Research On Some Topics Of Cryptographic Functions

Cryptographic functions especially Boolean functions play an important part in stream cipher and block cipher in private-key cryptosystem. Stream cipher can be rep-resented as a single-output Boolean function and block cipher can be represented as a multi-output Boolean function. Hence Boolean functions occupy an important position in cryptography. In this dissertation, we mainly research on some topics of crypto-graphic functions, and we get following results:1.We discuss the relationship between algebraic immunity and nonlinearity of Boolean functions.We give a result about the lower bound on nonlinearity with al-gebraic immunity, and a result about the upper bound on nonlinearity with algebraic immunity under some conditions.Comparing with previous related papers,our results are better than known results.2.We give a new lower bound on the rth(r≥2) order nonlinearity of Boolean functions via the algebraic immunity, which is better than previous known results.We also give a lower bound on the rth order nonlinearity of Boolean functions via the algebraic immunity and Hamming weight for the first time.3.We give a new construction of Boolean functions with maximum algebraic immunity on even number of variables, we also give a construction of balanced rota-tion symmetric Boolean functions with maximum algebraic immunity on even number of variables for the first time.We use some results of linear algebra and enumerative combinatorics in our constructions.These functions have strong resistance against al-gebraic attacks.The balanced rotation symmetric Boolean functions constructed can also be used in the construction of safer hashing functions.4.We obtain a construction of Boolean functions with maximum algebraic immu-nity on finite fields(with characteristic 2).5.we present partial results towards the conjecture "nonexistence of homogeneous rotation symmetric bent functions having degree>2", our results are better than known results. 6.We get a new result between the algebraic immunity and nonlinearity of q(q> 2)-valued functions on general finite fields for the first time.And we also give a result about the rth order nonlinearity of q(q>2)-valued functions via the algebraic immunity for the first time.7.We present a construction of general bent functions, and we also give the equiv-alence forms between the function values and its coefficients for q valued functions on finite fields.