Functions with difference uniformity have important applications in cryptography. Some planar functions and almost perfect nonlinear（APN） functions are presented in the note. In addition, an upper bound of the unifo...Functions with difference uniformity have important applications in cryptography. Some planar functions and almost perfect nonlinear（APN） functions are presented in the note. In addition, an upper bound of the uniformity of some power mappings is provided by using an interesting identity on Dickson polynomials. When the character of the finite field is less than 11, the upper bound is proved to be the best possibility.展开更多
A new construction of constant-composition codes based on all known perfect nonlinear functions from Fqm to itself is presented, which provides a kind of unified constructions of constant-composition codes based on al...A new construction of constant-composition codes based on all known perfect nonlinear functions from Fqm to itself is presented, which provides a kind of unified constructions of constant-composition codes based on all known perfect nonlinear functions from F qm to itself.It is proved that the new constant-composition codes are optimal with respect to the Luo-Fu-Vinck-Chen bound, when m is an odd positive integer greater than 1.Finally, we point out that two constructions of constant-composition codes, proposed by Ding Cunsheng et al.in 2005, are equivalent to two special types of the new constant-composition codes.展开更多
Let F_(p)^(m) be a finite field with p^(m) elements,where p is an odd prime and m is a positive integer.Recently,[17]and[35]determined the weight distributions of subfield codes with the form C f={((T r(a f(x)+b x)+c)...Let F_(p)^(m) be a finite field with p^(m) elements,where p is an odd prime and m is a positive integer.Recently,[17]and[35]determined the weight distributions of subfield codes with the form C f={((T r(a f(x)+b x)+c)_(x∈F_(p)^(m)),T r(a)):a,b∈F_(p)^(m),c∈F_(p)}for f(x)=x^(2) and f(x)=x p k+1,respectively,where Tr(⋅)is the trace function from F_(p)^(m) to F_(p),and k is a nonnegative integer.In this paper,we further investigate the subfield code C f for f(x)being a known perfect nonlinear function over F_(p)^(m) and generalize some results in[17,35].The weight distributions of the constructed codes are determined by applying the theory of quadratic forms and the properties of perfect nonlinear functions over finite fields.In addition,the parameters of the duals of these codes are also determined.Several examples show that some of our codes and their duals have the best known parameters according to the code tables in[16].The duals of some proposed codes are optimal according to the Sphere Packing bound if p≥5.展开更多
We study the functions with low differential uniformity,and concentrates mainly on the properties of perfect nonlinear（PN） functions,including the properties of the derivative of the components of those functions.So...We study the functions with low differential uniformity,and concentrates mainly on the properties of perfect nonlinear（PN） functions,including the properties of the derivative of the components of those functions.Some sufficient and necessary conditions have been explored to judge when a function is a PN function.These conditions may be useful in constructing new PN functions.We also construct some functions with differential 4-uniformity that have rarely been studied in the literature.Some of the constructed functions with differential 4-uniformity have high nonlinearity as well.Finally,a class of functions with differential 4-uniformity which are not extended affine equivalent to any power functions are constructed.展开更多
In this paper, we propose a construction of functions with low differential uniformity based on known perfect nonlinear functions over finite fields of odd characteristic. For an odd prime power q, it is proved that t...In this paper, we propose a construction of functions with low differential uniformity based on known perfect nonlinear functions over finite fields of odd characteristic. For an odd prime power q, it is proved that the proposed functions over the finite field Fq are permutations if and only if q≡3(mod 4).展开更多
Perfect adaptation describes the ability of a biological system to restore its biological function precisely to the pre-perturbation level after being affected by the environmental disturbances.Mathematically,a biolog...Perfect adaptation describes the ability of a biological system to restore its biological function precisely to the pre-perturbation level after being affected by the environmental disturbances.Mathematically,a biological system with perfect adaptation can be modelled as an input-output nonlinear system whose output,usually determining the biological function,is asymptotically stable under all input disturbances concerned.In this paper,a quite general input-output mathematical model is employed and the 'functional' of biological function(FBF)- output Lyapunov function- is explored to investigate its perfect adaptation ability.Sufficient condition is established for the systems with FBF to achieve perfect adaptation.Then a sufficient and necessary condition is obtained for the linear systems to possess an output Lyapunov function.Furthermore,it is shown that the 'functional'of receptors activity exists in the perfect adaptation model of E.coh chemotaxis.Different with the existing mathematical surveys on perfect adaptation,most of which are based on the standpoint of control theory,we first investigate this problem using ways of nonlinear systems analysis.展开更多
