Fast Algebraic Attacks and Decomposition of Symmetric Boolean Functions

Algebraic and fast algebraic attacks are power tools to analyze stream ciphers. A class of symmetric Boolean functions with maximum algebraic immunity were found vulnerable to fast algebraic attacks at EUROCRYPT'06. Recently, the notion of AAR (algebraic attack resistant) functions was introduced as a unified measure of protection against both classical algebraic and fast algebraic attacks. In this correspondence, we first give a decomposition of symmetric Boolean functions, then we show that almost all symmetric Boolean functions, including these functions with good algebraic immunity, behave badly against fast algebraic attacks, and we also prove that no symmetric Boolean functions are AAR functions. Besides, we improve the relations between algebraic degree and algebraic immunity of symmetric Boolean functions.Comment: 13 pages, submitted to IEEE Transactions on Information Theor

On the Boolean functions With Maximum Possible Algebraic Immunity : Construction and A Lower Bound of the Count

This paper gives a construction method which can get a large class of Boolean functions with maximum algebraic immunity(AI) from one such giving function. Our constructions get more functions than any previous construction. The cryptographic properties, such as balance, algebraic degree etc, of those functions are studied. It shows that we can construct Boolean functions with better cryptographic properties, which gives the guidance for the design of Boolean functions to resist algebraic attack, and helps to design good cryptographic primitives of cryptosystems. From these constructions, we show that the count of the Boolean functions with maximum AI is bigger than ${2^{2^{n-1}}}$ for $n$ odd, bigger than ${2^{2^{n-1}+\frac{1}{2}\binom{n}{\frac{n}{2}} }}$ for $n$ even, which confirms the computer simulation result that such boolean functions are numerous. As far as we know, this is the first bound about this count

Modifying Boolean Functions to Ensure Maximum Algebraic Immunity

The algebraic immunity of cryptographic Boolean functions is studied in this paper. Proper modifications of functions achieving maximum algebraic immunity are proved, in order to yield new functions of also maximum algebraic immunity. It is shown that the derived results apply to known classes of functions. Moreover, two new efficient algorithms to produce functions of guaranteed maximum algebraic immunity are developed, which further extend and generalize known constructions of functions with maximum algebraic immunity