Algebraic Attacks Research Articles

S-Box on Subgroup of Galois Field

In substitution–permutation network as a cryptosystem, substitution boxes play the role of the only nonlinear part. It would be easy for adversaries to compromise the security of the system without them. 8-bit S-boxes are the most used cryptographic components. So far, cryptographers were constructing 8-bit S-boxes used in cryptographic primitives by exhaustive search of permutations of order 256. However, now for cryptographic techniques with 8-bit S-boxes as confusion layers, researchers are trying to reduce the size of S-box by working with a small unit of data. The aim is to make the techniques compact, fast and elegant. The novelty of this research is the construction of S-box on the elements of the multiplicative subgroup of the Galois field instead of the entire Galois field. The sturdiness of the proposed S-box against algebraic attacks was hashed out by employing the renowned analyses, including balance, nonlinearity, strict avalanche criterion, and approximation probabilities. Furthermore, the statistical strength of the S-box was tested by the majority logic criterion. The fallouts show that the S-box is appropriate for applications for secure data communications. The S-box was also used for watermarking of grayscale images with good outcomes.

Designing secure substitution boxes based on permutation of symmetric group

The strength of cryptosystems heavily relies on the substitution boxes. Cryptosystems with weak substitution boxes cannot resist algebraic attacks, linear and differential cryptanalysis. In this paper, first, we propose a strong algebraic structure for the construction of substitution boxes. The proposed substitution boxes have good algebraic properties and are able to resist against algebraic attacks. Second, we propose a new method for creating multiple substitution boxes with the same algebraic properties using permutation of symmetric group on a set of size 8 and bitwise XOR operation. Third, the proposed substitution boxes with the same algebraic properties are then applied to images and it is observed that the statistical properties of substituted images are different from each other. The simulation results and statistical and security analysis for the proposed substitution boxes are very competitive. Also, it is shown in this work that the proposed substitution boxes can resist differential and linear cryptanalysis and sustain algebraic attacks.

Constructing odd-variable RSBFs with optimal algebraic immunity, good nonlinearity and good behavior against fast algebraic attacks

Rotation symmetric Boolean functions have raised widespread attention because of the good properties in cryptosystem. This paper presents a new construction of odd-variable rotation symmetric Boolean functions with optimal algebraic immunity. The nonlinearity of the new functions is higher than some existing theoretical constructions of rotation symmetric Boolean functions with optimal algebraic immunity. Further, it is also checked that such functions have almost optimal immunity against fast algebraic attacks for small variables. Besides, the algebraic degree of the constructed functions is also high enough.

Efficient public-key operation in multivariate schemes

The public-key operation in multivariate encryption and signature schemes evaluates \begin{document}$ m $\end{document} quadratic polynomials in \begin{document}$ n $\end{document} variables. In this paper we analyze how fast this simple operation can be made. We optimize it for different finite fields on modern architectures. We provide an objective and inherent efficiency measure of our implementations, by comparing their performance with the peak performance of the CPU. In order to provide a fair comparison for different parameter sets, we also analyze the expected security based on the algebraic attack taking into consideration the hybrid approach. We compare the attack's efficiency for different finite fields and establish trends. We detail the role that the field equations play in the attack. We then provide a broad picture of efficiency of MQ-public-key operation against security.

Fast Algebraic Immunity of $2^m+2$ & $2^m+3$ Variables Majority Function

Boolean functions should possess high fast algebraic immunity (FAI) when it is used in stream ciphers in order to stand up to fast algebraic attacks. However, in previous research, the FAI of Boolean functions was usually calculated by computer. But, as everyone knows, it is very difficult to calculate the FAI of a given Boolean function with high algebraic degree when the variable number is greater than 18. In 2016, Tang et al. gave that the exact value of FAI of the majority function on 2 m and 2 m + 1 (m ≥ 2) variables is 2 m -1 + 2. This paper proves that the FAI of 2 m + 2 and 2 m + 3 (m ≥ 2) variables majority function equals 2 m -1 + 4.