In this paper,we propose a conjecture that endogenous security without any prior knowledge is similar to perfect secrecy without any prior knowledge.To prove the conjecture,we first establish a cryptography model of i...In this paper,we propose a conjecture that endogenous security without any prior knowledge is similar to perfect secrecy without any prior knowledge.To prove the conjecture,we first establish a cryptography model of instinct function security to transform the security problem in the network domain into an encryption problem in the cryptographic domain.Then,we inherit and apply the established ideas and means of Perfect Secrecy,and propose the concept,definition and corollaries of the perfect instinct function security(PIFS)corresponding to Perfect Secrecy.Furthermore,we take the DHR system as a concrete implementation of PIFS and propose the DHR Perfect Security Theorem corresponding to Shannon’s Perfect Secrecy Theorem.Finally,we prove that the DHR satisfying the“OneTime Reconstruction”constraint is the sufficient and necessary condition to achieve perfect security.This means that the existence of PIFS is also proven.The analysis shows that any reconfigurable system can be encrypted by its construct and that the PIFS converts the oneway transparent superiority of the attacker into a double-blind problem for both the attacker and the defender,which leads to that the attacker is impossible to obtain useful construction information from the attacks and unable to find a better way than blind trial-and-error or brute-force attacks.Since the attackers are required to have the new powerful ability to crack the structure cryptogram,the threshold of cyber security is raised to at least the same level as cryptogram deciphering,thereafter the ubiquitous cyber threats are destined to be significantly reduced.展开更多
This paper gives a full classification of Dembowski-Ostrom polynomials derived from the compositions of reversed Dickson polynomials and monomials over finite fields of characteristic 2.The authors also classify almos...This paper gives a full classification of Dembowski-Ostrom polynomials derived from the compositions of reversed Dickson polynomials and monomials over finite fields of characteristic 2.The authors also classify almost perfect nonlinear functions among all such Dembowski-Ostrom polynomials based on a general result describing when the composition of an arbitrary linearized polynomial and a monomial of the form x^(2+2^α) is almost perfect nonlinear.It turns out that almost perfect nonlinear functions derived from reversed Dickson polynomials are all extended affine equivalent to the well-known Gold functions.展开更多
文摘Functions with difference uniformity have important applications in cryptography. Some planar functions and almost perfect nonlinear（APN） functions are presented in the note. In addition, an upper bound of the uniformity of some power mappings is provided by using an interesting identity on Dickson polynomials. When the character of the finite field is less than 11, the upper bound is proved to be the best possibility.
基金Supported in part by the National Natural Science Foundation of China (Grant Nos 60573028, 60803156)the Open Research Fund of the National Mobile Communications Research Laboratory of Southeast University (Grant No W200805)in part by Singapore Ministry of Education Academic Research Fund (Grant No T206B2204)
文摘A new construction of constant-composition codes based on all known perfect nonlinear functions from Fqm to itself is presented, which provides a kind of unified constructions of constant-composition codes based on all known perfect nonlinear functions from F qm to itself.It is proved that the new constant-composition codes are optimal with respect to the Luo-Fu-Vinck-Chen bound, when m is an odd positive integer greater than 1.Finally, we point out that two constructions of constant-composition codes, proposed by Ding Cunsheng et al.in 2005, are equivalent to two special types of the new constant-composition codes.
基金This work was supported in part by the National Natural Science Foundation of China(NSFC)under Grants 11971156 and 12001175.
文摘Let F_(p)^(m) be a finite field with p^(m) elements,where p is an odd prime and m is a positive integer.Recently,[17]and[35]determined the weight distributions of subfield codes with the form C f={((T r(a f(x)+b x)+c)_(x∈F_(p)^(m)),T r(a)):a,b∈F_(p)^(m),c∈F_(p)}for f(x)=x^(2) and f(x)=x p k+1,respectively,where Tr(⋅)is the trace function from F_(p)^(m) to F_(p),and k is a nonnegative integer.In this paper,we further investigate the subfield code C f for f(x)being a known perfect nonlinear function over F_(p)^(m) and generalize some results in[17,35].The weight distributions of the constructed codes are determined by applying the theory of quadratic forms and the properties of perfect nonlinear functions over finite fields.In addition,the parameters of the duals of these codes are also determined.Several examples show that some of our codes and their duals have the best known parameters according to the code tables in[16].The duals of some proposed codes are optimal according to the Sphere Packing bound if p≥5.
