In this study,we proposed a novel method that integrates density functional theory(DFT)with the finite field method to accurately estimate the polarizability and dielectric constant of polymers.Our approach effectivel...In this study,we proposed a novel method that integrates density functional theory(DFT)with the finite field method to accurately estimate the polarizability and dielectric constant of polymers.Our approach effectively accounts for the influence of electronic and geometric conformation changed on the dielectric constant.We validated our method using polyethylene(PE)and polytetrafluoroethylene(PTFE)as benchmark materials,and found that it reliably predicted their dielectric constants.Furthermore,we explored the impact of conformation variations in poly(vinylidene fluoride)(PVDF)on its dielectric constant and polarizability.The resulting dielectric constants ofα-andγ-PVDF(3.0)showed excellent agreement with crystalline PVDF in experiments.Our findings illuminate the relationship between PVDF’s structural properties and its electrical behavior,offering valuable insights for material design and applications.展开更多
Systolic implementation of multiplication over GF(2m) is usually very efficient in area-time complexity,but its latency is usually very large.Thus,two low latency systolic multipliers over GF(2m) based on general irre...Systolic implementation of multiplication over GF(2m) is usually very efficient in area-time complexity,but its latency is usually very large.Thus,two low latency systolic multipliers over GF(2m) based on general irreducible polynomials and irreducible pentanomials are presented.First,a signal flow graph(SFG) is used to represent the algorithm for multiplication over GF(2m).Then,the two low latency systolic structures for multiplications over GF(2m) based on general irreducible polynomials and pentanomials are presented from the SFG by suitable cut-set retiming,respectively.Analysis indicates that the proposed two low latency designs involve at least one-third less area-delay product when compared with the existing designs,To the authors' knowledge,the time-complexity of the structures is the lowest found in literature for systolic GF(2m) multipliers based on general irreducible polynomials and pentanomials.The proposed low latency designs are regular and modular,and therefore they are suitable for many time critical applications.展开更多
Let Fqbe the finite field,q=p^(k),with p being a prime and k being a positive integer.Let F_(q)^(*)be the multiplicative group of Fq,that is F_(q)^(*)=F_(q){0}.In this paper,by using the Jacobi sums and an analog of H...Let Fqbe the finite field,q=p^(k),with p being a prime and k being a positive integer.Let F_(q)^(*)be the multiplicative group of Fq,that is F_(q)^(*)=F_(q){0}.In this paper,by using the Jacobi sums and an analog of Hasse-Davenport theorem,an explicit formula for the number of solutions of cubic diagonal equation x_(1)^(3)+x_(2)^(3)+…+x_(n)^(3)=c over Fqis given,where c∈F_(q)^(*)and p≡1(mod 3).This extends earlier results.展开更多
In this paper, we present two explicit invalid-curve attacks on the genus 2 hyperelliptic curve over a finite field. First, we propose two explicit attack models by injecting a one-bit fault in a given divisor. Then, ...In this paper, we present two explicit invalid-curve attacks on the genus 2 hyperelliptic curve over a finite field. First, we propose two explicit attack models by injecting a one-bit fault in a given divisor. Then, we discuss the construction of an invalid curve based on the faulted divisor. Our attacks are based on the fact that the Hyperelliptic Curve Scalar Multiplication (HECSM) algorithm does not utilize the curve parameters and We consider three hyperelliptic curves as the attack targets. For curve with security level 186 (in bits), our attack method can get the weakest invalid curve with security level 42 (in bits); there are 93 invalid curves with security level less than 50. We also estimate the theoretical probability of getting a weak hyperelliptic curve whose cardinality is a smooth integer. Finally, we show that the complexity of the fault attack is subexponential if the attacker can freely inject a fault in the input divisor. Cryptosystems based on the genus 2 hyperelliptic curves cannot work against our attack algorithm in practice.展开更多
Let Fq be a finite field. In this paper, a construction of Cartesian au-thentication codes from the normal form of a class of nilpotent matrices over the field Fq is presented. Moreover, assume that the encoding rules...Let Fq be a finite field. In this paper, a construction of Cartesian au-thentication codes from the normal form of a class of nilpotent matrices over the field Fq is presented. Moreover, assume that the encoding rules are chosen according to a uniform probability distribution, the probabilities PI and PS, of a successful im-personation attack and of a successful substitution attack respectively, of these codes are also computed.展开更多
