science Foundations of China （Grant Nos. 61003299, 61202437）, the Natural Sci- ence Basic Research Plan in Shaanxi Province of China （2012JM8041）, the Fundamental Research Funds for the Central Universities （K5051201036） and the ＂l 1 l＂ Project （B08038）. Thanks are due to anonymous referees for a series of comment on this paper.
文摘The global avalanche characteristics （the sum- of-squares indicator and the absolute indicator） measure the overall avalanche characteristics of a cryptographic Boolean function. Sung et al. （1999） gave the lower bound on the sum- of-squares indicator for a balanced Boolean function satisfy- ing the propagation criterion with respect to some vectors. In this paper, if balanced Boolean functions satisfy the propaga- tion criterion with respect to some vectors, we give three nec- essary and sufficient conditions on the auto-correlation distri- bution of these functions reaching the minimum the bound on the sum-of-squares indicator. And we also find all Boolean functions with 3-variable, 4-variable, and 5-variable reaching the minimum the bound on the sum-of-squares indicator.
基金Supported by the National Natural Science Foundation of China(Grant No.60573028)the Open Founds of Key Lab of Fujian Province University Network Security and Cryptology(Grant No. 07A003)the Basic Research Foundation of National University of Defense Technology(Grant No.JC07-02-03)
文摘The properties of the 2m-variable symmetric Boolean functions with maximum al- gebraic immunity are studied in this paper. Their value vectors, algebraic normal forms, and algebraic degrees and weights are all obtained. At last, some necessary conditions for a symmetric Boolean function on even number variables to have maximum algebraic immunity are introduced.
基金the National Natural Science Foundation of China (Grant Nos. 69973034, 60373087, 60673071)
文摘By some basic transforms and invariant theory, we give two results： 1） an algorithm, which can be used to judge if two Boolean functions are affinely equivalent and to obtain the equivalence relationship if they are equivalent. This is useful in studying Boolean functions and in engineering. For example, we classify all 8-variable homogeneous bent functions of degree 3 into two classes; 2） Reed-Muller codes R（4,6）/R（1,6）, R（3,7）/R（1,7） are classified efficiently.
