Elliptic curves(ECs)are deemed one of the most solid structures against modern computational attacks because of their small key size and high security.In many well-known cryptosystems,the substitution box(Sbox)is used...Elliptic curves(ECs)are deemed one of the most solid structures against modern computational attacks because of their small key size and high security.In many well-known cryptosystems,the substitution box(Sbox)is used as the only nonlinear portion of a security system.Recently,it has been shown that using dynamic S-boxes rather than static S-boxes increases the security of a cryptosystem.The conferred study also extends the practical application of ECs in designing the nonlinear components of block ciphers in symmetric key cryptography.In this study,instead of the Mordell elliptic curve(MEC)over the prime field,the Galois field has been engaged in constructing the S-boxes,the main nonlinear component of the block ciphers.Also,the proposed scheme uses the coordinates of MEC and the operation of the Galois field to generate a higher number of S-boxes with optimal nonlinearity,which increases the security of cryptosystems.The proposed S-boxes resilience against prominent algebraic and statistical attacks is evaluated to determine its potential to induce confusion and produce acceptable results compared to other schemes.Also,the majority logic criteria(MLC)are used to assess the new S-boxes usage in the image encryption application,and the outcomes indicate that they have significant cryptographic strength.展开更多
Elliptic curve cryptography has been used in many security systems due to its small key size and high security compared with other cryptosystems. In many well-known security systems, a substitution box (S-box) is the ...Elliptic curve cryptography has been used in many security systems due to its small key size and high security compared with other cryptosystems. In many well-known security systems, a substitution box (S-box) is the only non-linear component. Recently, it has been shown that the security of a cryptosystem can be improved using dynamic S-boxes instead of a static S-box. This necessitates the construction of new secure S-boxes. We propose an efficient method to generate S-boxes that are based on a class of Mordell elliptic curves over prime fields and achieved by defining different total orders. The proposed scheme is devel-oped in such a way that for each input it outputs an S-box in linear time and constant space. Due to this property, our method takes less time and space than the existing S-box construction methods over elliptic curves. Computational results show that the pro-posed method is capable of generating cryptographically strong S-boxes with security comparable to some of the existing S-boxes constructed via different mathematical structures.展开更多
For a field F,let Gn(F) = {{a,Φn(a)} ∈ K2(F) | a,Φn(a) ∈ F*},where Φn(x) is the n-th cyclotomic polynomial.At first,by using Faltings' theorem on Mordell conjecture it is proved that if F is a number field an...For a field F,let Gn(F) = {{a,Φn(a)} ∈ K2(F) | a,Φn(a) ∈ F*},where Φn(x) is the n-th cyclotomic polynomial.At first,by using Faltings' theorem on Mordell conjecture it is proved that if F is a number field and if n = 4,8,12 is a positive integer having a square factor then Gn(F) is not a subgroup of K2(F),and then by using the results of Manin,Grauert,Samuel and Li on Mordell conjecture theorem for function fields,a similar result is established for function fields over an algebraically closed field.展开更多
We study the classification of elliptic curves E over the rationals Q according to the torsion subgroups E_(tors)(Q). More precisely, we classify those elliptic curves with E_(tors)(Q) being cyclic with even orders. W...We study the classification of elliptic curves E over the rationals Q according to the torsion subgroups E_(tors)(Q). More precisely, we classify those elliptic curves with E_(tors)(Q) being cyclic with even orders. We also give explicit formulas for generators of E_(tors)(Q). These results, together with the recent results of K. Ono for the non-cyclic E_(tors)(Q), completely solve the problem of the explicit classification and parameterization when E has a rational point of order 2.展开更多
Suppose that E：y2=x(x+M)(x+N) is an elliptic curve, where M<N arerational numbers (≠0,±1) and are relatively prime. Let K be a number field of type (2，…，2) with degree 2n. For arbitraryn, the structure of...Suppose that E：y2=x(x+M)(x+N) is an elliptic curve, where M<N arerational numbers (≠0,±1) and are relatively prime. Let K be a number field of type (2，…，2) with degree 2n. For arbitraryn, the structure of the torsion subgroup E(K)tors of the K-rational points (Mordell group) of E is completely determined here. Explicitly given are the classification, criteria and parameterization, as well as the groups E(K)tors themselves. The order of E(K)tors is also proved to be a power of 2 for any n. Besides, for any elliptic curve E over any number field F, it is shown that E(L)tors=E(F)tors holds for almost all extensions L/F of degree p(a prime number). These results have remarkably developed the recent results by Kwon about torsion subgroups over quadratic fields.展开更多
基金The authors extend their gratitude to the Deanship of Scientific Research at King Khalid University for funding this work through the research groups program under grant number R.G.P.2/109/43.
