On Hardware Implementation of Tang-Maitra Boolean Functions

Rotation Symmetric Boolean Functions –; Count and Cryptographic Properties

This paper investigates the optimum OBDD representation problem based on two classes of Boolean functions. The first class is defined by OBDDs, in which the number of non-terminal nodes is equal to the number of input variables. We refer to such OBDDs and their corresponding Boolean functions as thin OBDDs and thin Boolean functions. The second class is the thin factored Boolean functions, which is defined by factored forms with each variable appearing exactly once. Many interesting properties of these two classes of functions are presented. Based on which, a revised dynamic shortest cube first OBDD variable ordering algorithm is developed. This algorithm is shown to be optimum for thin factored Boolean functions.

We prove that polynomial size discrete synchronous Hopfield networks with hidden units compute exactly the class of Boolean functions PSPACE/poly, i.e., the same functions as are computed by polynomial space-bounded nonuniform Turing machines. As a corollary to the construction, we observe also that networks with polynomially bounded interconnection weights compute exactly the class of functions P/poly.

Several underlying structural and functional factors that determine the fault behavior of a network are not yet well understood. In this paper, we show that there exists a large class of Boolean functions, called root functions, which can never appear as faulty response in an irredundant two-level circuit even when any arbitrary multiple stuck-at faults are injected. Conversely, we show that any other Boolean function can appear as a faulty response in an irredundant realization of some root function under certain stuck-at faults. We characterize this new class of functions and show that for n variables, their number is exactly equal to the number of independent dominating sets (Harary and Livingston, Appl. Math. Lett., 1993) in a Boolean n-cube. Similar properties are observed for multiple-valued logic functions as well. Finally, we discuss its application to logic design and point out some open problems.

Recently, Tang, Carlet and Tang presented a combinatorial conjecture about binary strings, allowing proving that all balanced functions in some infinite class they introduced have optimal algebraic immunity. Later, Cohen and Flori completely proved that the conjecture is true. These functions have good (provable or at least observable) cryptographic properties but they are not 1-resilient, which represents a drawback for their use as filter functions in stream ciphers. We propose a construction of an infinite class of 1-resilient Boolean functions with optimal algebraic immunity by modifying the functions in this class. The constructed functions have optimal algebraic degree, that is, meet the Siegenthaler bound, and high nonlinearity. We prove a lower bound on their nonlinearity, but as for the Carlet-Feng functions and for the functions mentioned above, this bound is not enough for ensuring a nonlinearity sufficient for allowing resistance to the fast correlation attack. Nevertheless, as for previously found functions with the same features, there is a gap between the bound that we can prove and the actual values computed for small numbers of variables. Our computations show that the functions in this class have very good nonlinearity and also good immunity to fast algebraic attacks. This is the first time that an infinite class of functions gathers all of the main criteria allowing these functions to be used as filters in stream ciphers.

International audience

An analysis of root functions—A subclass of the Impossible Class of Faulty Functions (ICFF)

Recently, a new structure called butterfly introduced by Perrin et at. is attractive for that it has very good cryptographic properties: the differential uniformity is at most equal to 4 and algebraic degree is also very high when exponent $e=3$. It is conjecture that the nonlinearity is also optimal for every odd $k$, which was proposed as a open problem. In this paper, we further study the butterfly structures and show that these structure with exponent $e=2^i+1$ have also very good cryptographic properties. More importantly, we prove in theory the nonlinearity is optimal for every odd $k$, which completely solve the open problem. Finally, we study the butter structures with trivial coefficient and show these butterflies have also optimal nonlinearity. Furthermore, we show that the closed butterflies with trivial coefficient are bijective as well, which also can be used to serve as a cryptographic primitive.

Secondary constructions of RSBFs with good cryptographic properties

In 2008, Carlet and Feng studied a class of functions with good cryptographic properties. Based on that function, [18] proposed a family of cryptographically significant Boolean functions which contains the functions proposed by [28, 30]. However, their study is not in-depth. In this paper, we investigate the properties of those functions further, and find that they can be divided into some affine equivalent classes. The bent functions proposed by [18] are in fact in the same class with the function proposed by [30]. We then prove that those functions have optimum algebraic immunity if and only if a combinatorial conjecture is correct, which gives a new direction to prove the conjecture. Furthermore, we improve upon the lower bound on the nonlinearity, and our bound is higher than all other similar bounds. Finally, we extend the construction to a balanced function, and give an example of a 12-variable function which has the best cryptographic properties among all currently known functions.KeywordsBoolean functionAlgebraic immunityNonlinearity

This paper compares ASIC and FPGA implementations of two commonly used architectures for 2-dimensional discrete cosine transform (DCT), the parallel and folded architectures. The DCT has been designed for sizes 4x4, 8x8, and 16x16, and implemented on Silterra 180nm ASIC and Xilinx Kintex Ultrascale FPGA. The objective is to determine suitable low energy architectures to be used as their characteristics greatly differ in terms of cells usage, placement and routing methods on these platforms. The parallel and folded DCT architectures for all three sizes have been designed using Verilog HDL, including the basic serializer-deserializer input and output. Results show that for large size transform of 16x16, ASIC parallel architecture results in roughly 30% less energy compared to folded architecture. As for FPGAs, folded architecture results in roughly 34% less energy compared to parallel architecture. In terms of overall energy consumption between 180nm ASIC and Xilinx Ultrascale, ASIC implementation results in about 58% less energy compared to the FPGA.

Information processing in dendrites: II. Information theoretic complexity

SummaryThis paper presents the design and implementation of an integrated architecture for embedding security into quasi‐cyclic (QC) low‐density parity‐check (LDPC) code–based cryptographic system through a nonlinear function of low hardware complexity. Instead of using standardS‐boxes for implementation of nonlinear function, this paper considers a method on the basis of maximum length cellular automata (CA), so that enhanced security can be achieved with simple hardware structure. The proposed system adopts a lightweight random bit stream generator on the basis of linear feedback shift register (LFSR) for generating random error vectors, so that a large number of vectors with very good cryptographic properties can be made available with low hardware cost. Different permutation patterns generated for different message blocks help to provide good degrees of freedom for tuning security with reasonable key size. The hardware architecture for the proposed system is developed and validated through implementation on Xilinx Spartan 3S500E. Analytical and synthesis results show that the proposed scheme is lightweight and offers very high security through continuously changing parameters, thus making it highly suitable for resource‐constrained applications.

In quantum computing the decoherence time of the qubits determines the\ncomputation time available and this time is very limited when using current\nhardware. In this paper we minimize the execution time (the depth) for a class\nof circuits referred to as linear reversible circuits, which has many\napplications in quantum computing (e.g., stabilizer circuits, CNOT+T circuits,\netc.). We propose a practical formulation of a divide and conquer algorithm\nthat produces quantum circuits that are twice as shallow as those produced by\nexisting algorithms. We improve the theoretical upper bound of the depth in the\nworst case for some range of qubits. We also propose greedy algorithms based on\ncost minimization to find more optimal circuits for small or simple operators.\nOverall, we manage to consistently reduce the total depth of a class of\nreversible functions, with up to 92% savings in an ancilla-free case and up to\n99% when ancillary qubits are available.\n

Automatic parallelization of recursive functions with rewriting rules