Graph states are present in quantum information and found applications ranging from quantum network protocols (like secret sharing) to measurement based quantum computing. In this paper, we extend the notion of graph states, which can be regarded as pure quantum graph states, or as homogeneous quadratic Boolean functions associated to simple undirected graphs, to quantum states based on mixed graphs (graphs which allow both directed and undirected edges), obtaining mixed quantum states, which are defined by matrices associated to the measurement of homogeneous quadratic Boolean functions in some (ancillary) variables. In our main result, we describe the extended graph state as the sum of terms of a commutative subgroup of the stabilizer group of the corresponding mixed graph with the edges' directions reversed.
{"title":"Quantum states associated to mixed graphs and their algebraic characterization","authors":"Constanza Riera, M. Parker, P. Stănică","doi":"10.3934/AMC.2021015","DOIUrl":"https://doi.org/10.3934/AMC.2021015","url":null,"abstract":"Graph states are present in quantum information and found applications ranging from quantum network protocols (like secret sharing) to measurement based quantum computing. In this paper, we extend the notion of graph states, which can be regarded as pure quantum graph states, or as homogeneous quadratic Boolean functions associated to simple undirected graphs, to quantum states based on mixed graphs (graphs which allow both directed and undirected edges), obtaining mixed quantum states, which are defined by matrices associated to the measurement of homogeneous quadratic Boolean functions in some (ancillary) variables. In our main result, we describe the extended graph state as the sum of terms of a commutative subgroup of the stabilizer group of the corresponding mixed graph with the edges' directions reversed.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"14 1","pages":"660-680"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84538118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Kailu Yang, Xiaomiao Wang, Menglong Zhang, Lidong Wang
In this paper, we are concerned about bounds and constructions of optimal begin{document}$ 2 $end{document} -D begin{document}$ (ntimes m,3,2,1) $end{document} -optical orthogonal codes. The exact number of codewords of an optimal begin{document}$ 2 $end{document} -D begin{document}$ (ntimes m,3,2,1) $end{document} -optical orthogonal code is determined for begin{document}$ n = 2 $end{document} , begin{document}$ mequiv 1 pmod{2} $end{document} , and begin{document}$ nequiv 1 pmod{2} $end{document} , begin{document}$ mequiv 1,3,5 pmod{12} $end{document} , and begin{document}$ nequiv 4 pmod{6} $end{document} , begin{document}$ mequiv 8 pmod{16} $end{document} .
In this paper, we are concerned about bounds and constructions of optimal begin{document}$ 2 $end{document} -D begin{document}$ (ntimes m,3,2,1) $end{document} -optical orthogonal codes. The exact number of codewords of an optimal begin{document}$ 2 $end{document} -D begin{document}$ (ntimes m,3,2,1) $end{document} -optical orthogonal code is determined for begin{document}$ n = 2 $end{document} , begin{document}$ mequiv 1 pmod{2} $end{document} , and begin{document}$ nequiv 1 pmod{2} $end{document} , begin{document}$ mequiv 1,3,5 pmod{12} $end{document} , and begin{document}$ nequiv 4 pmod{6} $end{document} , begin{document}$ mequiv 8 pmod{16} $end{document} .
