Some valuable results over rings have a promising utilization in coding theory and error-correcting code theory. In this paper, we study character sums over a certain non-chain ring and their applications in codebooks. There are two major ingredients in this study. The first ingredient is to investigate Gaussian sums, hyper Eisenstein sums, Jacobi sums over a certain non-chain ring and study the properties of these character sums. For their applications, the second ingredient is to present three classes of asymptotically optimal codebooks with respect to the Welch bound and a family of optimal codebooks with respect to the Levenshtein bound, which are constructed from character sums over a certain non-chain ring.
{"title":"Character sums over a non-chain ring and their applications","authors":"Liqin Qian, X. Cao","doi":"10.3934/AMC.2020134","DOIUrl":"https://doi.org/10.3934/AMC.2020134","url":null,"abstract":"Some valuable results over rings have a promising utilization in coding theory and error-correcting code theory. In this paper, we study character sums over a certain non-chain ring and their applications in codebooks. There are two major ingredients in this study. The first ingredient is to investigate Gaussian sums, hyper Eisenstein sums, Jacobi sums over a certain non-chain ring and study the properties of these character sums. For their applications, the second ingredient is to present three classes of asymptotically optimal codebooks with respect to the Welch bound and a family of optimal codebooks with respect to the Levenshtein bound, which are constructed from character sums over a certain non-chain ring.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"89 1","pages":"381-403"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72815375","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":"Dual transform and projective self-dual codes","authors":"I. Bouyukliev, S. Bouyuklieva","doi":"10.3934/amc.2023032","DOIUrl":"https://doi.org/10.3934/amc.2023032","url":null,"abstract":"","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"227 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77469085","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 use symplectic self-dual additive codes over begin{document}$ mathbb{F}_4 $end{document} obtained from metacirculant graphs to construct, for the first time, begin{document}$ left[kern-0.15emleft[ {ell, 0, d} right]kern-0.15emright] $end{document} qubit codes with parameters begin{document}$ (ell,d) in {(78, 20), (90, 21), (91, 22), (93,21),(96,22)} $end{document}. Secondary constructions applied to the qubit codes result in many new qubit codes that perform better than the previous best-known.
We use symplectic self-dual additive codes over begin{document}$ mathbb{F}_4 $end{document} obtained from metacirculant graphs to construct, for the first time, begin{document}$ left[kern-0.15emleft[ {ell, 0, d} right]kern-0.15emright] $end{document} qubit codes with parameters begin{document}$ (ell,d) in {(78, 20), (90, 21), (91, 22), (93,21),(96,22)} $end{document}. Secondary constructions applied to the qubit codes result in many new qubit codes that perform better than the previous best-known.
{"title":"New quantum codes from metacirculant graphs via self-dual additive $mathbb{F}_4$-codes","authors":"P. Seneviratne, M. F. Ezerman","doi":"10.3934/amc.2021073","DOIUrl":"https://doi.org/10.3934/amc.2021073","url":null,"abstract":"<p style='text-indent:20px;'>We use symplectic self-dual additive codes over <inline-formula><tex-math id=\"M1\">begin{document}$ mathbb{F}_4 $end{document}</tex-math></inline-formula> obtained from metacirculant graphs to construct, for the first time, <inline-formula><tex-math id=\"M2\">begin{document}$ left[kern-0.15emleft[ {ell, 0, d} right]kern-0.15emright] $end{document}</tex-math></inline-formula> qubit codes with parameters <inline-formula><tex-math id=\"M3\">begin{document}$ (ell,d) in {(78, 20), (90, 21), (91, 22), (93,21),(96,22)} $end{document}</tex-math></inline-formula>. Secondary constructions applied to the qubit codes result in many new qubit codes that perform better than the previous best-known.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"46 1","pages":"288-297"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76819138","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 the context of digital signatures, the proxy signature holds a significant role of enabling an original signer to delegate its signing ability to another party (i.e., proxy signer). It has significant practical applications. Particularly it is useful in distributed systems, where delegation of authentication rights is quite common. For example, key sharing protocol, grid computing, and mobile communications. Currently, a large portion of existing proxy signature schemes are based on the hardness of problems like integer factoring, discrete logarithms, and/or elliptic curve discrete logarithms. However, with the rising of quantum computers, the problem of prime factorization and discrete logarithm will be solvable in polynomial-time, due to Shor's algorithm, which dilutes the security features of existing ElGamal, RSA, ECC, and the proxy signature schemes based on these problems. As a consequence, construction of secure and efficient post-quantum proxy signature becomes necessary. In this work, we develop a post-quantum proxy signature scheme Mult-proxy, relying on multivariate public key cryptography (MPKC), which is one of the most promising candidates of post-quantum cryptography. We employ a 5-pass identification protocol to design our proxy signature scheme. Our work attains the usual proxy criterion and a one-more-unforgeability criterion under the hardness of the Multivariate Quadratic polynomial (MQ) problem. It produces optimal size proxy signatures and optimal size proxy shares in the field of MPKC.