基金Supported by the National Natural Science Foundation of China (60673068)the Fundamental Research Funds for the Central Universities (2009B27414)the Natural Science Foundation of Hohai University (2084/409270)
文摘We study the functions with low differential uniformity,and concentrates mainly on the properties of perfect nonlinear（PN） functions,including the properties of the derivative of the components of those functions.Some sufficient and necessary conditions have been explored to judge when a function is a PN function.These conditions may be useful in constructing new PN functions.We also construct some functions with differential 4-uniformity that have rarely been studied in the literature.Some of the constructed functions with differential 4-uniformity have high nonlinearity as well.Finally,a class of functions with differential 4-uniformity which are not extended affine equivalent to any power functions are constructed.
基金supported by National Natural Science Foundation of China(Grant Nos.61070172,10990011 and 61170257)the External Science and Technology Cooperation Program of Hubei Province(Grant No.2012IHA01402)+1 种基金National Key Basic Research Program of China(Grant No.2013CB834203)the Strategic Priority Research Program of Chinese Academy of Sciences(Grant No.XDA06010702)
文摘In this paper, we propose a construction of functions with low differential uniformity based on known perfect nonlinear functions over finite fields of odd characteristic. For an odd prime power q, it is proved that the proposed functions over the finite field Fq are permutations if and only if q≡3(mod 4).
文摘Perfect adaptation describes the ability of a biological system to restore its biological function precisely to the pre-perturbation level after being affected by the environmental disturbances.Mathematically,a biological system with perfect adaptation can be modelled as an input-output nonlinear system whose output,usually determining the biological function,is asymptotically stable under all input disturbances concerned.In this paper,a quite general input-output mathematical model is employed and the 'functional' of biological function(FBF)- output Lyapunov function- is explored to investigate its perfect adaptation ability.Sufficient condition is established for the systems with FBF to achieve perfect adaptation.Then a sufficient and necessary condition is obtained for the linear systems to possess an output Lyapunov function.Furthermore,it is shown that the 'functional'of receptors activity exists in the perfect adaptation model of E.coh chemotaxis.Different with the existing mathematical surveys on perfect adaptation,most of which are based on the standpoint of control theory,we first investigate this problem using ways of nonlinear systems analysis.
基金supported by the National Natural Science Foundation of China(No.U22A2001)the National Key Research and Development Program under Grants 2022YFB2902205
文摘In this paper,we propose a conjecture that endogenous security without any prior knowledge is similar to perfect secrecy without any prior knowledge.To prove the conjecture,we first establish a cryptography model of instinct function security to transform the security problem in the network domain into an encryption problem in the cryptographic domain.Then,we inherit and apply the established ideas and means of Perfect Secrecy,and propose the concept,definition and corollaries of the perfect instinct function security(PIFS)corresponding to Perfect Secrecy.Furthermore,we take the DHR system as a concrete implementation of PIFS and propose the DHR Perfect Security Theorem corresponding to Shannon’s Perfect Secrecy Theorem.Finally,we prove that the DHR satisfying the“OneTime Reconstruction”constraint is the sufficient and necessary condition to achieve perfect security.This means that the existence of PIFS is also proven.The analysis shows that any reconfigurable system can be encrypted by its construct and that the PIFS converts the oneway transparent superiority of the attacker into a double-blind problem for both the attacker and the defender,which leads to that the attacker is impossible to obtain useful construction information from the attacks and unable to find a better way than blind trial-and-error or brute-force attacks.Since the attackers are required to have the new powerful ability to crack the structure cryptogram,the threshold of cyber security is raised to at least the same level as cryptogram deciphering,thereafter the ubiquitous cyber threats are destined to be significantly reduced.
基金supported by the National Basic Research Program of China under Grant No.2011CB302400
文摘This paper gives a full classification of Dembowski-Ostrom polynomials derived from the compositions of reversed Dickson polynomials and monomials over finite fields of characteristic 2.The authors also classify almost perfect nonlinear functions among all such Dembowski-Ostrom polynomials based on a general result describing when the composition of an arbitrary linearized polynomial and a monomial of the form x^(2+2^α) is almost perfect nonlinear.It turns out that almost perfect nonlinear functions derived from reversed Dickson polynomials are all extended affine equivalent to the well-known Gold functions.
