In this paper, we prove the following results： 1） A normal basis N over a finite field is equivalent to its dual basis if and only if the multiplication table of N is symmetric; 2） The normal basis N is self-dual i...In this paper, we prove the following results： 1） A normal basis N over a finite field is equivalent to its dual basis if and only if the multiplication table of N is symmetric; 2） The normal basis N is self-dual if and only if its multiplication table is symmetric and Tr（α^2） = 1, where α generates N; 3） An optimal normal basis N is self-dual if and only if N is a type-Ⅰ optimal normal basis with q = n = 2 or N is a type-Ⅱ optimal normal basis.展开更多
Intrusion detection is critical to guaranteeing the safety of the data in the network.Even though,since Internet commerce has grown at a breakneck pace,network traffic kinds are rising daily,and network behavior chara...Intrusion detection is critical to guaranteeing the safety of the data in the network.Even though,since Internet commerce has grown at a breakneck pace,network traffic kinds are rising daily,and network behavior characteristics are becoming increasingly complicated,posing significant hurdles to intrusion detection.The challenges in terms of false positives,false negatives,low detection accuracy,high running time,adversarial attacks,uncertain attacks,etc.lead to insecure Intrusion Detection System(IDS).To offset the existing challenge,the work has developed a secure Data Mining Intrusion detection system(DataMIDS)framework using Functional Perturbation(FP)feature selection and Bengio Nesterov Momentum-based Tuned Generative Adversarial Network(BNM-tGAN)attack detection technique.The data mining-based framework provides shallow learning of features and emphasizes feature engineering as well as selection.Initially,the IDS data are analyzed for missing values based on the Marginal Likelihood Fisher Information Matrix technique(MLFIMT)that identifies the relationship among the missing values and attack classes.Based on the analysis,the missing values are classified as Missing Completely at Random(MCAR),Missing at random(MAR),Missing Not at Random(MNAR),and handled according to the types.Thereafter,categorical features are handled followed by feature scaling using Absolute Median Division based Robust Scalar(AMDRS)and the Handling of the imbalanced dataset.The selection of relevant features is initiated using FP that uses‘3’Feature Selection(FS)techniques i.e.,Inverse Chi Square based Flamingo Search(ICS-FSO)wrapper method,Hyperparameter Tuned Threshold based Decision Tree(HpTT-DT)embedded method,and Xavier Normal Distribution based Relief(XavND-Relief)filter method.Finally,the selected features are trained and tested for detecting attacks using BNM-tGAN.The Experimental analysis demonstrates that the introduced DataMIDS framework produces an accurate diagnosis about the attack with low computation time.The work avoids false alarm rate of attacks and remains to be relatively robust against malicious attacks as compared to existing methods.展开更多
Let q be a prime or prime power and Fq^n the extension of q elements finite field Fq with degree n （n 〉 1）. Davenport, Lenstra and School proved that there exists a primitive element α ∈ Fq^n such that α generat...Let q be a prime or prime power and Fq^n the extension of q elements finite field Fq with degree n （n 〉 1）. Davenport, Lenstra and School proved that there exists a primitive element α ∈ Fq^n such that α generates a normal basis of Fq^n over Fq. Later, Mullin, Gao and Lenstra, etc., raised the definition of optimal normal bases and constructed such bases. In this paper, we determine all primitive type I optimal normal bases and all finite fields in which there exists a pair of reciprocal elements α and α^-1 such that both of them generate optimal normal bases of Fq^n over Fq. Furthermore, we obtain a sufficient condition for the existence of primitive type II optimal normal bases over finite fields and prove that all primitive optimal normal elements are conjugate to each other.展开更多
Reed-Solomon (RS) and Bose-Chaudhuri-Hocquenghem (BCH) error correcting codes are widely used in digital technology. An important problem in the implementation of RS and BCH decoding is the fast finding of the error p...Reed-Solomon (RS) and Bose-Chaudhuri-Hocquenghem (BCH) error correcting codes are widely used in digital technology. An important problem in the implementation of RS and BCH decoding is the fast finding of the error positions (the roots of error locator polynomials). Several fast root-finding algorithms for polynomials over finite fields have been proposed. In this paper we give a generalization of the Goertzel algorithm. Our algorithm is suitable for the parallel hardware implementation and the time of multiplications used is restricted by a constant.展开更多
文摘In this paper, we prove the following results： 1） A normal basis N over a finite field is equivalent to its dual basis if and only if the multiplication table of N is symmetric; 2） The normal basis N is self-dual if and only if its multiplication table is symmetric and Tr（α^2） = 1, where α generates N; 3） An optimal normal basis N is self-dual if and only if N is a type-Ⅰ optimal normal basis with q = n = 2 or N is a type-Ⅱ optimal normal basis.
