Patterson-Wiedemann type functions on 21 variables with Nonlinearity greater than Bent Concatenation bound

Nonlinearity is one of the most challenging combinatorial property in the domain of Boolean function research. Obtaining nonlinearity greater than the bent concatenation bound for odd number of variables continues to be one of the most sought after combinatorial research problems. The pioneering result in this direction has been discovered by Patterson and Wiedemann in 1983 (IEEE-IT), which considered Boolean functions on $5 \times 3 = 15$ variables that are invariant under the actions of the cyclic group ${GF(2^5)}^\ast \cdot {GF(2^3)}^\ast$ as well as the group of Frobenius authomorphisms. Some of these Boolean functions posses nonlinearity greater than the bent concatenation bound. The next possible option for exploring such functions is on $7 \times 3 = 21$ variables. However, obtaining such functions remained elusive for more than three decades even after substantial efforts as evident in the literature. In this paper, we exploit combinatorial arguments together with heuristic search to demonstrate such functions for the first time

Autocorrelations of Vectorial Boolean Functions

International audienceRecently, BarOn et al. introduced at Eurocrypt'19 a new tool, called the differential-linear connectivity table (DLCT), which allows for taking into account the dependency between the two subciphers E0 and E1 involved in differential-linear attacks. This paper presents a theoretical characterization of the DLCT, which corresponds to an autocorrelation table (ACT) of a vectorial Boolean function. We further provide some new theoretical results on ACTs of vectorial Boolean functions

Balanced Boolean Functions with (Almost) Optimal Algebraic Immunity and Very High Nonlinearity

In this paper, we present a class of $2k$-variable balanced Boolean functions and a class of $2k$-variable $1$-resilient Boolean functions for an integer $k\ge 2$, which both have the maximal algebraic degree and very high nonlinearity. Based on a newly proposed conjecture by Tu and Deng, it is shown that the proposed balanced Boolean functions have optimal algebraic immunity and the $1$-resilient Boolean functions have almost optimal algebraic immunity. Among all the known results of balanced Boolean functions and $1$-resilient Boolean functions, our new functions possess the highest nonlinearity. Based on the fact that the conjecture has been verified for all $k\le 29$ by computer, at least we have constructed a class of balanced Boolean functions and a class of $1$-resilient Boolean functions with the even number of variables $\le 58$, which are cryptographically optimal or almost optimal in terms of balancedness, algebraic degree, nonlinearity, and algebraic immunity

Mathematical aspects of the design and security of block ciphers

Block ciphers constitute a major part of modern symmetric cryptography. A mathematical analysis is necessary to ensure the security of the cipher. In this thesis, I develop several new contributions for the analysis of block ciphers. I determine cryptographic properties of several special cryptographically interesting mappings like almost perfect nonlinear functions. I also give some new results both on the resistance of functions against differential-linear attacks as well as on the efficiency of implementation of certain block ciphers

Contributions to Confidentiality and Integrity Algorithms for 5G

The confidentiality and integrity algorithms in cellular networks protect the transmission of user and signaling data over the air between users and the network, e.g., the base stations. There are three standardised cryptographic suites for confidentiality and integrity protection in 4G, which are based on the AES, SNOW 3G, and ZUC primitives, respectively. These primitives are used for providing a 128-bit security level and are usually implemented in hardware, e.g., using IP (intellectual property) cores, thus can be quite efficient. When we come to 5G, the innovative network architecture and high-performance demands pose new challenges to security. For the confidentiality and integrity protection, there are some new requirements on the underlying cryptographic algorithms. Specifically, these algorithms should: 1) provide 256 bits of security to protect against attackers equipped with quantum computing capabilities; and 2) provide at least 20 Gbps (Gigabits per second) speed in pure software environments, which is the downlink peak data rate in 5G. The reason for considering software environments is that the encryption in 5G will likely be moved to the cloud and implemented in software. Therefore, it is crucial to investigate existing algorithms in 4G, checking if they can satisfy the 5G requirements in terms of security and speed, and possibly propose new dedicated algorithms targeting these goals. This is the motivation of this thesis, which focuses on the confidentiality and integrity algorithms for 5G. The results can be summarised as follows.1. We investigate the security of SNOW 3G under 256-bit keys and propose two linear attacks against it with complexities 2172 and 2177, respectively. These cryptanalysis results indicate that SNOW 3G cannot provide the full 256-bit security level. 2. We design some spectral tools for linear cryptanalysis and apply these tools to investigate the security of ZUC-256, the 256-bit version of ZUC. We propose a distinguishing attack against ZUC-256 with complexity 2236, which is 220 faster than exhaustive key search. 3. We design a new stream cipher called SNOW-V in response to the new requirements for 5G confidentiality and integrity protection, in terms of security and speed. SNOW-V can provide a 256-bit security level and achieve a speed as high as 58 Gbps in software based on our extensive evaluation. The cipher is currently under evaluation in ETSI SAGE (Security Algorithms Group of Experts) as a promising candidate for 5G confidentiality and integrity algorithms. 4. We perform deeper cryptanalysis of SNOW-V to ensure that two common cryptanalysis techniques, guess-and-determine attacks and linear cryptanalysis, do not apply to SNOW-V faster than exhaustive key search. 5. We introduce two minor modifications in SNOW-V and propose an extreme performance variant, called SNOW-Vi, in response to the feedback about SNOW-V that some use cases are not fully covered. SNOW-Vi covers more use cases, especially some platforms with less capabilities. The speeds in software are increased by 50% in average over SNOW-V and can be up to 92 Gbps.Besides these works on 5G confidentiality and integrity algorithms, the thesis is also devoted to local pseudorandom generators (PRGs). 6. We investigate the security of local PRGs and propose two attacks against some constructions instantiated on the P5 predicate. The attacks improve existing results with a large gap and narrow down the secure parameter regime. We also extend the attacks to other local PRGs instantiated on general XOR-AND and XOR-MAJ predicates and provide some insight in the choice of safe parameters

Decrypting legal dilemmas

It has become a truism that the speed of technological progress leaves law and policy scrambling to keep up. But in addition to creating new challenges, technological advances also enable new improvements to issues at the intersection of law and technology. In this thesis, I develop new cryptographic tools for informing and improving our law and policy, including specific technical innovations and analysis of the limits of possible interventions. First, I present a cryptographic analysis of a legal question concerning the limits of the Fifth Amendment: can courts legally compel people to decrypt their devices? Our cryptographic analysis is useful not only for answering this specific question about encrypted devices, but also for analyzing questions about the wider legal doctrine. The second part of this thesis turns to algorithmic fairness. With the rise of automated decision-making, greater attention has been paid to statistical notions of fairness and equity. In this part of the work, I demonstrate technical limits of those notions and examine a relaxation of those notions; these analyses should inform legal or policy interventions. Finally, the third section of this thesis describes several methods for improving zero-knowledge proofs of knowledge, which allow a prover to convince a verifier of some property without revealing anything beyond the fact of the prover's knowledge. The methods in this work yield a concrete proof size reduction of two plausibly post-quantum styles of proof with transparent setup that can be made non-interactive via the Fiat-Shamir transform: "MPC-in-the-head," which is a linear-size proof that is fast, low-memory, and has few assumptions, and "Ligero," a sublinear-size proof achieving a balance between proof size and prover runtime. We will describe areas where zero-knowledge proofs in general can provide new, currently-untapped functionalities for resolving legal disputes, proving adherence to a policy, executing contracts, and enabling the sale of information without giving it away

Reflections on the number field sieve

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