{"title":"Some progress on optimal $ 2 $-D $ (ntimes m,3,2,1) $-optical orthogonal codes","authors":"Kailu Yang, Xiaomiao Wang, Menglong Zhang, Lidong Wang","doi":"10.3934/AMC.2021012","DOIUrl":"https://doi.org/10.3934/AMC.2021012","url":null,"abstract":"In this paper, we are concerned about bounds and constructions of optimal begin{document}$ 2 $end{document} -D begin{document}$ (ntimes m,3,2,1) $end{document} -optical orthogonal codes. The exact number of codewords of an optimal begin{document}$ 2 $end{document} -D begin{document}$ (ntimes m,3,2,1) $end{document} -optical orthogonal code is determined for begin{document}$ n = 2 $end{document} , begin{document}$ mequiv 1 pmod{2} $end{document} , and begin{document}$ nequiv 1 pmod{2} $end{document} , begin{document}$ mequiv 1,3,5 pmod{12} $end{document} , and begin{document}$ nequiv 4 pmod{6} $end{document} , begin{document}$ mequiv 8 pmod{16} $end{document} .","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"2016 1","pages":"605-625"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86130061","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Some invariants related to threshold and chain graphs","authors":"R. Raja, Samir Ahmad Wagay","doi":"10.3934/amc.2023020","DOIUrl":"https://doi.org/10.3934/amc.2023020","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"3 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87321439","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A Boolean function in begin{document}$ n $end{document} variables is 2-rotation symmetric if it is invariant under even powers of begin{document}$ rho(x_1, ldots, x_n) = (x_2, ldots, x_n, x_1) $end{document} , but not under the first power (ordinary rotation symmetry); we call such a function a 2-function. A 2-function is called monomial rotation symmetric (MRS) if it is generated by applying powers of begin{document}$ rho^2 $end{document} to a single monomial. If the quartic MRS 2-function in begin{document}$ 2n $end{document} variables has a monomial begin{document}$ x_1 x_q x_r x_s $end{document} , then we use the notation begin{document}$ {2-}(1,q,r,s)_{2n} $end{document} for the function. A detailed theory of equivalence of quartic MRS 2-functions in begin{document}$ 2n $end{document} variables was given in a begin{document}$ 2020 $end{document} paper by Cusick, Cheon and Dougan. This theory divides naturally into two classes, called begin{document}$ mf1 $end{document} and begin{document}$ mf2 $end{document} in the paper. After describing the equivalence classes, the second major problem is giving details of the linear recursions that the Hamming weights for any sequence of functions begin{document}$ {2-}(1,q,r,s)_{2n} $end{document} (with begin{document}$ q say), begin{document}$ n = s, s+1, ldots $end{document} can be shown to satisfy. This problem was solved for the begin{document}$ mf1 $end{document} case only in the begin{document}$ 2020 $end{document} paper. Using new ideas about "short" functions, Cusick and Cheon found formulas for the begin{document}$ mf2 $end{document} weights in a begin{document}$ 2021 $end{document} sequel to the begin{document}$ 2020 $end{document} paper. In this paper the actual recursions for the weights in the begin{document}$ mf2 $end{document} case are determined.
{"title":"The weight recursions for the 2-rotation symmetric quartic Boolean functions","authors":"T. Cusick, Younhwan Cheon","doi":"10.3934/AMC.2021011","DOIUrl":"https://doi.org/10.3934/AMC.2021011","url":null,"abstract":"A Boolean function in begin{document}$ n $end{document} variables is 2-rotation symmetric if it is invariant under even powers of begin{document}$ rho(x_1, ldots, x_n) = (x_2, ldots, x_n, x_1) $end{document} , but not under the first power (ordinary rotation symmetry); we call such a function a 2-function. A 2-function is called monomial rotation symmetric (MRS) if it is generated by applying powers of begin{document}$ rho^2 $end{document} to a single monomial. If the quartic MRS 2-function in begin{document}$ 2n $end{document} variables has a monomial begin{document}$ x_1 x_q x_r x_s $end{document} , then we use the notation begin{document}$ {2-}(1,q,r,s)_{2n} $end{document} for the function. A detailed theory of equivalence of quartic MRS 2-functions in begin{document}$ 2n $end{document} variables was given in a begin{document}$ 2020 $end{document} paper by Cusick, Cheon and Dougan. This theory divides naturally into two classes, called begin{document}$ mf1 $end{document} and begin{document}$ mf2 $end{document} in the paper. After describing the equivalence classes, the second major problem is giving details of the linear recursions that the Hamming weights for any sequence of functions begin{document}$ {2-}(1,q,r,s)_{2n} $end{document} (with begin{document}$ q say), begin{document}$ n = s, s+1, ldots $end{document} can be shown to satisfy. This problem was solved for the begin{document}$ mf1 $end{document} case only in the begin{document}$ 2020 $end{document} paper. Using new ideas about \"short\" functions, Cusick and Cheon found formulas for the begin{document}$ mf2 $end{document} weights in a begin{document}$ 2021 $end{document} sequel to the begin{document}$ 2020 $end{document} paper. In this paper the actual recursions for the weights in the begin{document}$ mf2 $end{document} case are determined.