{"title":"Delegating signing rights in a multivariate proxy signature scheme","authors":"Sumit Kumar Debnath, Tanmay Choudhury, P. Stănică, Kunal Dey, Nibedita Kundu","doi":"10.3934/AMC.2021016","DOIUrl":"https://doi.org/10.3934/AMC.2021016","url":null,"abstract":"In the context of digital signatures, the proxy signature holds a significant role of enabling an original signer to delegate its signing ability to another party (i.e., proxy signer). It has significant practical applications. Particularly it is useful in distributed systems, where delegation of authentication rights is quite common. For example, key sharing protocol, grid computing, and mobile communications. Currently, a large portion of existing proxy signature schemes are based on the hardness of problems like integer factoring, discrete logarithms, and/or elliptic curve discrete logarithms. However, with the rising of quantum computers, the problem of prime factorization and discrete logarithm will be solvable in polynomial-time, due to Shor's algorithm, which dilutes the security features of existing ElGamal, RSA, ECC, and the proxy signature schemes based on these problems. As a consequence, construction of secure and efficient post-quantum proxy signature becomes necessary. In this work, we develop a post-quantum proxy signature scheme Mult-proxy, relying on multivariate public key cryptography (MPKC), which is one of the most promising candidates of post-quantum cryptography. We employ a 5-pass identification protocol to design our proxy signature scheme. Our work attains the usual proxy criterion and a one-more-unforgeability criterion under the hardness of the Multivariate Quadratic polynomial (MQ) problem. It produces optimal size proxy signatures and optimal size proxy shares in the field of MPKC.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"24 1","pages":"681-696"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82896646","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 characterize the connection between begin{document}$ p $end{document} -ary linear codes and Ramanujan Cayley graphs. We explicitly determine an equivalence between begin{document}$ t $end{document} -weight linear codes over the finite field begin{document}$ Bbb F_p $end{document} and Ramanujan Cayley graphs with begin{document}$ t+1 $end{document} eigenvalues. In particular, we get an explicit criterion on the equivalence between two-weight linear codes and Ramanujan strongly regular graphs with explicit parameters. Using this characterization, we construct several families of Ramanujan Cayley graphs with two or three eigenvalues from known linear codes with two or three weights, respectively.
We characterize the connection between begin{document}$ p $end{document} -ary linear codes and Ramanujan Cayley graphs. We explicitly determine an equivalence between begin{document}$ t $end{document} -weight linear codes over the finite field begin{document}$ Bbb F_p $end{document} and Ramanujan Cayley graphs with begin{document}$ t+1 $end{document} eigenvalues. In particular, we get an explicit criterion on the equivalence between two-weight linear codes and Ramanujan strongly regular graphs with explicit parameters. Using this characterization, we construct several families of Ramanujan Cayley graphs with two or three eigenvalues from known linear codes with two or three weights, respectively.
