On The Cryptographic Properties And Security Analysis Of T-functions

T-functions firstly proposed by Klimov and Shamir in 2002 are a new class of nonlinear cryptographic functions.T-functions combine basic algebraic operations with logical operations that are highly efficient for modern processors.They play an important role in cryptography for their fast speed,high efficiency,long period and complex nonlinear structure.T-functions are a new attempt in the field of nonlinear research,and also establish a bridge for the cryptographic components from linear to nonlinear.In this paper,we study the cryptographic properties and analyze the security of stream cipher based on T-functions.The main conclusions are as follows:Firstly,research on the autocorrelation properties of the sequences generated by the single cycle T-functions.(1)This paper introduces the autocorrelation properties of the sequences generated by single cycle T-functions,and points out the flaws about them existing in the conclusions got by Yan Wang et al.(2)This paper studies the autocorrelation properties of the coordinate sequences generated by single cycle polynomial T-functions.According to the conjecture provided by Kolokotronis in 2009 regarding the values of the autocorrelation function when the shift is the 1/8 period,we find the counterexamples which prove that the conjecture does not hold in general.Then we use the p-adic analysis theory to give the autocorrelation calculation formula with which the conjecture holding conditions are given.Secondly,research on the properties of uniformly differentiable T-functions.(1)This paper studies the invertibility and single cycle of the uniformly differentiable T-functions.The necessary and sufficient conditions for the invertibility of the modulo 2 uniformly differentiable T-functions and the necessary and sufficient conditions for the single cycle of the modulo 4 uniformly differentiable T-functions are given.Moreover,we study a kind of T-functions with excellent cryptographic properties,and give their necessary and sufficient conditions for being invertibility and single cycle respectively.The cryptographic properties of T-functions are easier to be explained and judged.And they also provide an important reference for further research on other types of T-functions.(2)The relationships between parameters and uniform differentiability are studied.At present,there are two main methods on T-functions studying,one is the traditional T-functions theory which uses parameters as its tool,the other is the non-Archimedes T-functions theory which uses the uniform differentiability.The two theories have their own advantages.In this paper,we descript the definitions and properties of parameters in the non-Archimedes theory,and establish the relationships between the two kinds of theories by means of uniform differentiability,which provides new ideas for the research of T-functions.(3)The generalized entropy preservability of the most significant bit sequences generated by modulo 4 uniformly differentiable T-functions is studied.For the modulo 4 uniformly differentiable single cycle T-functions,the generalized entropy preservability of the most significant bit sequences is equivalent to the generalized entropy preservability of sequences in small scale.Moreover,we present the conditions for generalized entropy preservability of most significant bit sequences and show the transitivity.Thirdly,research on the security of stream cipher based on T-functions.This paper analyzes the security of synchronous stream cipher ABC based on T-functions against differential fault attack.Firstly,according to the characteristics of ABC stream cipher,in order to make full use of the differential fault properties,we establish the bit oriented fault model.By injecting single bit fault into the internal state,we establish the intermediate state equations in different time.Then,we use the properties of the single cycle T-functions to design the algorithm of solving the equations,and determine the values bit by bit.The analysis results show that ABC is not secure against differential fault attack.This attack algorithm can further explore the design weakness of ABC and has an important reference to other stream cipher based on T-functions.