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"21 1","pages":"589-604"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78042776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<p style='text-indent:20px;'>An <inline-formula><tex-math id="M1">begin{document}$ N times k $end{document}</tex-math></inline-formula> array <inline-formula><tex-math id="M2">begin{document}$ A $end{document}</tex-math></inline-formula> with entries from <inline-formula><tex-math id="M3">begin{document}$ v $end{document}</tex-math></inline-formula>-set <inline-formula><tex-math id="M4">begin{document}$ mathcal{V} $end{document}</tex-math></inline-formula> is said to be an <i>orthogonal array</i> with <inline-formula><tex-math id="M5">begin{document}$ v $end{document}</tex-math></inline-formula> levels, strength <inline-formula><tex-math id="M6">begin{document}$ t $end{document}</tex-math></inline-formula> and index <inline-formula><tex-math id="M7">begin{document}$ lambda $end{document}</tex-math></inline-formula>, denoted by OA<inline-formula><tex-math id="M8">begin{document}$ (N,k,v,t) $end{document}</tex-math></inline-formula>, if every <inline-formula><tex-math id="M9">begin{document}$ Ntimes t $end{document}</tex-math></inline-formula> sub-array of <inline-formula><tex-math id="M10">begin{document}$ A $end{document}</tex-math></inline-formula> contains each <inline-formula><tex-math id="M11">begin{document}$ t $end{document}</tex-math></inline-formula>-tuple based on <inline-formula><tex-math id="M12">begin{document}$ mathcal{V} $end{document}</tex-math></inline-formula> exactly <inline-formula><tex-math id="M13">begin{document}$ lambda $end{document}</tex-math></inline-formula> times as a row. An OA<inline-formula><tex-math id="M14">begin{document}$ (N,k,v,t) $end{document}</tex-math></inline-formula> is called <i>irredundant</i>, denoted by IrOA<inline-formula><tex-math id="M15">begin{document}$ (N,k,v,t) $end{document}</tex-math></inline-formula>, if in any <inline-formula><tex-math id="M16">begin{document}$ Ntimes (k-t ) $end{document}</tex-math></inline-formula> sub-array, all of its rows are different. Goyeneche and <inline-formula><tex-math id="M17">begin{document}$ dot{Z} $end{document}</tex-math></inline-formula>yczkowski firstly introduced the definition of an IrOA and showed that an IrOA<inline-formula><tex-math id="M18">begin{document}$ (N,k,v,t) $end{document}</tex-math></inline-formula> corresponds to a <inline-formula><tex-math id="M19">begin{document}$ t $end{document}</tex-math></inline-formula>-uniform state of <inline-formula><tex-math id="M20">begin{document}$ k $end{document}</tex-math></inline-formula> subsystems with local dimension <inline-formula><tex-math id="M21">begin{document}$ v $end{document}</tex-math></inline-formula> (Physical Review A. 90 (2014), 022316). In this paper, we present some new constructions of irredundant orthogonal arrays by using difference matrices and some special matrices over finite fields, respectively, as a consequence, many infinite families of irredundant orthogonal arrays are obtained. Furthermore, several infinite classes of <inline-formula><tex-math id="M22">begin{document}$ t $en
An begin{document}$ N times k $end{document} array begin{document}$ A $end{document} with entries from begin{document}$ v $end{document}-set begin{document}$ mathcal{V} $end{document} is said to be an orthogonal array with begin{document}$ v $end{document} levels, strength begin{document}$ t $end{document} and index begin{document}$ lambda $end{document}, denoted by OAbegin{document}$ (N,k,v,t) $end{document}, if every begin{document}$ Ntimes t $end{document} sub-array of begin{document}$ A $end{document} contains each begin{document}$ t $end{document}-tuple based on begin{document}$ mathcal{V} $end{document} exactly begin{document}$ lambda $end{document} times as a row. An OAbegin{document}$ (N,k,v,t) $end{document} is called irredundant, denoted by IrOAbegin{document}$ (N,k,v,t) $end{document}, if in any begin{document}$ Ntimes (k-t ) $end{document} sub-array, all of its rows are different. Goyeneche and begin{document}$ dot{Z} $end{document}yczkowski firstly introduced the definition of an IrOA and showed that an IrOAbegin{document}$ (N,k,v,t) $end{document} corresponds to a begin{document}$ t $end{document}-uniform state of begin{document}$ k $end{document} subsystems with local dimension begin{document}$ v $end{document} (Physical Review A. 90 (2014), 022316). In this paper, we present some new constructions of irredundant orthogonal arrays by using difference matrices and some special matrices over finite fields, respectively, as a consequence, many infinite families of irredundant orthogonal arrays are obtained. Furthermore, several infinite classes of begin{document}$ t $end{document}-uniform states arise from these irredundant orthogonal arrays.
