Algorithm 959: VBF: A Library of C plus plus Classes for Vector Boolean Functions in Cryptography

VBF is a collection of C++ classes designed for analyzing vector Boolean functions (functions that map a Boolean vector to another Boolean vector) from a cryptographic perspective. This implementation uses the NTL library from Victor Shoup, adding new modules that call NTL functions and complement the existing ones, making it better suited to cryptography. The class representing a vector Boolean function can be initialized by several alternative types of data structures such as Truth Table, Trace Representation, and Algebraic Normal Form (ANF), among others. The most relevant cryptographic criteria for both block and stream ciphers as well as for hash functions can be evaluated with VBF: it obtains the nonlinearity, linearity distance, algebraic degree, linear structures, and frequency distribution of the absolute values of the Walsh Spectrum or the Autocorrelation Spectrum, among others. In addition, operations such as equality testing, composition, inversion, sum, direct sum, bricklayering (parallel application of vector Boolean functions as employed in Rijndael cipher), and adding coordinate functions of two vector Boolean functions are presented. Finally, three real applications of the library are described: the first one analyzes the KASUMI block cipher, the second one analyzes the Mini-AES cipher, and the third one finds Boolean functions with very high nonlinearity, a key property for robustness against linear attacks

A C++ Class for Analyzing Vector Boolean Functions from a Cryptographic Perspective

In this paper, a C++ class for analising Vector Boolean Functions from a cryptographic perspective is presented. This implementation uses the NTL library from Victor Shoup, replacing some of the general purpose modules of this library by some more specialized and better suited to cryptography, and adding new modules that complement the existing ones. With this class, we can obtain the classical representation of Vector Boolean Function such as its Truth Table and Algebraic Normal Form (ANF). It is possible to calculate mathematical structures such as the Walsh Spectrum, Linear Profile, Differential Profile and Autocorrelation Spectrum. Cryptographic criteria such as nonlinearity, linearity distance, order of correlation immunity, bal-ancedness, algebraic degree and propagation criterion can be obtained with this class. It permits to find out some interesting cryptologic parameters such as linear structures, linear potential, differential potential and the maximum possible nonlinearity or linearity distance of a Vector Boolean Function with the same dimensions. Finally, operations such as to identify if two Vector Boolean Functions are equal, their sum, direct sum, composition, bricklayering, adding coordinate functions and obtaining the polynomial representation over GF(2n) of a Vector Boolean Function given the irreducible polynomial and its Truth Table are presented

New Family of Stream Ciphers as Physically Clone-Resistant VLSI-Structures

A new large class of $2^{100}$ possible stream ciphers as keystream generators KSGs, is presented. The sample cipher-structure-concept is based on randomly selecting a set of 16 maximum-period Nonlinear Feedback Shift Registers (NLFSRs). A non-linear combining function is merging the 16 selected sequences. All resulting stream ciphers with a total state-size of 223 bits are designed to result with the same security level and have a linear complexity exceeding $2^{81}$ and a period exceeding $2^{161}$. A Secret Unknown Cipher (SUC) is created randomly by selecting one cipher from that class of $2^{100}$ ciphers. SUC concept was presented recently as a physical security anchor to overcome the drawbacks of the traditional analog Physically Unclonable Functions (PUFs). Such unknown ciphers may be permanently self-created within System-on-Chip SoC non-volatile FPGA devices to serve as a digital clone-resistant structure. Moreover, a lightweight identification protocol is presented in open networks for physically identifying such SUC structures in FPGA-devices. The proposed new family may serve for lightweight realization of clone-resistant identities in future self-reconfiguring SoC non-volatile FPGAs. Such self-reconfiguring FPGAs are expected to be emerging in the near future smart VLSI systems. The security analysis and hardware complexities of the resulting clone-resistant structures are evaluated and shown to exhibit scalable security levels even for post-quantum cryptography.Comment: 24 pages, 7 Figures, 3 Table

On the Immunity of Rotation Symmetric Boolean Functions Against Fast Algebraic Attacks

In this paper, it is shown that an $n$-variable rotation symmetric Boolean function $f$ with $n$ even but not a power of 2 admits a rotation symmetric function $g$ of degree at most $e\leq n/3$ such that the product $gf$ has degree at most $n-e-1$

Parameterization of Boolean functions by vectorial functions and associated constructions

Despite intensive research on Boolean functions for cryptography for over thirty years, there are very few known general constructions allowing to satisfy all the necessary criteria for ensuring the resistance against all the main known attacks on the stream ciphers using them. In this paper, we investigate the general construction of Boolean functions $f$ from vectorial functions, in which the support of $f$ equals the image set of an injective vectorial function $F$, that we call a parameterization of $f$. Any Boolean function whose Hamming weight is a power of 2, and in particular, every balanced Boolean function, can be obtained this way. We study five illustrations of this general construction. The three first correspond to known classes of functions (Maiorana-McFarland, majority functions and balanced functions in odd numbers of variables with optimal algebraic immunity). The two last correspond to new classes of Boolean functions: - sums of indicators of disjoint graphs of $(k,n-k$)-functions, - functions parameterized by highly nonlinear injective vectorial $(n-1,n)$-functions derived from functions due to Beelen and Leander. We study the cryptographic parameters (corresponding to the main criteria) of balanced Boolean functions, according to those of their parameterizations: the algebraic degree of $f$, that we relate to the algebraic degrees of $F$ and of its graph indicator, the nonlinearity of $f$, that we relate by a bound to the nonlinearity of $F$, and the algebraic immunity (AI), whose optimality is related to a natural question in linear algebra, and which may be handled (in two ways) by means of the graph indicator of $F$. We show how the algebraic degree and the nonlinearity of the parameterized function can be controlled. We revisit each of the five classes for each criterion. We show that the fourth class is very promising, thanks to a lower bound on the nonlinearity by means of the nonlinearity of the chosen $(k,n-k$)-functions. Its sub-class made of the sums of indicators of affine functions, for which we prove an upper bound on the nonlinearity, seems also interesting. The fifth class includes functions with optimal algebraic degree, good nonlinearity and good AI. We leave for future works the determination of simple effective sufficient conditions on $F$ ensuring that $f$ has a good AI, the completion of the study of the fourth class, the mathematical study of the AI and fast algebraic immunity of the functions in the fifth class, and the introduction and study of a class of parameterized functions having good parameters and whose output is fast to compute