文摘Intrusion detection is critical to guaranteeing the safety of the data in the network.Even though,since Internet commerce has grown at a breakneck pace,network traffic kinds are rising daily,and network behavior characteristics are becoming increasingly complicated,posing significant hurdles to intrusion detection.The challenges in terms of false positives,false negatives,low detection accuracy,high running time,adversarial attacks,uncertain attacks,etc.lead to insecure Intrusion Detection System(IDS).To offset the existing challenge,the work has developed a secure Data Mining Intrusion detection system(DataMIDS)framework using Functional Perturbation(FP)feature selection and Bengio Nesterov Momentum-based Tuned Generative Adversarial Network(BNM-tGAN)attack detection technique.The data mining-based framework provides shallow learning of features and emphasizes feature engineering as well as selection.Initially,the IDS data are analyzed for missing values based on the Marginal Likelihood Fisher Information Matrix technique(MLFIMT)that identifies the relationship among the missing values and attack classes.Based on the analysis,the missing values are classified as Missing Completely at Random(MCAR),Missing at random(MAR),Missing Not at Random(MNAR),and handled according to the types.Thereafter,categorical features are handled followed by feature scaling using Absolute Median Division based Robust Scalar(AMDRS)and the Handling of the imbalanced dataset.The selection of relevant features is initiated using FP that uses‘3’Feature Selection(FS)techniques i.e.,Inverse Chi Square based Flamingo Search(ICS-FSO)wrapper method,Hyperparameter Tuned Threshold based Decision Tree(HpTT-DT)embedded method,and Xavier Normal Distribution based Relief(XavND-Relief)filter method.Finally,the selected features are trained and tested for detecting attacks using BNM-tGAN.The Experimental analysis demonstrates that the introduced DataMIDS framework produces an accurate diagnosis about the attack with low computation time.The work avoids false alarm rate of attacks and remains to be relatively robust against malicious attacks as compared to existing methods.
基金Supported by the National Natural Science Foundation of China (Grant No10990011)Special Research Found for the Doctoral Program Issues New Teachers of Higher Education (Grant No20095134120001)the Found of Sichuan Province (Grant No09ZA087)
文摘Let q be a prime or prime power and Fq^n the extension of q elements finite field Fq with degree n （n 〉 1）. Davenport, Lenstra and School proved that there exists a primitive element α ∈ Fq^n such that α generates a normal basis of Fq^n over Fq. Later, Mullin, Gao and Lenstra, etc., raised the definition of optimal normal bases and constructed such bases. In this paper, we determine all primitive type I optimal normal bases and all finite fields in which there exists a pair of reciprocal elements α and α^-1 such that both of them generate optimal normal bases of Fq^n over Fq. Furthermore, we obtain a sufficient condition for the existence of primitive type II optimal normal bases over finite fields and prove that all primitive optimal normal elements are conjugate to each other.
基金This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 60433050, 90607005)
文摘Reed-Solomon (RS) and Bose-Chaudhuri-Hocquenghem (BCH) error correcting codes are widely used in digital technology. An important problem in the implementation of RS and BCH decoding is the fast finding of the error positions (the roots of error locator polynomials). Several fast root-finding algorithms for polynomials over finite fields have been proposed. In this paper we give a generalization of the Goertzel algorithm. Our algorithm is suitable for the parallel hardware implementation and the time of multiplications used is restricted by a constant.