{"title":"Constructions of irredundant orthogonal arrays","authors":"Guangzhou Chen, Xiaotong Zhang","doi":"10.3934/amc.2021051","DOIUrl":"https://doi.org/10.3934/amc.2021051","url":null,"abstract":"<p style='text-indent:20px;'>An <inline-formula><tex-math id=\"M1\">begin{document}$ N times k $end{document}</tex-math></inline-formula> array <inline-formula><tex-math id=\"M2\">begin{document}$ A $end{document}</tex-math></inline-formula> with entries from <inline-formula><tex-math id=\"M3\">begin{document}$ v $end{document}</tex-math></inline-formula>-set <inline-formula><tex-math id=\"M4\">begin{document}$ mathcal{V} $end{document}</tex-math></inline-formula> is said to be an <i>orthogonal array</i> with <inline-formula><tex-math id=\"M5\">begin{document}$ v $end{document}</tex-math></inline-formula> levels, strength <inline-formula><tex-math id=\"M6\">begin{document}$ t $end{document}</tex-math></inline-formula> and index <inline-formula><tex-math id=\"M7\">begin{document}$ lambda $end{document}</tex-math></inline-formula>, denoted by OA<inline-formula><tex-math id=\"M8\">begin{document}$ (N,k,v,t) $end{document}</tex-math></inline-formula>, if every <inline-formula><tex-math id=\"M9\">begin{document}$ Ntimes t $end{document}</tex-math></inline-formula> sub-array of <inline-formula><tex-math id=\"M10\">begin{document}$ A $end{document}</tex-math></inline-formula> contains each <inline-formula><tex-math id=\"M11\">begin{document}$ t $end{document}</tex-math></inline-formula>-tuple based on <inline-formula><tex-math id=\"M12\">begin{document}$ mathcal{V} $end{document}</tex-math></inline-formula> exactly <inline-formula><tex-math id=\"M13\">begin{document}$ lambda $end{document}</tex-math></inline-formula> times as a row. An OA<inline-formula><tex-math id=\"M14\">begin{document}$ (N,k,v,t) $end{document}</tex-math></inline-formula> is called <i>irredundant</i>, denoted by IrOA<inline-formula><tex-math id=\"M15\">begin{document}$ (N,k,v,t) $end{document}</tex-math></inline-formula>, if in any <inline-formula><tex-math id=\"M16\">begin{document}$ Ntimes (k-t ) $end{document}</tex-math></inline-formula> sub-array, all of its rows are different. Goyeneche and <inline-formula><tex-math id=\"M17\">begin{document}$ dot{Z} $end{document}</tex-math></inline-formula>yczkowski firstly introduced the definition of an IrOA and showed that an IrOA<inline-formula><tex-math id=\"M18\">begin{document}$ (N,k,v,t) $end{document}</tex-math></inline-formula> corresponds to a <inline-formula><tex-math id=\"M19\">begin{document}$ t $end{document}</tex-math></inline-formula>-uniform state of <inline-formula><tex-math id=\"M20\">begin{document}$ k $end{document}</tex-math></inline-formula> subsystems with local dimension <inline-formula><tex-math id=\"M21\">begin{document}$ v $end{document}</tex-math></inline-formula> (Physical Review A. 90 (2014), 022316). In this paper, we present some new constructions of irredundant orthogonal arrays by using difference matrices and some special matrices over finite fields, respectively, as a consequence, many infinite families of irredundant orthogonal arrays are obtained. Furthermore, several infinite classes of <inline-formula><tex-math id=\"M22\">begin{document}$ t $en","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"45 1","pages":"1314-1337"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79439330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, for each of six families of three-valued begin{document}$ m $end{document}-sequence correlation, we construct an infinite family of five-weight codes from trace codes over the ring begin{document}$ R = mathbb{F}_2+umathbb{F}_2 $end{document}, where begin{document}$ u^2 = 0. $end{document} The trace codes have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Their support structure is determined. An application to secret sharing schemes is given. The parameters of the binary image are begin{document}$ [2^{m+1}(2^m-1),4m,2^{m}(2^m-2^r)] $end{document} for some explicit begin{document}$ r. $end{document}
In this paper, for each of six families of three-valued begin{document}$ m $end{document}-sequence correlation, we construct an infinite family of five-weight codes from trace codes over the ring begin{document}$ R = mathbb{F}_2+umathbb{F}_2 $end{document}, where begin{document}$ u^2 = 0. $end{document} The trace codes have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Their support structure is determined. An application to secret sharing schemes is given. The parameters of the binary image are begin{document}$ [2^{m+1}(2^m-1),4m,2^{m}(2^m-2^r)] $end{document} for some explicit begin{document}$ r. $end{document}