n-to-1 mappings have many applications in combinatorial design,coding theory and cryptography.In this paper,by using piecewise method and monomials on subsets of q+1-th roots of unity,we show a class of n-to-1 binomia...n-to-1 mappings have many applications in combinatorial design,coding theory and cryptography.In this paper,by using piecewise method and monomials on subsets of q+1-th roots of unity,we show a class of n-to-1 binomials having the form x^(r)(a+x^(s(q-1)))over F_(q^(2)).展开更多
Let s be a positive integer,p be an odd prime,q=p^(s),and let F_(q)be a finite field of q elements.Let N_(q)be the number of solutions of the following equations:(x_(1)^(m_(1))+x_(2)^(m_(2))+…+x_(n)^(m_(n)))^(k)=x_(1...Let s be a positive integer,p be an odd prime,q=p^(s),and let F_(q)be a finite field of q elements.Let N_(q)be the number of solutions of the following equations:(x_(1)^(m_(1))+x_(2)^(m_(2))+…+x_(n)^(m_(n)))^(k)=x_(1)x_(2)…x_(n)x^(k_(n+1))_(n+1)…x^(k_(t))_(t)over the finite field F_(q),with n≥2,t>n,k,and k_(j)(n+1≤j≤t),m_(i)(1≤i≤n)are positive integers.In this paper,we find formulas for N_(q)when there is a positive integer l such that dD|(p^(l)+1),where D=1 cm[d_(1),…,d_(n)],d=gcd(n∑i=1M/m_(i)-kM,(q-1)/D),M=1 cm[m_(1),…,m_(n)],d_(j)=gcd(m_(j),q-1),1≤j≤n.And we determine N_(q)explicitly under certain cases.This extends Markoff-Hurwitz-type equations over finite field.展开更多
Projective Reed-Solomon code is an important class of maximal distance separable codes in reliable communication and deep holes play important roles in its decoding.In this paper,we obtain two classes of deep holes of...Projective Reed-Solomon code is an important class of maximal distance separable codes in reliable communication and deep holes play important roles in its decoding.In this paper,we obtain two classes of deep holes of projective Reed-Solomon codes over finite fields with even characteristic.That is,let F_(q) be finite field with even characteristic,k∈{2,q-2},and let u(x)be the Lagrange interpolation polynomial of the first q components of the received vector u∈F_(q)+1 q Suppose that the(q+1)-th component of u is 0,and u(x)=λx^(k)+f_(≤k-2)(x),λx^(q-2)+f_(≤k-2)(x),where λ∈F^(*)_(q) and f_(≤k-2)(x)is a polynomial over F_(q) with degree no more than k-2.Then the received vector u is a deep hole of projective Reed-Solomon codes PRS(F_(q),k).In fact,our result partially solved an open problem on deep holes of projective Reed-Solomon codes proposed by Wan in 2020.展开更多
By using the minimal polynomial of ergodic matrix and the property of polynomial over finite field,we present a polynomial time algorithm for the two-side exponentiation problem about ergodic matrices over finite fie...By using the minimal polynomial of ergodic matrix and the property of polynomial over finite field,we present a polynomial time algorithm for the two-side exponentiation problem about ergodic matrices over finite field （TSEPEM）,and analyze the time and space complexity of the algorithm.According to this algorithm,the public key scheme based on TSEPEM is not secure.展开更多
Let Fq stand for the finite field of odd characteristic p with q elements（q=pn,n∈N）and Fq＊ denote the set of all the nonzero elements of Fq.In this paper,by using the augmented degree matrix and the result given b...Let Fq stand for the finite field of odd characteristic p with q elements（q=pn,n∈N）and Fq＊ denote the set of all the nonzero elements of Fq.In this paper,by using the augmented degree matrix and the result given by Cao,we obtain a formula for the number of rational points of the following equation over Fq：f（x 1,x 2,...,x n）=（a1 x1 x2d＋a2 x2 x3d...＋a（n-1）x（n-1）xnd＋an xn x1d）λ-bx1（d1）x2d2...xn（dn）,with ai,b∈Fq＊,n≥2,λ〉0 being positive integers,and d,di being nonnegative integers for 1≤i n.This technique can be applied to the polynomials of the form h1λ=h2 with λ being positive integer and h1,h2∈Fq[x 1,x 2,...,x n].It extends the results of the Markoff-Hurwitz-type equations.展开更多
Let F q be a finite field with qelements where q=p~α. In the present paper, the authors study the existence and structure of Carter subgroups of singular symplectic group Sp (n+t,n)(F q), singular unitary group U (n+...Let F q be a finite field with qelements where q=p~α. In the present paper, the authors study the existence and structure of Carter subgroups of singular symplectic group Sp (n+t,n)(F q), singular unitary group U (n+t,n)(F (q^2)) and singular orthogonal group O (n+t,n)(F q)(n is even) over finite fields F q.展开更多
