Construction And Application Of High Nonlinear Strong S-Box Based On Hyper Chaos

With the rapid development of information technology,information security has attracted wide attention.To improve the security of information,chaotic cryptography has become one of the research hotspots.As the only nonlinear component,an S-Box is widely used in encryption algorithms such as AES,SM4.0 and Whirlpool,which can enhance the security strength of the cryptosystem by improving the design level of an S-Box.The chaotic system can be very suitable for constructing S-Boxes due to its ergodicity,highly sensitive dependence on initial conditions and nonlinearity.Therefore,the construction of chaotic S-Boxes has become a hot research topic in the field of information security.This paper primarily investigates the construction of non-degenerate multi-dimensional hyper chaotic maps,the construction of strong S-Boxes with high nonlinearity based on hyper chaos and its application.To eliminate the weaknesses of 1-dimensional chaotic maps,a non-degenerate multi-dimensional hyper chaotic map can be constructed to enhance its dynamic performance.Based on the excellent properties of a non-degenerate hyper chaotic map,the S-Box with high nonlinearity can be constructed and applied to parallel irreversible key expansion algorithm.The main research contents of this paper are as follows:（1）A non-degenerate multi-dimensional digital domain hyper chaotic map is constructed.First,we design a general n-dimensional（n≥2）discrete hyper chaotic map（nD-DHCM）model that can generate non-degenerate n-dimensional discrete chaotic map with Lyapunov exponents of any desired size.Furthermore,to verify the effectiveness of the nD-DHCM,we provide two illustrative examples such as a 6D-DHCM and a 7D-DHCM,and analyze their dynamic characteristics.In addition,due to the finite precision effect of the simulation platform,we further analyze the relationship between the randomness of a chaotic sequence and the size of a Lyapunov exponent of the nD-DHCM.Finally,to evaluate its practicability,a keyed parallel Hash function is designed based on a 6D-DHCM,and its performance can be used to evaluate by the simulation experiment.（2）A non-degenerate multi-dimensional integer domain hyper chaotic map is constructed.First,a non-degenerate n-dimensional（n≥2）integer domain chaotic map（nD-IDCM）model is proposed,which can construct n-dimensional integer domain hyper chaotic maps.The nD-IDCM generates chaotic sequences in integer domain,which fundamentally solves the problem of finite precision effect when digital domain chaotic maps are implemented by computers or digital devices.Furthermore,to verify the effectiveness of the nD-IDCM,we construct a 3D-IDCM and a 6D-IDCM,and compare them to digital domain chaotic maps.Finally,to evaluate its practicability,a keyed pseudo random number generator（PRNG）is designed based on a 3D-IDCM,and its performance can be evaluated by a series of statistical tests.（3）S-Boxes with high nonlinearity are constructed based on hyper chaos.Cryptanalysis results of the S-Box construction in AES revealed that,（1）the number of irreducible polynomials can be extend to 30,（2）if we ignore the existence of fixed point（s）and reverse fixed point（s）in an S-Box,the affine transformation constant c can select from[01,FF],and（3）the S-Box in AES is fixed,which may be an exploitable weakness by attacker.Based on the above analysis,first we construct a non-degenerate 2-dimensional enhanced Quadratic map（2D-EQM）with ergodicity and better randomness,and then use it to generate different affine transformation matrices and affine transformation constants under different initial conditions,and construct keyed strong S-Boxes based on 30 seed S-Boxes with high nonlinearity.Furthermore,we use 6 evaluation criteria to analyze the performance of the generated S-Boxes.Finally,we count the number of S-Boxes with the nonlinearity of 112generated by the proposed algorithm,about 4.0913×10²²,among which about 5.4414×10²⁰strong S-Boxes still keep the nonlinearity of 112.In addition,based on a seed S-Box with the nonlinearity of 116,the proposed algorithm can also quickly generate a strong S-Box with no weaknesses and the nonlinearity of 116.Experimental results verify the effectiveness and practicability of the proposed algorithm.（4）A parallel irreversible key expansion algorithm is designed based on hyper chaos and a dynamic S-Box.Cryptanalysis results of key expansion algorithms in AES revealed that:（1）there exist weaknesses in the S-Box,（2）the key expansion algorithm is serial,and（3）the round key expansion algorithm is reversible,i.e.,the initial key can be recovered from any round key,which may be an exploitable weakness by attacker.To solve the above problems,first a non-degenerate 2-dimensional exponential hyper chaotic map（2D-ECM）is constructed,and the experimental analysis results show that it has better dynamic characteristics.A general parallel irreversible key expansion algorithm is designed based on the 2D-ECM and a dynamic S-Box.Security and statistical analysis results demonstrate that the algorithm can transform the initial key into relatively independent round keys,and resist side channel power attacks and differential attacks.