Recursive construction method of Boolean permutations

S-boxes are often the core nonlinear component in many encryption algorithms. By using vector Boolean functions to represent S-boxes, cryptographic properties as well as constructions can be made possible. This chapter studies the S-boxes by the view of vector Boolean functions, with focus being on Boolean permutations, which are a special class of vector Boolean functions. Properties and constructions of Boolean permutations are studied; computation of inverses of Boolean functions is also studied. The concept of one-way trapdoor Boolean permutation is proposed. Construction of Boolean permutations using function composition is studied which enables the construction of one-way trapdoor Boolean permutations.

S-boxes, typically the only nonlinear part of a block cipher, are the heart of symmetric cryptographic primitives. They significantly impact the cryptographic strength and the implementation characteristics of an algorithm. Due to their simplicity, quadratic vectorial Boolean functions are preferred when efficient implementations for a variety of applications are of concern. Many characteristics of a function stay invariant under affine equivalence. So far, all 6-bit Boolean functions, 3- and 4-bit permutations have been classified up to affine equivalence. At FSE 2017, Bozoliv et al. presented the first classification of 5-bit quadratic permutations. In this work, we propose an adaptation of their work resulting in a highly efficient algorithm to classify n x m functions for n ≥ m. Our algorithm enables for the first time a complete classification of 6-bit quadratic permutations as well as all balanced quadratic functions for n ≤ 6. These functions can be valuable for new cryptographic algorithm designs with efficient multi-party computation or side-channel analysis resistance as goal. In addition, we provide a second tool for finding decompositions of length two. We demonstrate its use by decomposing existing higher degree S-boxes and constructing new S-boxes with good cryptographic and implementation properties.

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

Modern block ciphers can be used for designing security schemes to meet the security requirements in underwater acoustic sensor networks. The substitution boxes (S-boxes) used in block ciphers must have good cryptographic properties to resist various attacks. The difficulty in the design of S-boxes is that the cryptographic properties of S-boxes have no obvious distribution. In this paper, we investigate the certain subgraph of the Cayley graph of the symmetric group to get some information about the distribution of the cryptographic properties of S-boxes. Based on the information obtained, an algorithm for designing S-boxes with good cryptographic properties is proposed. The security analysis shows that the preferred S-box constructed by the proposed algorithm has good cryptographic properties.

Rotation Symmetric Boolean Functions –; Count and Cryptographic Properties

Cellular Automata (CA) represent an interesting approach to design Substitution Boxes (S-boxes) having good cryptographic properties and low implementation costs. From the cryptographic perspective, up to now there have been only ad-hoc studies about specific kinds of CA, the best known example being the χ nonlinear transformation used in Keccak. In this paper, we undertake a systematic investigation of the cryptographic properties of S-boxes defined by CA, proving some upper bounds on their nonlinearity and differential uniformity. Next, we extend some previous published results about the construction of CAbased S-boxes by means of a heuristic technique, namely Genetic Programming (GP). In particular, we propose a “reverse engineering” method based on De Bruijn graphs to determine whether a specific S-box is expressible through a single CA rule. Then, we use GP to assess if some CA-based S-box with optimal cryptographic properties can be described by a smaller CA. The results show that GP is able to find much smaller CA rules defining the same reference S-boxes up to the size 7 × 7, suggesting that our method could be used to find more efficient representations of CA-based S-boxes for hardware implementations. Finally, we classify up to affine equivalence all 3 × 3 and 4 × 4 CA-based S-boxes.

The strict avalanche criterion (SAC) was introduced by Webster and Tavares [10] in a study of cryptographic design criteria. This is an indicator for local property. In order to improve the global analysis of cryptographically strong functions, Zhang and Zheng [11] introduced the global avalanche characteristics (GAC). The sum-of-squares indicator related to the GAC is defined as σfe∑vΔf2(v), where Δf(v)e∑x (−1)f(x)⊕f(x}⊕ v). In this paper, we give a few methods to construct Boolean functions controlling five good cryptographic properties, namely balancedness, good local and GAC, high nonlinearity and high algebraic degree. We improve upon the results of Stanica [8] and Zhang and Zheng [11].

In this paper, the analysis of cyclotomic coset has been done for construction of Boolean functions. Boolean function used in cryptographic systems should have good cryptographic properties against standard or fast algebraic attacks. The cyclotomic coset plays very important role in cryptography. By using cyclotomic coset leader and minimal polynomial, we construct more balanced Boolean function. In this paper, cryptographic properties; balancedness, algebraic-degree, algebraic-immunity and non-linearity are also defined in preliminaries.

To further enhance the nonlinear and cryptographic properties of chaotic S-boxes, a novel S-box design is proposed based on the discrete chaotic map and improved artificial bee colony algorithm. Firstly, the S-box generated by chaotic sequence is initialized by the opposition-based optimization. Then, the dual transposition artificial bee colony algorithm is used to optimize the population, where the Gaussian swap is engaged to strengthen the optimum efficiency, and make the population acquire a better quality with each other. With the Gaussian mutation operator, the opposition-based optimization performs self-learning to obtain better population. Comparing with some recent S-boxes based on chaos. The S-boxes generated by the proposed scheme can achieve better performance in bijection, nonlinearity, strict avalanche criterion(SAC), bit independence criterion(BIC), I / O XOR distribution and linear approximation probability(LP). It provides a new and effective method for designing S-boxes 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

We present a new Authenticated Encryption (AE) scheme based on finite automata using the method Encrypt-then-MAC. The scheme is constructed from two cryptographic primitives, a symmetric encryption scheme based on finite automata known as FASKC and a Message Authentication Code (MAC) constructed from a finite automaton. The paper shows how the scheme works and how to construct a MAC from a finite automaton using a boolean function with good cryptographic properties. Due to the construction method and the assumption that the symmetric encryption scheme is secure under Chosen Plaintext Attack (CPA), it is shown that the MAC is unforgeable under Chosen Message Attack (UF-CMA) making the scheme secure under Chosen Ciphertext Attack (CCA).

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 this paper, we investigate the hardware circuit complexity of the class of Boolean functions recently introduced by Tang and Maitra (IEEE-TIT 64(1): 393–402, 2018). While this class of functions has very good cryptographic properties, the exact hardware requirement is an immediate concern as noted in the paper itself. In this direction, we consider different circuit architectures based on finite field arithmetic and Boolean optimization. An estimation of the circuit complexity is provided for such functions given any input size n. We study different candidate architectures for implementing these functions, all based on the finite field arithmetic. We also show different implementations for both ASIC and FPGA, providing further analysis on the practical aspects of the functions in question and the relation between these implementations and the theoretical bound. The practical results show that the Tang-Maitra functions are quite competitive in terms of area, while still maintaining an acceptable level of throughput performance for both ASIC and FPGA implementations.

Substitution boxes (S-boxes) are an important part of the design of block ciphers. They provide nonlinearity and so the security of the cipher depends strongly on them. Some block ciphers use S-boxes given by lookup tables (e.g., DES) where as others use S-boxes obtained from finite field operations (e.g., AES). As a generalization of the latter, finite semifields (i.e., finite nonassociative division rings) have been suggested as algebraic structures from which S-boxes with good cryptographic properties might be obtained. In this paper we present the results of experiments on the construction of S-boxes from finite semifields of orders 256 and 64, using the left and right inverses of these rings.