In the present paper, we compute the number of the symplectic involaLions over the finite field F with chafF = 2, and also one Cartesian authentication code is obtained.Furthermore, its size parameters are computed co...In the present paper, we compute the number of the symplectic involaLions over the finite field F with chafF = 2, and also one Cartesian authentication code is obtained.Furthermore, its size parameters are computed completely. If assume that the coding rules are chosen according to a uniform probability, PI and Ps denote the largest probabilities of a successful impersonation attack and a successful substitution attack respectively, then PI and Ps are also computed.展开更多
This paper proves that if qn is large enough, for each element a and primitive element b of Fq, there etists a primitive polynomial of degree n ≥5 over the finite field Fq having a as the coefficient of xn-1 and b as...This paper proves that if qn is large enough, for each element a and primitive element b of Fq, there etists a primitive polynomial of degree n ≥5 over the finite field Fq having a as the coefficient of xn-1 and b as the constant term. This proves that if qn is large enongh, for each element a ∈Fq, there exists a primitive polynomial of degree n ≥ 5 over Fq having a as the coefficient of x.展开更多
By establishing the connection between graph colouring and the solution of some equation systems in finite fields, we obtain some formulas to the number of solutions of some equation systems in finite fields, in terms...By establishing the connection between graph colouring and the solution of some equation systems in finite fields, we obtain some formulas to the number of solutions of some equation systems in finite fields, in terms of chromatic polynomial of a graph.展开更多
A （t, n）--secret sharing scheme is a method of distribution of information among n participants such that t 〉 1 can reconstruct the secret but （t - 1） cannot. We explore some （k, n）--secret sharing schemes base...A （t, n）--secret sharing scheme is a method of distribution of information among n participants such that t 〉 1 can reconstruct the secret but （t - 1） cannot. We explore some （k, n）--secret sharing schemes based on the finite fields.展开更多
In this paper, we study about trigonometry in finite field, we know that , the field with p elements, where p is a prime number if and only if p = 8k + 1 or p = 8k -1. Let F and K be two fields, we say that F is an ex...In this paper, we study about trigonometry in finite field, we know that , the field with p elements, where p is a prime number if and only if p = 8k + 1 or p = 8k -1. Let F and K be two fields, we say that F is an extension of K, if K⊆F or there exists a monomorphism f: K→F. Recall that , F[x] is the ring of polynomial over F. If (means that F is an extension of K), an element is algebraic over K if there exists such that f(u) = 0 (see [1]-[4]). The algebraic closure of K in F is , which is the set of all algebraic elements in F over K.展开更多
Wan and Zhang(2021) obtained a nontrivial lower bound for the number of zeros of complete symmetric polynomials over finite fields,and proposed a problem whether their bound can be improved.In this paper,the author im...Wan and Zhang(2021) obtained a nontrivial lower bound for the number of zeros of complete symmetric polynomials over finite fields,and proposed a problem whether their bound can be improved.In this paper,the author improves Wan-Zhang's bound from three aspects.The proposed results are based on the estimates related to the number of certain permutations and the value sets of non-permutation polynomials associated to the complete symmetric polynomial.And the author believes that there are still possibilities to improve the bounds and hence Wan-Zhang's bound.展开更多
In this paper,the approximate synchronization of leader-follower multiagent systems(MASs) over finite fields is studied in regard to local and global synchronization.First,the approximately synchronous state set(ASSS)...In this paper,the approximate synchronization of leader-follower multiagent systems(MASs) over finite fields is studied in regard to local and global synchronization.First,the approximately synchronous state set(ASSS) is obtained.Second,combined with ASSS and transient periods,some criteria for the local and global approximate synchronization of systems are given.Moreover,the algorithms for calculating the maximum approximately synchronous basin(MASB) and the maximum control invariant set(MCIS) are presented.Third,the global approximate synchronization of the system is achieved by designing the state feedback control,and a design algorithm of the controller using the truth matrix method is proposed.Moreover,the results of approximate synchronization are degenerated to complete synchronization.Last,two examples are shown to demonstrate the results of this paper.展开更多
基金This work was financially supported by the Project of Shenzhen Science and Technology(Nos.JCYJ20210324095210028 and JSGGZD20220822095201003)the National Natural Science Foundation of China(No.U21A2087).