A Lower Bound of Fast Algebraic Immunity of a Class of 1-Resilient Boolean Functions

Boolean functions should possess high fast algebraic immunity when used in stream ciphers in order to stand up to fast algebraic attacks. However, in previous research, the fast algebraic immunity of Boolean functions was usually calculated by the computer. In 2017, Tang, Carlet, and Tang first mathematically proved that every function belonging to a class of 1-resilient Boolean functions has the fast algebraic immunity no less than n - 6. Inspired by the Tang's method, we also demonstrate that the fast algebraic immunity of another class of the 1-resilient Boolean functions is no less than n - 6. Meanwhile, we also prove some combinator facts originated from the Tu-Deng Conjecture.

Constructing Two Classes of Boolean Functions With Good Cryptographic Properties

Wu et al. proposed a generalized Tu-Deng conjecture over $\mathbb {F}_{2^{rm}}\times {\mathbb {F}_{2^{m}}}$ , and constructed Boolean functions with good properties. However the proof of the generalized conjecture is still open. Based on Wu’s work and assuming that the conjecture is true, we come up with a new class of balanced Boolean functions which has optimal algebraic degree, high nonlinearity and optimal algebraic immunity. The Boolean function also behaves well against fast algebraic attacks. Meanwhile we construct another class of Boolean functions by concatenation, which is 1-resilient and also has other good cryptographic properties.

Constructing Odd‐Variable Rotation Symmetric Boolean Functions with Optimal Algebraic Immunity and High Nonlinearity

Rotation symmetric Boolean functions (RSBFs) have attracted widespread attention due to their good cryptographic properties. We present a new construction of RSBFs with optimal algebraic immunity on odd number of variables. The nonlinearity of the new function is much higher than other best known RSBFs with optimal algebraic immunity. The algebraic degree of the constructed n-variable RSBF can achieve the upper bound n-1 when n/2 is odd or when n/2 is a power of 2 for n-11. In addition, the constructed function can possess almost perfect immunity to fast algebraic attacks for n=11, 13, 15

Findings Annihilator(s) via Fault Injection Attack (FIA) on Boolean Function of Grain v0

In developing stream cipher algorithms, Boolean function is one of vital elements. Attacks on LFSR-based stream cipher is the challenge for the cryptanalyst to get low-degree annihilator(s). In this paper, we proposed Fault Injection Attack (FIA) on Boolean function of Grain v0, which is the original variant of Grain family algorithm. Fault injection attack (FIA) is used on Boolean function of Grain v0 by replacing certain coefficient with value of one (1) which results in the generation of several injected Boolean functions. With these injected Boolean function, we proceed using HAO’s algorithm to find annihilator(s). As a result, we obtained several new annihilator(s) of Grain v0’s Boolean function. This new annihilator(s) will be utilized to launch algebraic attacks upon Grain v0.

Type-preserving matrices and security of block ciphers

We introduce a new property for mixing layers which guarantees protection against algebraic attacks based on the imprimitivity of the group generated by the round functions. Mixing layers satisfying this property are called non-type-preserving. Our main result is to characterize such mixing layers by providing a list of necessary and sufficient conditions on the structure of their underlying binary matrices. Then we show how several families of linear maps are non-type-preserving, including the mixing layers of AES, GOST and PRESENT. Finally we prove that the group generated by the round functions of an SPN cipher with addition modulo \begin{document}$ 2^n $\end{document} as key mixing function is primitive if its mixing layer satisfies this property.

Automated Analysis of Cryptographic Assumptions in Generic Group Models

We initiate the study of principled, automated, methods for analyzing hardness assumptions in generic group models, following the approach of symbolic cryptography. We start by defining a broad class of generic and symbolic group models for different settings—symmetric or asymmetric (leveled) k-linear groups—and by proving “computational soundness” theorems for the symbolic models. Based on this result, we formulate a very general master theorem that formally relates the hardness of a (possibly interactive) assumption in these models to solving problems in polynomial algebra. Then, we systematically analyze these problems. We identify different classes of assumptions and obtain decidability and undecidability results. Then, we develop and implement automated procedures for verifying the conditions of master theorems, and thus the validity of hardness assumptions in generic group models. The concrete outcome of this work is an automated tool which takes as input the statement of an assumption, and outputs either a proof of its generic hardness or shows an algebraic attack against the assumption.

