Yusong Du

Identification and construction of Boolean functions with maximum algebraic immunity

AbstractBoolean functions with maximum algebraic immunity have been considered as one class of cryptographically significant functions. It is known that Boolean functions on odd variables have maximum algebraic immunity if and only if a correlative matrix has column full rank, and Boolean functions on even variables have maximum algebraic immunity if and only if two correlative matrices have column full rank. Recently, a smaller matrix was used in the odd case. We find that one or two smaller matrices can be used in the even case and consequently present several sufficient and necessary conditions for Boolean functions with maximum algebraic immunity. This result advances the ability to identify whether Boolean functions on even variables achieve maximum algebraic immunity. We also present a construction algorithm for n-variable Boolean functions with maximum algebraic immunity, specially with the Hamming weights of

Construction of Boolean functions with maximum algebraic immunity and count of their annihilators at lowest degree

\sum {_{i = 0}^{\left\lceil {\frac{n} {2}} \right\rceil - 1} } \left( {\begin{array}{*{20}c} n \\ i \\ \end{array} } \right)

A rejection sampling algorithm for off-centered discrete Gaussian distributions over the integers

. It is easily realized for not too large n and helps construct balanced Boolean functions with maximum algebraic immunity on even variables. Furthermore, we present a sufficient and necessary condition for balanced Boolean functions to achieve maximum algebraic immunity and optimum algebraic degree, and modify the construction algorithm to construct Boolean functions on odd variables with maximum algebraic immunity, optimum algebraic degree and high nonlinearity.

Fair Electronic Voting via Bitcoin Deposits

Boolean functions with large algebraic immunity resist algebraic attacks to a certain degree, but they may not resist fast algebraic attacks (FAAs). It is necessary to study the resistance of Boolean functions against FAAs. In this paper, we localize the optimal resistance of Boolean functions against FAAs and introduce the concept of e-fast algebraic immunity (e-FAI) for n-variable Boolean functions against FAAs, where e is a positive integer and

On Computing the Immunity of Boolean Power Functions Against Fast Algebraic Attacks

1 \leq e<\lceil\frac{n}{2}\rceil

Identity Based Threshold Ring Signature from Lattices

. We give the sufficient and necessary condition of e-FAI. With e-FAI the problem of deciding the resistance of an n-variable Boolean function against FAAs can be converted into the problem of observing the properties of one given matrix. An algorithm for deciding e-FAI and the optimal resistance against FAAs is also described.

On designated-weight Boolean functions with highest algebraic immunity

Boolean functions used in stream ciphers against algebraic attacks are required to have a necessary cryptographic property-high algebraic immunity (AI). In this paper, Boolean functions of even variables with the maximum AI are investigated. The number of independent annihilators at the lowest degree of Boolean functions of even variables with the maximum AI is determined. It is shown that when n is even, one can get an (n + 1)-variable Boolean function with the maximum AI from two n-variable Boolean functions with the maximum AI only if the Hamming weights of the two functions satisfy the given conditions. The nonlinearity of the Boolean functions obtained in this way is computed. Similarly, one can get an (n + 2)-variable Boolean function with the maximum AI from four n-variable Boolean functions with the maximum AI. The nonlinearity of a class of Boolean functions with the maximum AI is determined such that their Hamming weights are either the maximum or the minimum.

Self-bilinear Map from One Way Encoding System and Indistinguishability Obfuscation.

We propose a non-delegatable strong designated verifier signature on elliptic curves. The size of the signature is less than 500 bits considering an 80 bits security strength. It provably satisfies the non-delegatability and signer ambiguity properties. The construction method is a combination of the Schnorr signature and the elliptic curve Diffie-Hellman problem.