{"title":"Connection of $ p $-ary $ t $-weight linear codes to Ramanujan Cayley graphs with $ t+1 $ eigenvalues","authors":"J. Hyun, Yoonjin Lee, Yansheng Wu","doi":"10.3934/AMC.2020133","DOIUrl":"https://doi.org/10.3934/AMC.2020133","url":null,"abstract":"We characterize the connection between begin{document}$ p $end{document} -ary linear codes and Ramanujan Cayley graphs. We explicitly determine an equivalence between begin{document}$ t $end{document} -weight linear codes over the finite field begin{document}$ Bbb F_p $end{document} and Ramanujan Cayley graphs with begin{document}$ t+1 $end{document} eigenvalues. In particular, we get an explicit criterion on the equivalence between two-weight linear codes and Ramanujan strongly regular graphs with explicit parameters. Using this characterization, we construct several families of Ramanujan Cayley graphs with two or three eigenvalues from known linear codes with two or three weights, respectively.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"92 1","pages":"367-380"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79555643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, a new class of generalized cyclotomic binary sequences with period begin{document}$ 4p^n $end{document} is proposed. These sequences are almost balanced, and the explicit formulas of their linear complexity and autocorrelation are presented.
In this paper, a new class of generalized cyclotomic binary sequences with period begin{document}$ 4p^n $end{document} is proposed. These sequences are almost balanced, and the explicit formulas of their linear complexity and autocorrelation are presented.
{"title":"On the linear complexity and autocorrelation of generalized cyclotomic binary sequences with period $ 4p^n $","authors":"Lin Yi, Xiangyong Zeng, Zhimin Sun, Shasha Zhang","doi":"10.3934/AMC.2021019","DOIUrl":"https://doi.org/10.3934/AMC.2021019","url":null,"abstract":"In this paper, a new class of generalized cyclotomic binary sequences with period begin{document}$ 4p^n $end{document} is proposed. These sequences are almost balanced, and the explicit formulas of their linear complexity and autocorrelation are presented.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"14 1","pages":"733-756"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88889992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Recently, Ivanov et al. proposed a new approach to construct code-based cryptosystems, namely the begin{document}$ {sf IKKR} $end{document} public-key encryptions (PKE) in the International Workshop on Code-Based Cryptography (CBCrypto 2020) [ 9 ]. Unlike the usual construction in code-based encryption schemes which has restrictions on the Hamming weight of the error introduced into the ciphertext, the begin{document}$ {sf IKKR} $end{document} approach allows error vectors of arbitrary weight being introduced into the ciphertext. Using this new approach, Ivanov et al. constructed two cryptosystems, namely the modified and the upgraded begin{document}$ {sf IKKR} $end{document} -PKE. This paper aims to discuss the practical security of the begin{document}$ {sf IKKR} $end{document} -PKE. In particular, we describe the weaknesses in the design of the public key used in the begin{document}$ {sf IKKR} $end{document} -PKE. We exploit such weaknesses and propose two attacks to recover the plaintext in the begin{document}$ {sf IKKR} $end{document} -PKE. The approach of our first attack is similar to the LCKN attack [ 12 ], whilst our second attack is more efficient than the LCKN attack. Our experimental results show that we can recover the plaintext from a given ciphertext in less than 176 milliseconds for schemes based on random Goppa codes and BCH codes.