A novel block cipher for enhancing data security in healthcare internet of things

The modern technologies and their contributions have made monitoring and communication possible from everything around us with each other with no or little human intervention. Experts named these connected devices as Internet of Things (IoT), or the Internet of Everything which is considered as the next industrial revolution. This open network posts quite good number of security challenges because of the nature of the IoT environment. Thus, this paper proposes a lightweight block cipher called, JAC_Jo, to enhance the security of data transmitted in the healthcare environment with reduced hardware footprints. The master key for data encryption is derived using soft set based key generation algorithm SoftKeyGen() and encrypts 32-bit data using 64 bit key. The proposed block cipher occupies 2109 GEs of area and requires 360.05mW power without compromising the security. The FPGA implementation of the proposed block cipher is presented and the performance of the block cipher is evaluated using Atmel STK-600 device with 128 kb app / boot memory. The security evolution is presented with respect to differential, linear, slide and algebraic attacks.

Single Trace Side Channel Analysis on NTRU Implementation

As researches on the quantum computer have progressed immensely, interests in post-quantum cryptography have greatly increased. NTRU is one of the well-known algorithms due to its practical key sizes and fast performance along with the resistance against the quantum adversary. Although NTRU has withstood various algebraic attacks, its side-channel resistance must also be considered for secure implementation. In this paper, we proposed the first single trace attack on NTRU. Previous side-channel attacks on NTRU used numerous power traces, which increase the attack complexity and limit the target algorithm. There are two versions of NTRU implementation published in succession. We demonstrated our attack on both implementations using a single power consumption trace obtained in the decryption phase. Furthermore, we propose a countermeasure to prevent the proposed attacks. Our countermeasure does not degrade in terms of performance.

An Algorithm for Improving Algebraic Degree of S-Box Coordinate Boolean Functions Based on Affine Equivalence Transformation

The Substitution box (\(S\)-box) plays an important role in a block cipher as it is the only nonlinear part of the cipher in most cases. \(S\)-box \(S\) can be considered as a vectorial Boolean function consisting of \(m\) individual Boolean functions \(f_1,f_2,\dots,f_m\), where \(f_i: GF (2^n)\to GF (2)\) and \( f_i(x)=y_i\) for \(i=1,2,\dots,m\). These functions are called coordinate Boolean functions of the \(S\)-box \(S\). To avoid various attacks on the ciphers and for efficient software implementation, the coordinate Boolean functions of \(S\)-boxes are required to satisfy a lot of properties, for instance being a permutation defined on the fields with even degrees, with a high nonlinearity, a low differential uniformity and a high algebraic degree, etc. However, it seems very difficult to find an \(S\)-box with the coordinate Boolean functions to satisfy all the criteria. The \(S\)-box with low algebraic degree of the coordinate Boolean functions is vulnerable to many attacks such as linear and differential cryptanalysis, for instance higher-order differential attacks, algebraic attacks or cube attacks. In this paper we propose an algorithm for improving algebraic degree of the $S$-box coordinate Boolean functions while not affecting its other important properties. The algorithm is based on affine equivalence transformation of the \(S\)-boxes.

MRHS solver based on linear algebra and exhaustive search

Abstract We show how to build a binary matrix from the MRHS representation of a symmetric-key cipher. The matrix contains the cipher represented as an equation system and can be used to assess a cipher’s resistance against algebraic attacks. We give an algorithm for solving the system and compute its complexity. The complexity is normally close to exhaustive search on the variables representing the user-selected key. Finally, we show that for some variants of LowMC, the joined MRHS matrix representation can be used to speed up regular encryption in addition to exhaustive key search.

Optimal annihilator immunity with suboptimal bound of other properties

In this paper we present constructions of Boolean functions based on the basic theory of annihilator immunity and bent functions. These constructions provide functions with the maximum possible annihilator immunity, and the nonlinearity and resiliency of the functions can also be calculated with specific cases. Here we have also described certain properties of Boolean functions that are necessary for resistance against algebraic and fast algebraic attacks. Annihilator immunity, nonlinearity and resiliency are important properties of a Boolean function to resist these attacks. There was no systematic attempt made to construct Boolean functions with maximum annihilator immunity that have good bound of nonlinearity and resiliency. We have constructed two different classes of desired Boolean functions through our newly developed method based on multiobjective optimization method(NSGA-II). In the first class, we have constructed bent functions(optimal nonlinearity) with sub-optimal bound of annihilator immunity while in other, Boolean functions having optimal bound of annihilator immunity with good bound of nonlinearity were developed. Our problem was multiobjective(nonlinearity, annihilator immunity and resiliency), hence a heuristic multiobjective optimization technique(NSGA-II) was used.