文摘In this study,we proposed a novel method that integrates density functional theory(DFT)with the finite field method to accurately estimate the polarizability and dielectric constant of polymers.Our approach effectively accounts for the influence of electronic and geometric conformation changed on the dielectric constant.We validated our method using polyethylene(PE)and polytetrafluoroethylene(PTFE)as benchmark materials,and found that it reliably predicted their dielectric constants.Furthermore,we explored the impact of conformation variations in poly(vinylidene fluoride)(PVDF)on its dielectric constant and polarizability.The resulting dielectric constants ofα-andγ-PVDF(3.0)showed excellent agreement with crystalline PVDF in experiments.Our findings illuminate the relationship between PVDF’s structural properties and its electrical behavior,offering valuable insights for material design and applications.
基金Project(61174132) supported by the National Natural Science Foundation of ChinaProject(09JJ6098) supported by the Natural Science Foundation of Hunan Province,China
文摘Systolic implementation of multiplication over GF(2m) is usually very efficient in area-time complexity,but its latency is usually very large.Thus,two low latency systolic multipliers over GF(2m) based on general irreducible polynomials and irreducible pentanomials are presented.First,a signal flow graph(SFG) is used to represent the algorithm for multiplication over GF(2m).Then,the two low latency systolic structures for multiplications over GF(2m) based on general irreducible polynomials and pentanomials are presented from the SFG by suitable cut-set retiming,respectively.Analysis indicates that the proposed two low latency designs involve at least one-third less area-delay product when compared with the existing designs,To the authors' knowledge,the time-complexity of the structures is the lowest found in literature for systolic GF(2m) multipliers based on general irreducible polynomials and pentanomials.The proposed low latency designs are regular and modular,and therefore they are suitable for many time critical applications.
基金Supported by the Natural Science Foundation of Henan Province(232300420123)the National Natural Science Foundation of China(12026224)the Research Center of Mathematics and Applied Mathematics,Nanyang Institute of Technology。
文摘Let Fqbe the finite field,q=p^(k),with p being a prime and k being a positive integer.Let F_(q)^(*)be the multiplicative group of Fq,that is F_(q)^(*)=F_(q){0}.In this paper,by using the Jacobi sums and an analog of Hasse-Davenport theorem,an explicit formula for the number of solutions of cubic diagonal equation x_(1)^(3)+x_(2)^(3)+…+x_(n)^(3)=c over Fqis given,where c∈F_(q)^(*)and p≡1(mod 3).This extends earlier results.
基金supported by the National Basic Research Program （973 Program）under Grant No.2013CB834205 the National Natural Science Foundation of China under Grant No.61272035 the Independent Innovation Foundation of Shandong University under Grant No.2012JC020
文摘In this paper, we present two explicit invalid-curve attacks on the genus 2 hyperelliptic curve over a finite field. First, we propose two explicit attack models by injecting a one-bit fault in a given divisor. Then, we discuss the construction of an invalid curve based on the faulted divisor. Our attacks are based on the fact that the Hyperelliptic Curve Scalar Multiplication (HECSM) algorithm does not utilize the curve parameters and We consider three hyperelliptic curves as the attack targets. For curve with security level 186 (in bits), our attack method can get the weakest invalid curve with security level 42 (in bits); there are 93 invalid curves with security level less than 50. We also estimate the theoretical probability of getting a weak hyperelliptic curve whose cardinality is a smooth integer. Finally, we show that the complexity of the fault attack is subexponential if the attacker can freely inject a fault in the input divisor. Cryptosystems based on the genus 2 hyperelliptic curves cannot work against our attack algorithm in practice.
