Probability advantages of two linear expressions in symmetric ciphers
Zhang Hai-na, Shaohui Wang, Xiaoyun Wang · IACR Cryptology ePrint Archive · 2006
The paper proves the probability advantages of two linear expressions which are summarized from the ABC stream cipher submitted to ECRPYT Estream Project.Two linear expressions with probability advan- tages reflect the linear correlations among Modular Addition equations.Corresponding to each linear expression and its advantage,a large amount of weak keys are derived under which all the ABC main keys can be re- trieved successively.The first linear expression is a generic bit linear correlation between two Modular Addi- tion equations.The second is a linear correlation of bit carries derived from three Modular Addition equations and the linearequation of LFSR in ABC. It is remarked that the second is found by Wu and Preneel,and has been used to find 2% weak keys.In the cryptanalysis of ABC,Wu and Preneel only utilized its estimated probability advantage which is concluded by experimental data,and they did not give its strict proof. Modular Addition and XOR operations are widely used in designing symmetric ciphers.It is believed that these types of linear expressions with probability advantages not only can be used to analyze some other sym- metric ciphers,but also are important criteria in designing secure symmetric ciphers.