Constructions of balanced odd-variable rotation symmetric Boolean functions with optimal algebraic immunity and high nonlinearity

Rotation symmetric Boolean functions have been used as components of different cryptosystems. In this paper, two classes of balanced rotation symmetric Boolean functions having optimal algebraic immunity on odd number of variables are constructed. We give a lower bound on the algebraic degree of the first class of functions, and prove that the n-variable functions in the second class has optimal algebraic degree if n≠2m+1 for m>2. Moreover, it is shown that both classes of functions have much better nonlinearity than all the previously obtained rotation symmetric Boolean functions with optimal algebraic immunity, and have good behavior against fast algebraic attacks at least for small numbers of input variables.

A Method of Generating 8 × 8 Substitution Boxes Based on Elliptic Curves

Elliptic curve cryptography provides better security and is more efficient as compared to other public key cryptosystems with identical key size. In this article, we present a new method for the construction of substitution boxes(S-boxes) based on points on elliptic curve over prime field. The resistance of the newly generated S-box against common attacks such as linear, differential and algebraic attacks is analyzed by calculating their non-linearity, linear approximation, strict avalanche, bit independence, differential approximation and algebraic complexity. The experimental results are further compared with some of the prevailing S-boxes presented in Shi et al. (Int Conf Inf Netw Appl 2:689–693, 1997), Jakimoski and Kocarev (IEEE Trans Circuits Syst I 48:163–170, 2001), Guoping et al. (Chaos, Solitons Fractals 23:413–419, 2005), Guo (Chaos, Solitons Fractals 36:1028–1036, 2008), Kim and Phan (Cryptologia 33: 246–270, 2009), Neural et al. (2010 sixth international conference on natural computation (ICNC 2010), 2010), Hussain et al. (Neural Comput Appl. https://doi.org/10.1007/s00521-012-0914-5 , 2012). Comparison reveals that the proposed algorithm generates cryptographically strong S-boxes as compared to some of the other exiting techniques.

Cryptanalysis of a cryptosystem with non-commutative platform groups

A cryptosystem with non-commutative platform groups based on conjugator search problem was recently introduced at Neural Computing and Applications 2016. Its versatility was illustrated by building a public-key encryption scheme. We propose an algebraic key-recovery attack in the polynomial computational complexity. Furthermore, we peel off the encryption and decryption process and propose attack methods for solving the conjugator search problem over the given non-abelian group. Finally, we provide corresponding practical attack examples to illustrate the attack methods in our cryptanalysis, and provide some improved suggestions.

Cryptanalysis of an asymmetric cipher protocol using a matrix decomposition problem: revisited

With the development of quantum computing technology, quantum computers pose a serious threat to the widely used public key cryptography. This is because there are effective quantum algorithms to solve many difficult problems based on commutative algebra structures such as factorization or discrete logarithms. It is generally believed that many public key crytosystems based on non-commutative cryptosystem algebraic structures have the potential to resist quantum computing attacks. Since multiplication of matrices has non-commutative properties, the cryptography based on matrix-based has the potential to resist quantum computing attacks. The security of matrix-based cryptography is closely related to the difficulty of matrix decomposition. An asymmetric cipher protocol based on matrix decomposition problem has been proposed by Raulynaitis et al. to meet the requirements of public key cryptography in the post quantum era. Liu et al. identified some weak keys in this scheme, through which an attacker can solve the equivalent secret key and crack the scheme by solving simultaneous linear equations. Liu et al. proposed an improved scheme to avoid weak keys. However, Raulynaitis and Liu schemes are not fully secured because a special structure of matrix is used to make some matrics commutative. The analysis presented in this paper demostrates that regardless of whether the private key is weak key or not, the equivalent keys from an associated public key can be solved in a reasonable time by a linear algebra attack. For this purpose, the linear equations with coefficients n2 × n2are needed to solve. The equation coefficients are much less than the coefficients 5n2 × 2n2 in the attack methods of Liu et al. Thus, the proposed attack method is not only more general and but also more efficient.