Recently, Ivanov et al. proposed a new approach to construct code-based cryptosystems, namely the begin{document}$ {sf IKKR} $end{document} public-key encryptions (PKE) in the International Workshop on Code-Based Cryptography (CBCrypto 2020) [ 9 ]. Unlike the usual construction in code-based encryption schemes which has restrictions on the Hamming weight of the error introduced into the ciphertext, the begin{document}$ {sf IKKR} $end{document} approach allows error vectors of arbitrary weight being introduced into the ciphertext. Using this new approach, Ivanov et al. constructed two cryptosystems, namely the modified and the upgraded begin{document}$ {sf IKKR} $end{document} -PKE. This paper aims to discuss the practical security of the begin{document}$ {sf IKKR} $end{document} -PKE. In particular, we describe the weaknesses in the design of the public key used in the begin{document}$ {sf IKKR} $end{document} -PKE. We exploit such weaknesses and propose two attacks to recover the plaintext in the begin{document}$ {sf IKKR} $end{document} -PKE. The approach of our first attack is similar to the LCKN attack [ 12 ], whilst our second attack is more efficient than the LCKN attack. Our experimental results show that we can recover the plaintext from a given ciphertext in less than 176 milliseconds for schemes based on random Goppa codes and BCH codes.
{"title":"Polynomial-time plaintext recovery attacks on the IKKR code-based cryptosystems","authors":"T. Lau, C. H. Tan","doi":"10.3934/AMC.2020132","DOIUrl":"https://doi.org/10.3934/AMC.2020132","url":null,"abstract":"Recently, Ivanov et al. proposed a new approach to construct code-based cryptosystems, namely the begin{document}$ {sf IKKR} $end{document} public-key encryptions (PKE) in the International Workshop on Code-Based Cryptography (CBCrypto 2020) [ 9 ]. Unlike the usual construction in code-based encryption schemes which has restrictions on the Hamming weight of the error introduced into the ciphertext, the begin{document}$ {sf IKKR} $end{document} approach allows error vectors of arbitrary weight being introduced into the ciphertext. Using this new approach, Ivanov et al. constructed two cryptosystems, namely the modified and the upgraded begin{document}$ {sf IKKR} $end{document} -PKE. This paper aims to discuss the practical security of the begin{document}$ {sf IKKR} $end{document} -PKE. In particular, we describe the weaknesses in the design of the public key used in the begin{document}$ {sf IKKR} $end{document} -PKE. We exploit such weaknesses and propose two attacks to recover the plaintext in the begin{document}$ {sf IKKR} $end{document} -PKE. The approach of our first attack is similar to the LCKN attack [ 12 ], whilst our second attack is more efficient than the LCKN attack. Our experimental results show that we can recover the plaintext from a given ciphertext in less than 176 milliseconds for schemes based on random Goppa codes and BCH codes.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"29 1","pages":"353-366"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85433503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
where begin{document}$ mathcal{A} = left{a_{1},a_{2},cdots,a_{k}right}(kin mathbb{N},kgeq 2) $end{document} is a finite set of begin{document}$ k $end{document} symbols. Later many pseudorandom sequences of begin{document}$ k $end{document} symbols have been given and studied by using number theoretic methods. In this paper we study the pseudorandom properties of the begin{document}$ k $end{document}-ary Sidel'nikov sequences with length begin{document}$ q-1 $end{document} by using the estimates for certain character sums with exponential function, where begin{document}$ q $end{document} is a prime power. Our results show that Sidel'nikov sequences enjoy good well-distribution measure and correlation measure. Furthermore, we prove that the set of size begin{document}$ phi(q-1) $end{document} of begin{document}$ k $end{document}-ary Sidel'nikov sequences is collision free and possesses the strict avalanche effect property provided that begin{document}$ k = o(q^{frac{1}{4}}) $end{document}, where begin{document}$ phi $end{document} denotes Euler's totient function.
