Algebraic Attacks on Summation Generators

We apply the algebraic attacks on stream ciphers with memories to the summation generator. For a summation generator that uses $n$ LFSRs, the algebraic equation relating the key stream bits and LFSR output bits can be made to be of degree less than or equal to $2^{\lceil\log_2 n \rceil}$, using $\lceil\log_2 n \rceil + 1$ consecutive key stream bits. This is much lower than the upper bound given by previous general results

Algebraic construction of semi bent function via known power function

The study of semi bent functions (2- plateaued Boolean function) has attracted the attention of many researchers due to their cryptographic and combinatorial properties. In this paper, we have given the algebraic construction of semi bent functions defined over the finite field F₂ⁿ (n even) using the notion of trace function and Gold power exponent. Algebraically constructed semi bent functions have some special cryptographical properties such as high nonlinearity, algebraic immunity, and low correlation immunity as expected to use them effectively in cryptosystems. We have illustrated the existence of these properties with suitable examples.Publisher's Versio

ALGEBRAIC COUNTERMEASURE TO ENHANCE THE IMPROVED SUMMATION GENERATOR WITH 2-BIT MEMORY

Recently proposed algebraic attack has been shown to be very effective on several stream ciphers. In this paper, we have investigated the resistance of PingPong family of stream ciphers against algebraic attacks. This stream cipher was proposed in 2008 to enhance the security of the improved summation generator against the algebraic attack. In particular, we focus on the PingPong-128 stream cipher’s resistance against algebraic attack in this paper. In our analysis, it is found that an algebraic attack on PingPong family of stream ciphers require much more operations compare to the exhaustive key search on the internal state of the LFSRs. It will be shown that due to the irregular and mutual clock controlling in PingPong stream cipher the degree of the generated equation tends to grow up with each successive clock which in turn increases the overall complexity of an algebraic attack. Along with the PingPong 128 stream cipher the other instances of PingPong family stream ciphers are also investigated against the algebraic attack. Our analysis shows that, PingPong family stream ciphers are highly resistant against the algebraic attack due to their mutual and irregular clocking function

Topics on Register Synthesis Problems

Pseudo-random sequences are ubiquitous in modern electronics and information technology. High speed generators of such sequences play essential roles in various engineering applications, such as stream ciphers, radar systems, multiple access systems, and quasi-Monte-Carlo simulation. Given a short prefix of a sequence, it is undesirable to have an efficient algorithm that can synthesize a generator which can predict the whole sequence. Otherwise, a cryptanalytic attack can be launched against the system based on that given sequence. Linear feedback shift registers (LFSRs) are the most widely studied pseudorandom sequence generators. The LFSR synthesis problem can be solved by the Berlekamp-Massey algorithm, by constructing a system of linear equations, by the extended Euclidean algorithm, or by the continued fraction algorithm. It is shown that the linear complexity is an important security measure for pseudorandom sequences design. So we investigate lower bounds of the linear complexity of different kinds of pseudorandom sequences. Feedback with carry shift registers (FCSRs) were first described by Goresky and Klapper. They have many good algebraic properties similar to those of LFSRs. FCSRs are good candidates as building blocks of stream ciphers. The FCSR synthesis problem has been studied in many literatures but there are no FCSR synthesis algorithms for multi-sequences. Thus one of the main contributions of this dissertation is to adapt an interleaving technique to develop two algorithms to solve the FCSR synthesis problem for multi-sequences. Algebraic feedback shift registers (AFSRs) are generalizations of LFSRs and FCSRs. Based on a choice of an integral domain R and π ∈ R, an AFSR can produce sequences whose elements can be thought of elements of the quotient ring R/(π). A modification of the Berlekamp-Massey algorithm, Xu\u27s algorithm solves the synthesis problem for AFSRs over a pair (R, π) with certain algebraic properties. We propose two register synthesis algorithms for AFSR synthesis problem. One is an extension of lattice approximation approach but based on lattice basis reduction and the other one is based on the extended Euclidean algorithm

Algebraic Attacks on Summation Generators

We apply the algebraic attacks on stream ciphers with memories to the summation generator. For a summation generator that uses n LFSRs, an algebraic equation relating the key stream bits and LFSR output bits can be made to be of degree less than or equal to 2 , using dlog 2 ne + 1 consecutive key stream bits. This is much lower than the upper bound given by previous general results. We also show that the techniques of [5] can be applied to summation generators using 2 LFSRs to reduce the eective degree of the algebraic equation