{"title":"Five-weight codes from three-valued correlation of M-sequences","authors":"M. Shi, Liqin Qian, T. Helleseth, P. Solé","doi":"10.3934/amc.2021022","DOIUrl":"https://doi.org/10.3934/amc.2021022","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, for each of six families of three-valued <inline-formula><tex-math id=\"M1\">begin{document}$ m $end{document}</tex-math></inline-formula>-sequence correlation, we construct an infinite family of five-weight codes from trace codes over the ring <inline-formula><tex-math id=\"M2\">begin{document}$ R = mathbb{F}_2+umathbb{F}_2 $end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=\"M3\">begin{document}$ u^2 = 0. $end{document}</tex-math></inline-formula> The trace codes have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using character sums. Their support structure is determined. An application to secret sharing schemes is given. The parameters of the binary image are <inline-formula><tex-math id=\"M4\">begin{document}$ [2^{m+1}(2^m-1),4m,2^{m}(2^m-2^r)] $end{document}</tex-math></inline-formula> for some explicit <inline-formula><tex-math id=\"M5\">begin{document}$ r. $end{document}</tex-math></inline-formula></p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"4 1","pages":"799-814"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91397712","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Tight security analysis of the public Permutation-based $ {{textsf{PMAC_Plus}}} $","authors":"Avijit Dutta, M. Nandi, Suprita Talnikar","doi":"10.3934/amc.2023025","DOIUrl":"https://doi.org/10.3934/amc.2023025","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"9 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90000371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Constructing Boolean functions with good cryptographic properties over a subset of vectors with fixed Hamming weight $ E_{n,k} subset {{rm{I!F}}}_2^n $ is significant in lightweight stream ciphers like FLIP [14]. In this article, we have given a construction for a class of $ n $-variable weightwise almost perfectly balanced (WAPB) Boolean functions from known support of an $ n_0 $-variable WAPB Boolean function where $ n_0 < n $. This is a generalization of constructing a weightwise perfectly balanced (WPB) Boolean function by Mesnager and Su [16]. We have studied some cryptographic properties like ANF, nonlinearity, weightwise nonlinearities, and algebraic immunity of the functions. The ANF of this function is obtained recursively, which would be a low-cost implementation in a lightweight stream cipher. Further, we have presented another class of WAPB Boolean functions by modifying the earlier function, and we studied some of its cryptographic properties. The nonlinearity and weightwise nonlinearities of the modified functions improve substantially.
{"title":"A class of weightwise almost perfectly balanced Boolean functions","authors":"Deepak Kumar Dalai, Krishna Mallick","doi":"10.3934/amc.2023048","DOIUrl":"https://doi.org/10.3934/amc.2023048","url":null,"abstract":"Constructing Boolean functions with good cryptographic properties over a subset of vectors with fixed Hamming weight $ E_{n,k} subset {{rm{I!F}}}_2^n $ is significant in lightweight stream ciphers like FLIP [14]. In this article, we have given a construction for a class of $ n $-variable weightwise almost perfectly balanced (WAPB) Boolean functions from known support of an $ n_0 $-variable WAPB Boolean function where $ n_0 < n $. This is a generalization of constructing a weightwise perfectly balanced (WPB) Boolean function by Mesnager and Su [16]. We have studied some cryptographic properties like ANF, nonlinearity, weightwise nonlinearities, and algebraic immunity of the functions. The ANF of this function is obtained recursively, which would be a low-cost implementation in a lightweight stream cipher. Further, we have presented another class of WAPB Boolean functions by modifying the earlier function, and we studied some of its cryptographic properties. The nonlinearity and weightwise nonlinearities of the modified functions improve substantially.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135709288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose an effective version of the lift by derivation, an invariant that allows us to provide the classification of $ B(5,6,8) = RM(6, 8)/RM(4, 8) $. The main consequence is to establish that the covering radius of the Reed-Muller $ RM(4,8) $ is equal to 26.
{"title":"Covering radius of $ RM(4,8) $","authors":"Valérie Gillot, Philippe Langevin","doi":"10.3934/amc.2023038","DOIUrl":"https://doi.org/10.3934/amc.2023038","url":null,"abstract":"We propose an effective version of the lift by derivation, an invariant that allows us to provide the classification of $ B(5,6,8) = RM(6, 8)/RM(4, 8) $. The main consequence is to establish that the covering radius of the Reed-Muller $ RM(4,8) $ is equal to 26.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"2011 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136258527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Equivalence for generalized Boolean functions","authors":"Ayça Çesmelioglu, W. Meidl","doi":"10.3934/amc.2023009","DOIUrl":"https://doi.org/10.3934/amc.2023009","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"77 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76836648","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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