In 2002 Mauduit and Sárközy started to study finite sequences of begin{document}$ k $end{document} symbols begin{document}$ E_{N} = left(e_{1},e_{2},cdots,e_{N}right)in mathcal{A}^{N}, $end{document} where begin{document}$ mathcal{A} = left{a_{1},a_{2},cdots,a_{k}right}(kin mathbb{N},kgeq 2) $end{document} is a finite set of begin{document}$ k $end{document} symbols. Later many pseudorandom sequences of begin{document}$ k $end{document} symbols have been given and studied by using number theoretic methods. In this paper we study the pseudorandom properties of the begin{document}$ k $end{document}-ary Sidel'nikov sequences with length begin{document}$ q-1 $end{document} by using the estimates for certain character sums with exponential function, where begin{document}$ q $end{document} is a prime power. Our results show that Sidel'nikov sequences enjoy good well-distribution measure and correlation measure. Furthermore, we prove that the set of size begin{document}$ phi(q-1) $end{document} of begin{document}$ k $end{document}-ary Sidel'nikov sequences is collision free and possesses the strict avalanche effect property provided that begin{document}$ k = o(q^{frac{1}{4}}) $end{document}, where begin{document}$ phi $end{document} denotes Euler's totient function.
{"title":"On the pseudorandom properties of $ k $-ary Sidel'nikov sequences","authors":"Huaning Liu, Yixin Ren","doi":"10.3934/amc.2021038","DOIUrl":"https://doi.org/10.3934/amc.2021038","url":null,"abstract":"<p style='text-indent:20px;'>In 2002 Mauduit and Sárközy started to study finite sequences of <inline-formula><tex-math id=\"M2\">begin{document}$ k $end{document}</tex-math></inline-formula> symbols</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ E_{N} = left(e_{1},e_{2},cdots,e_{N}right)in mathcal{A}^{N}, $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id=\"M3\">begin{document}$ mathcal{A} = left{a_{1},a_{2},cdots,a_{k}right}(kin mathbb{N},kgeq 2) $end{document}</tex-math></inline-formula> is a finite set of <inline-formula><tex-math id=\"M4\">begin{document}$ k $end{document}</tex-math></inline-formula> symbols. Later many pseudorandom sequences of <inline-formula><tex-math id=\"M5\">begin{document}$ k $end{document}</tex-math></inline-formula> symbols have been given and studied by using number theoretic methods. In this paper we study the pseudorandom properties of the <inline-formula><tex-math id=\"M6\">begin{document}$ k $end{document}</tex-math></inline-formula>-ary Sidel'nikov sequences with length <inline-formula><tex-math id=\"M7\">begin{document}$ q-1 $end{document}</tex-math></inline-formula> by using the estimates for certain character sums with exponential function, where <inline-formula><tex-math id=\"M8\">begin{document}$ q $end{document}</tex-math></inline-formula> is a prime power. Our results show that Sidel'nikov sequences enjoy good well-distribution measure and correlation measure. Furthermore, we prove that the set of size <inline-formula><tex-math id=\"M9\">begin{document}$ phi(q-1) $end{document}</tex-math></inline-formula> of <inline-formula><tex-math id=\"M10\">begin{document}$ k $end{document}</tex-math></inline-formula>-ary Sidel'nikov sequences is collision free and possesses the strict avalanche effect property provided that <inline-formula><tex-math id=\"M11\">begin{document}$ k = o(q^{frac{1}{4}}) $end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=\"M12\">begin{document}$ phi $end{document}</tex-math></inline-formula> denotes Euler's totient function.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"44 1","pages":"1072-1085"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82655278","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let begin{document}$ mathbb{F}_{q} $end{document} be a finite field with character begin{document}$ p $end{document}. In this paper, the multiplicative group begin{document}$ mathbb{F}_{q}^{*} = mathbb{F}_{q}setminus{0} $end{document} is decomposed into a mutually disjoint union of begin{document}$ gcd(6l^mp^n,q-1) $end{document} cosets over subgroup begin{document}$ $end{document}, where begin{document}$ xi $end{document} is a primitive element of begin{document}$ mathbb{F}_{q} $end{document}. Based on the decomposition, the structure of constacyclic codes of length begin{document}$ 6l^mp^n $end{document} over finite field begin{document}$ mathbb{F}_{q} $end{document} and their duals is established in terms of their generator polynomials, where begin{document}$ pneq{3} $end{document} and begin{document}$ lneq{3} $end{document} are distinct odd primes, begin{document}$ m $end{document} and begin{document}$ n $end{document} are positive integers. In addition, we determine the characterization and enumeration of all linear complementary dual(LCD) negacyclic codes and self-dual constacyclic codes of length begin{document}$ 6l^mp^n $end{document} over begin{document}$ mathbb{F}_{q} $end{document}.