文摘Let Fq be a finite field. In this paper, a construction of Cartesian au-thentication codes from the normal form of a class of nilpotent matrices over the field Fq is presented. Moreover, assume that the encoding rules are chosen according to a uniform probability distribution, the probabilities PI and PS, of a successful im-personation attack and of a successful substitution attack respectively, of these codes are also computed.
基金Supported by the National Natural Science Foundation of China(11926344)。
文摘n-to-1 mappings have many applications in combinatorial design,coding theory and cryptography.In this paper,by using piecewise method and monomials on subsets of q+1-th roots of unity,we show a class of n-to-1 binomials having the form x^(r)(a+x^(s(q-1)))over F_(q^(2)).
基金Supported by the National Natural Science Foundation of China(12026224)
文摘Let s be a positive integer,p be an odd prime,q=p^(s),and let F_(q)be a finite field of q elements.Let N_(q)be the number of solutions of the following equations:(x_(1)^(m_(1))+x_(2)^(m_(2))+…+x_(n)^(m_(n)))^(k)=x_(1)x_(2)…x_(n)x^(k_(n+1))_(n+1)…x^(k_(t))_(t)over the finite field F_(q),with n≥2,t>n,k,and k_(j)(n+1≤j≤t),m_(i)(1≤i≤n)are positive integers.In this paper,we find formulas for N_(q)when there is a positive integer l such that dD|(p^(l)+1),where D=1 cm[d_(1),…,d_(n)],d=gcd(n∑i=1M/m_(i)-kM,(q-1)/D),M=1 cm[m_(1),…,m_(n)],d_(j)=gcd(m_(j),q-1),1≤j≤n.And we determine N_(q)explicitly under certain cases.This extends Markoff-Hurwitz-type equations over finite field.
基金Supported by Foundation of Sichuan Tourism University(20SCTUTY01)Initial Scientific Research Fund of Doctors in Sichuan Tourism University。
文摘Projective Reed-Solomon code is an important class of maximal distance separable codes in reliable communication and deep holes play important roles in its decoding.In this paper,we obtain two classes of deep holes of projective Reed-Solomon codes over finite fields with even characteristic.That is,let F_(q) be finite field with even characteristic,k∈{2,q-2},and let u(x)be the Lagrange interpolation polynomial of the first q components of the received vector u∈F_(q)+1 q Suppose that the(q+1)-th component of u is 0,and u(x)=λx^(k)+f_(≤k-2)(x),λx^(q-2)+f_(≤k-2)(x),where λ∈F^(*)_(q) and f_(≤k-2)(x)is a polynomial over F_(q) with degree no more than k-2.Then the received vector u is a deep hole of projective Reed-Solomon codes PRS(F_(q),k).In fact,our result partially solved an open problem on deep holes of projective Reed-Solomon codes proposed by Wan in 2020.
基金Supported by the National Natural Science Foundation of China (70671096)Jiangsu Teachers University of Technology (KYY08004,KYQ09002)
文摘By using the minimal polynomial of ergodic matrix and the property of polynomial over finite field,we present a polynomial time algorithm for the two-side exponentiation problem about ergodic matrices over finite field （TSEPEM）,and analyze the time and space complexity of the algorithm.According to this algorithm,the public key scheme based on TSEPEM is not secure.
基金Supported partially by the Key Program of Universities of Henan Province(17A110010)Science and Technology Department of Henan Province(152300410180,142300410107,182102210379)+1 种基金China Postdoctoral Science Foundation Funded Project(2016M602251)the National Natural Science Foundation of China(11501387,U1504105)
文摘Let Fq stand for the finite field of odd characteristic p with q elements（q=pn,n∈N）and Fq＊ denote the set of all the nonzero elements of Fq.In this paper,by using the augmented degree matrix and the result given by Cao,we obtain a formula for the number of rational points of the following equation over Fq：f（x 1,x 2,...,x n）=（a1 x1 x2d＋a2 x2 x3d...＋a（n-1）x（n-1）xnd＋an xn x1d）λ-bx1（d1）x2d2...xn（dn）,with ai,b∈Fq＊,n≥2,λ〉0 being positive integers,and d,di being nonnegative integers for 1≤i n.This technique can be applied to the polynomials of the form h1λ=h2 with λ being positive integer and h1,h2∈Fq[x 1,x 2,...,x n].It extends the results of the Markoff-Hurwitz-type equations.