Time-delayed models of gene regulatory networks

We discuss different mathematical models of gene regulatory networks as relevant to the onset and development of cancer. After discussion of alternativemodelling approaches, we use a paradigmatic two-gene network to focus on the role played by time delays in the dynamics of gene regulatory networks. We contrast the dynamics of the reduced model arising in the limit of fast mRNA dynamics with that of the full model. The review concludes with the discussion of some open problems

On the Design and Analysis of Stream Ciphers

This thesis presents new cryptanalysis results for several different stream cipher constructions. In addition, it also presents two new stream ciphers, both based on the same design principle. The first attack is a general attack targeting a nonlinear combiner. A new class of weak feedback polynomials for linear feedback shift registers is identified. By taking samples corresponding to the linear recurrence relation, it is shown that if the feedback polynomial has taps close together an adversary to take advantage of this by considering the samples in a vector form. Next, the self-shrinking generator and the bit-search generator are analyzed. Both designs are based on irregular decimation. For the self-shrinking generator, it is shown how to recover the internal state knowing only a few keystream bits. The complexity of the attack is similar to the previously best known but uses a negligible amount of memory. An attack requiring a large keystream segment is also presented. It is shown to be asymptotically better than all previously known attacks. For the bit-search generator, an algorithm that recovers the internal state is given as well as a distinguishing attack that can be very efficient if the feedback polynomial is not carefully chosen. Following this, two recently proposed stream cipher designs, Pomaranch and Achterbahn, are analyzed. Both stream ciphers are designed with small hardware complexity in mind. For Pomaranch Version 2, based on an improvement of previous analysis of the design idea, a key recovery attack is given. Also, for all three versions of Pomaranch, a distinguishing attack is given. For Achterbahn, it is shown how to recover the key of the latest version, known as Achterbahn-128/80. The last part of the thesis introduces two new stream cipher designs, namely Grain and Grain-128. The ciphers are designed to be very small in hardware. They also have the distinguishing feature of allowing users to increase the speed of the ciphers by adding extra hardware

Design and Analysis of Cryptographic Pseudorandom Number/Sequence Generators with Applications in RFID

This thesis is concerned with the design and analysis of strong de Bruijn sequences and span n sequences, and nonlinear feedback shift register (NLFSR) based pseudorandom number generators for radio frequency identification (RFID) tags. We study the generation of span n sequences using structured searching in which an NLFSR with a class of feedback functions is employed to find span n sequences. Some properties of the recurrence relation for the structured search are discovered. We use five classes of functions in this structured search, and present the number of span n sequences for 6 <= n <= 20. The linear span of a new span n sequence lies between near-optimal and optimal. According to our empirical studies, a span n sequence can be found in the structured search with a better probability of success. Newly found span n sequences can be used in the composited construction and in designing lightweight pseudorandom number generators. We first refine the composited construction based on a span n sequence for generating long de Bruijn sequences. A de Bruijn sequence produced by the composited construction is referred to as a composited de Bruijn sequence. The linear complexity of a composited de Bruijn sequence is determined. We analyze the feedback function of the composited construction from an approximation point of view for producing strong de Bruijn sequences. The cycle structure of an approximated feedback function and the linear complexity of a sequence produced by an approximated feedback function are determined. A few examples of strong de Bruijn sequences with the implementation issues of the feedback functions of an (n+16)-stage NLFSR are presented. We propose a new lightweight pseudorandom number generator family, named Warbler family based on NLFSRs for smart devices. Warbler family is comprised of a combination of modified de Bruijn blocks (CMDB) and a nonlinear feedback Welch-Gong (WG) generator. We derive the randomness properties such as period and linear complexity of an output sequence produced by the Warbler family. Two instances, Warbler-I and Warbler-II, of the Warbler family are proposed for passive RFID tags. The CMDBs of both Warbler-I and Warbler-II contain span n sequences that are produced by the structured search. We analyze the security properties of Warbler-I and Warbler-II by considering the statistical tests and several cryptanalytic attacks. Hardware implementations of both instances in VHDL show that Warbler-I and Warbler-II require 46 slices and 58 slices, respectively. Warbler-I can be used to generate 16-bit random numbers in the tag identification protocol of the EPC Class 1 Generation 2 standard, and Warbler-II can be employed as a random number generator in the tag identification as well as an authentication protocol for RFID systems.1 yea