Let begin{document}$ mathbb{F}_{q} $end{document} be a finite field with character begin{document}$ p $end{document}. In this paper, the multiplicative group begin{document}$ mathbb{F}_{q}^{*} = mathbb{F}_{q}setminus{0} $end{document} is decomposed into a mutually disjoint union of begin{document}$ gcd(6l^mp^n,q-1) $end{document} cosets over subgroup begin{document}$ $end{document}, where begin{document}$ xi $end{document} is a primitive element of begin{document}$ mathbb{F}_{q} $end{document}. Based on the decomposition, the structure of constacyclic codes of length begin{document}$ 6l^mp^n $end{document} over finite field begin{document}$ mathbb{F}_{q} $end{document} and their duals is established in terms of their generator polynomials, where begin{document}$ pneq{3} $end{document} and begin{document}$ lneq{3} $end{document} are distinct odd primes, begin{document}$ m $end{document} and begin{document}$ n $end{document} are positive integers. In addition, we determine the characterization and enumeration of all linear complementary dual(LCD) negacyclic codes and self-dual constacyclic codes of length begin{document}$ 6l^mp^n $end{document} over begin{document}$ mathbb{F}_{q} $end{document}.
{"title":"Repeated-root constacyclic codes of length 6lmpn","authors":"Tingting Wu, Shixin Zhu, Li Liu, Lanqiang Li","doi":"10.3934/amc.2021044","DOIUrl":"https://doi.org/10.3934/amc.2021044","url":null,"abstract":"<p style='text-indent:20px;'>Let <inline-formula><tex-math id=\"M1\">begin{document}$ mathbb{F}_{q} $end{document}</tex-math></inline-formula> be a finite field with character <inline-formula><tex-math id=\"M2\">begin{document}$ p $end{document}</tex-math></inline-formula>. In this paper, the multiplicative group <inline-formula><tex-math id=\"M3\">begin{document}$ mathbb{F}_{q}^{*} = mathbb{F}_{q}setminus{0} $end{document}</tex-math></inline-formula> is decomposed into a mutually disjoint union of <inline-formula><tex-math id=\"M4\">begin{document}$ gcd(6l^mp^n,q-1) $end{document}</tex-math></inline-formula> cosets over subgroup <inline-formula><tex-math id=\"M5\">begin{document}$ <xi^{6l^mp^n}> $end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id=\"M6\">begin{document}$ xi $end{document}</tex-math></inline-formula> is a primitive element of <inline-formula><tex-math id=\"M7\">begin{document}$ mathbb{F}_{q} $end{document}</tex-math></inline-formula>. Based on the decomposition, the structure of constacyclic codes of length <inline-formula><tex-math id=\"M8\">begin{document}$ 6l^mp^n $end{document}</tex-math></inline-formula> over finite field <inline-formula><tex-math id=\"M9\">begin{document}$ mathbb{F}_{q} $end{document}</tex-math></inline-formula> and their duals is established in terms of their generator polynomials, where <inline-formula><tex-math id=\"M10\">begin{document}$ pneq{3} $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M11\">begin{document}$ lneq{3} $end{document}</tex-math></inline-formula> are distinct odd primes, <inline-formula><tex-math id=\"M12\">begin{document}$ m $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M13\">begin{document}$ n $end{document}</tex-math></inline-formula> are positive integers. In addition, we determine the characterization and enumeration of all linear complementary dual(LCD) negacyclic codes and self-dual constacyclic codes of length <inline-formula><tex-math id=\"M14\">begin{document}$ 6l^mp^n $end{document}</tex-math></inline-formula> over <inline-formula><tex-math id=\"M15\">begin{document}$ mathbb{F}_{q} $end{document}</tex-math></inline-formula>.</p>","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"23 1","pages":"1154-1180"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82766987","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}