文摘Let F q be a finite field with qelements where q=p~α. In the present paper, the authors study the existence and structure of Carter subgroups of singular symplectic group Sp (n+t,n)(F q), singular unitary group U (n+t,n)(F (q^2)) and singular orthogonal group O (n+t,n)(F q)(n is even) over finite fields F q.
基金Supported by the National Natural Science Foundation of China（10771023）
文摘In the present paper, we compute the number of the symplectic involaLions over the finite field F with chafF = 2, and also one Cartesian authentication code is obtained.Furthermore, its size parameters are computed completely. If assume that the coding rules are chosen according to a uniform probability, PI and Ps denote the largest probabilities of a successful impersonation attack and a successful substitution attack respectively, then PI and Ps are also computed.
基金This work is supported by project number 1998-015-D00015.
文摘This paper proves that if qn is large enough, for each element a and primitive element b of Fq, there etists a primitive polynomial of degree n ≥5 over the finite field Fq having a as the coefficient of xn-1 and b as the constant term. This proves that if qn is large enongh, for each element a ∈Fq, there exists a primitive polynomial of degree n ≥ 5 over Fq having a as the coefficient of x.
文摘By establishing the connection between graph colouring and the solution of some equation systems in finite fields, we obtain some formulas to the number of solutions of some equation systems in finite fields, in terms of chromatic polynomial of a graph.
文摘A （t, n）--secret sharing scheme is a method of distribution of information among n participants such that t 〉 1 can reconstruct the secret but （t - 1） cannot. We explore some （k, n）--secret sharing schemes based on the finite fields.
文摘In this paper, we study about trigonometry in finite field, we know that , the field with p elements, where p is a prime number if and only if p = 8k + 1 or p = 8k -1. Let F and K be two fields, we say that F is an extension of K, if K⊆F or there exists a monomorphism f: K→F. Recall that , F[x] is the ring of polynomial over F. If (means that F is an extension of K), an element is algebraic over K if there exists such that f(u) = 0 (see [1]-[4]). The algebraic closure of K in F is , which is the set of all algebraic elements in F over K.
基金supported by the Natural Science Foundation of Fujian Province,China under Grant No.2022J02046Fujian Key Laboratory of Granular Computing and Applications (Minnan Normal University)Institute of Meteorological Big Data-Digital Fujian and Fujian Key Laboratory of Data Science and Statistics。
文摘Wan and Zhang(2021) obtained a nontrivial lower bound for the number of zeros of complete symmetric polynomials over finite fields,and proposed a problem whether their bound can be improved.In this paper,the author improves Wan-Zhang's bound from three aspects.The proposed results are based on the estimates related to the number of certain permutations and the value sets of non-permutation polynomials associated to the complete symmetric polynomial.And the author believes that there are still possibilities to improve the bounds and hence Wan-Zhang's bound.
基金supported by the National Natural Science Foundation of China under Grant Nos.62373178,62273201,and 62103176the Research Fundfor the Taishan Scholar Project of Shandong Province of China under Grant Nos.tstp20221103 and tstp20221103。
文摘In this paper,the approximate synchronization of leader-follower multiagent systems(MASs) over finite fields is studied in regard to local and global synchronization.First,the approximately synchronous state set(ASSS) is obtained.Second,combined with ASSS and transient periods,some criteria for the local and global approximate synchronization of systems are given.Moreover,the algorithms for calculating the maximum approximately synchronous basin(MASB) and the maximum control invariant set(MCIS) are presented.Third,the global approximate synchronization of the system is achieved by designing the state feedback control,and a design algorithm of the controller using the truth matrix method is proposed.Moreover,the results of approximate synchronization are degenerated to complete synchronization.Last,two examples are shown to demonstrate the results of this paper.
