Study On Several Problems In The Advanced Encryption Standard

An investigation of the Rijndael algorithm which is the Advanced Encryption Standard of USA is taken in this thesis. We have focused on developing the cryptographic properties of the Rijndael Sbox from the viewpoint of Boolean function, Walsh spectrum and algebraic structures, on the attacks against the reduced variants of Rijndael, and on the optimized implementations of Rijndael. The key contributions follow below.1.Two new approaches to determine the algebraic representation of a Boolean function are proposed with their applications on the DES and Rijndael Sboxes.2. Based on the theory of Walsh spectrum, such properties as linearity, strict avalanche, propagation and correlation immunity are analysed.3. The calculating table of the autocorrelation functions are constructed and their essence and abilities are revealed. Furthermore, the new concept of general autocorrelations is presented for the calculation of the order of SAC(strict avalanche criteria) and PC(propagation criteria).4. Based on the partition of equivalence classes, the resolving of linear equations and the calculation of the dual basis, three methodologies are proposed to determine the algebraic representations of coordinates of finite field elements with the elements themselves as the variable. With those above, an essential explanation to the simplicity of the representation of Rijndael S-box is given, and a direct proof to the equivalence between any two coordinate functions of Rijndael S-box is provided, with the equivalence described by only one matrix of order eight over GF(28).5. A fact is pointed out that no proper A set could be built in the Square-6 attack and therefore the overall attack would fail without doubt. Based on the technique of the partial sums, a correctional Square-6 attack independent of the initial round key is described.6. It is discovered that the change of the activity always happens before the change of the balance, which builds the foundation for a new activity attack based on the birthday paradox. Better performance are gained when the properties of balance and activity are combined in an attack of reduced variants of Rijndael.7. The optimisations of Rijndael Sbox, of ColumnMix and its inverse, and of the round trans formation are thoroughly studied for purpose of better performance. Meanwhile, the implementation of such block cipher as Rijndael with shifting registers is designed with performance as good as that of he method of table-lookup.