文摘Elliptic curves(ECs)are deemed one of the most solid structures against modern computational attacks because of their small key size and high security.In many well-known cryptosystems,the substitution box(Sbox)is used as the only nonlinear portion of a security system.Recently,it has been shown that using dynamic S-boxes rather than static S-boxes increases the security of a cryptosystem.The conferred study also extends the practical application of ECs in designing the nonlinear components of block ciphers in symmetric key cryptography.In this study,instead of the Mordell elliptic curve(MEC)over the prime field,the Galois field has been engaged in constructing the S-boxes,the main nonlinear component of the block ciphers.Also,the proposed scheme uses the coordinates of MEC and the operation of the Galois field to generate a higher number of S-boxes with optimal nonlinearity,which increases the security of cryptosystems.The proposed S-boxes resilience against prominent algebraic and statistical attacks is evaluated to determine its potential to induce confusion and produce acceptable results compared to other schemes.Also,the majority logic criteria(MLC)are used to assess the new S-boxes usage in the image encryption application,and the outcomes indicate that they have significant cryptographic strength.
基金Project supported by the JSPS KAKENHI(No.18J23484)
文摘Elliptic curve cryptography has been used in many security systems due to its small key size and high security compared with other cryptosystems. In many well-known security systems, a substitution box (S-box) is the only non-linear component. Recently, it has been shown that the security of a cryptosystem can be improved using dynamic S-boxes instead of a static S-box. This necessitates the construction of new secure S-boxes. We propose an efficient method to generate S-boxes that are based on a class of Mordell elliptic curves over prime fields and achieved by defining different total orders. The proposed scheme is devel-oped in such a way that for each input it outputs an S-box in linear time and constant space. Due to this property, our method takes less time and space than the existing S-box construction methods over elliptic curves. Computational results show that the pro-posed method is capable of generating cryptographically strong S-boxes with security comparable to some of the existing S-boxes constructed via different mathematical structures.
基金supported by the National Natural Science Foundation of China (Grant No.10371061)
文摘For a field F,let Gn(F) = {{a,Φn(a)} ∈ K2(F) | a,Φn(a) ∈ F*},where Φn(x) is the n-th cyclotomic polynomial.At first,by using Faltings' theorem on Mordell conjecture it is proved that if F is a number field and if n = 4,8,12 is a positive integer having a square factor then Gn(F) is not a subgroup of K2(F),and then by using the results of Manin,Grauert,Samuel and Li on Mordell conjecture theorem for function fields,a similar result is established for function fields over an algebraically closed field.
基金Project supported by the National Natural Science Foundation of China (Grant No. 19771052)
文摘We study the classification of elliptic curves E over the rationals Q according to the torsion subgroups E_(tors)(Q). More precisely, we classify those elliptic curves with E_(tors)(Q) being cyclic with even orders. We also give explicit formulas for generators of E_(tors)(Q). These results, together with the recent results of K. Ono for the non-cyclic E_(tors)(Q), completely solve the problem of the explicit classification and parameterization when E has a rational point of order 2.
基金This work was supported by the National Natural Science Foundation of China (Grant No. 19771052) .
文摘Suppose that E：y2=x(x+M)(x+N) is an elliptic curve, where M<N arerational numbers (≠0,±1) and are relatively prime. Let K be a number field of type (2，…，2) with degree 2n. For arbitraryn, the structure of the torsion subgroup E(K)tors of the K-rational points (Mordell group) of E is completely determined here. Explicitly given are the classification, criteria and parameterization, as well as the groups E(K)tors themselves. The order of E(K)tors is also proved to be a power of 2 for any n. Besides, for any elliptic curve E over any number field F, it is shown that E(L)tors=E(F)tors holds for almost all extensions L/F of degree p(a prime number). These results have remarkably developed the recent results by Kwon about torsion subgroups over quadratic fields.