Vikas Srivastava, Sumit Kumar Debnath, P. Stănică, S. Pal
When Kevin Ashton proposed the catchword 'Internet of Things' in 1999, little did he know that technology will become an indispensable part of human lives in just two decades. In short, the Internet of Things (IoT), is a catch-all terminology used to describe devices connected to the internet. These devices can share and receive data as well as provide instructions over a network. By design itself, the IoT system requires multicasting data and information to a set of designated devices, securely. Taking everything into account, Broadcast Encryption (BE) seems to be the natural choice to address the problem. BE allows an originator to broadcast ciphertexts to a big group of receivers in a well-organized and competent way, while ensuring that only designated people can decrypt the data. In this work, we put forward the first Identity-Based Broadcast Encryption scheme based on multivariate polynomials that achieves post-quantum security. Multivariate public key cryptosystems (MPKC), touted as one of the most promising post-quantum cryptography candidates, forms the foundation on which our scheme relies upon, which allows it to be very cost-effective and faster when implemented. In addition, it also provides resistance to collusion attack, and as a consequence our scheme can be utilized to form an efficient and robust IoT system.
当凯文·阿什顿(Kevin Ashton)在1999年提出“物联网”(Internet of Things)这个口号时，他根本不知道，在短短20年的时间里，技术将成为人类生活中不可或缺的一部分。简而言之，物联网(IoT)是一个包罗万象的术语，用于描述连接到互联网的设备。这些设备可以通过网络共享和接收数据以及提供指令。根据设计本身，物联网系统需要将数据和信息安全地广播到一组指定设备。考虑到所有因素，广播加密(BE)似乎是解决这个问题的自然选择。BE允许发端者以一种组织良好、胜任的方式向一大群接收者广播密文，同时确保只有指定的人才能解密数据。在这项工作中，我们提出了第一个基于多元多项式的基于身份的广播加密方案，实现了后量子安全。多元公钥密码系统(MPKC)被吹捧为最有前途的后量子密码候选者之一，它构成了我们方案所依赖的基础，这使得它在实现时非常具有成本效益和速度。此外，它还提供了抵抗合谋攻击的能力，因此我们的方案可以用来形成一个高效和强大的物联网系统。
{"title":"A multivariate identity-based broadcast encryption with applications to the internet of things","authors":"Vikas Srivastava, Sumit Kumar Debnath, P. Stănică, S. Pal","doi":"10.3934/amc.2021050","DOIUrl":"https://doi.org/10.3934/amc.2021050","url":null,"abstract":"When Kevin Ashton proposed the catchword 'Internet of Things' in 1999, little did he know that technology will become an indispensable part of human lives in just two decades. In short, the Internet of Things (IoT), is a catch-all terminology used to describe devices connected to the internet. These devices can share and receive data as well as provide instructions over a network. By design itself, the IoT system requires multicasting data and information to a set of designated devices, securely. Taking everything into account, Broadcast Encryption (BE) seems to be the natural choice to address the problem. BE allows an originator to broadcast ciphertexts to a big group of receivers in a well-organized and competent way, while ensuring that only designated people can decrypt the data. In this work, we put forward the first Identity-Based Broadcast Encryption scheme based on multivariate polynomials that achieves post-quantum security. Multivariate public key cryptosystems (MPKC), touted as one of the most promising post-quantum cryptography candidates, forms the foundation on which our scheme relies upon, which allows it to be very cost-effective and faster when implemented. In addition, it also provides resistance to collusion attack, and as a consequence our scheme can be utilized to form an efficient and robust IoT system.","PeriodicalId":50859,"journal":{"name":"Advances in Mathematics of Communications","volume":"79 1","pages":"1302-1313"},"PeriodicalIF":0.9,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82188953","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：