Equivalent keys in Multivariate Quadratic public key systems

ℳultivariate uadratic public key schemes have been suggested as early as 1985 by Matsumoto and Imai as an alternative for the RSA scheme. Since then, several schemes have been proposed, for example hidden field equations, unbalanced oil and vinegar schemes, and stepwise triangular schemes. All these schemes have a rather large key space for a secure choice of parameters. Surprisingly, the question of equivalent keys has not been discussed in the open literature until recently. In this article, we show that for all basic classes mentioned above, it is possible to reduce the private – and hence the public – key space by several orders of magnitude, i.e. the size of the set of possible private and hence public keys can be reduced. For the Matsumoto–Imai scheme, we are even able to show that the reductions we found are the only ones possible, i.e. that these reductions are tight. While the theorems developed in this article are of independent interest themselves as they broaden our understanding of ℳultivariate uadratic public key systems, we see applications of our results both in cryptanalysis and in memory efficient implementations of -schemes.

Weil descent methods have recently been applied to attack the Hidden Field Equation (HFE) public key systems and solve the elliptic curve discrete logarithm problem (ECDLP) in small characteristic. However the claims of quasi-polynomial time attacks on the HFE systems and the subexponential time algorithm for the ECDLP depend on various heuristic assumptions.In this paper we introduce the notion of the last fall degree of a polynomial system, which is independent of choice of a monomial order. We then develop complexity bounds on solving polynomial systems based on this last fall degree.We prove that HFE systems have a small last fall degree, by showing that one can do division with remainder after Weil descent. This allows us to solve HFE systems unconditionally in polynomial time if the degree of the defining polynomial and the cardinality of the base field are fixed. For the ECDLP over a finite field of characteristic 2, we provide computational evidence that raises doubt on the validity of the first fall degree assumption, which was widely adopted in earlier works and which promises sub-exponential algorithms for ECDLP. In addition, we construct a Weil descent system from a set of summation polynomials in which the first fall degree assumption is unlikely to hold. These examples suggest that greater care needs to be exercised when applying this heuristic assumption to arrive at complexity estimates.These results taken together underscore the importance of rigorously bounding last fall degrees of Weil descent systems, which remains an interesting but challenging open problem.KeywordsHFEECDLPWeil descentSolving equationsFirst fall degreeLast fall degree

In this paper we show that solving systems coming from the public key of the Unbalanced Oil and Vinegar (UOV) signature scheme is on average at least as hard as solving a certain quadratic system with completely random quadratic part. In providing lower bounds on direct attack complexity we rely on the empirical fact that complexity of solving a non-linear polynomial system is determined by the homogeneous part of this system of the highest degree. Our reasoning explains, in particular, the results on solving the UOV systems presented by J.-C. Faugere and L. Perret at the SCC conference in 2008.

In this article, we investigate the question of equivalent keys for two ${\mathcal M}$ultivariate ${\mathcal Q}$uadratic public key schemes HFE and C*−− and improve over a previously known result, which appeared at PKC 2005. Moreover, we show a new non-trivial extension of these results to the classes HFE-, HFEv, HFEv-, and C*−−, which are cryptographically stronger variants of the original HFE and C* schemes. In particular, we are able to reduce the size of the private — and hence the public — key space by at least one order of magnitude and several orders of magnitude on average. While the results are of independent interest themselves as they broaden our understanding of ${\mathcal M}$ultivariate ${\mathcal Q}$uadratic schemes, we also see applications both in cryptanalysis and in memory efficient implementations.

A new image cipher in time and frequency domains

Mercury removal is an integral part of gas-processing and coal combustion plants due to its implication on health, environment and process equipment. Utilization of impregnated carbons and metal sulfide-based adsorbents for mercury removal are common, however with limitations in adsorption capacity and life time. Continued efforts to develop porous carbon sorbents with better mercury adsorption capacity and kinetics are evident from the open literature with the recent focus being on solid support ionic liquid (SSIL)-based adsorbents with ionic liquid and chelating agents. However, reports on application of SSILs-based adsorbents for gas-phase mercury removal are not available in the open literature. Toward this objective, the present work attempts to synthesize three different SSILs adsorbents identified as SSIL (porous carbon + IL), SSILM (porous carbon + IL + MPTS), SSILA (porous carbon + IL + APTS–MBT) utilizing ultrasound-induced wet impregnation method. The adsorbents were subjected to characterization utilizing BET, XRD, FT-IR and SEM and were tested for its mercury adsorption capacity. The SSIL sorbents were found to possess higher order of magnitude adsorption capacity as compared to high-surface-area porous carbons and metal sulfide-based porous carbons. The adsorption capacity of the SSILs increased orders of magnitude with increase in adsorption temperature from 30 to 50 °C attributed to the predominance of the chemisorption. Among the three SSIL adsorbents, SSILA was found to exhibit highest equilibrium adsorption capacity of 36.9 mg/g at 50 °C. ΔH°, ΔS° and ΔG° indicate that elemental mercury adsorption on SSILA is a spontaneous and endothermic process.

In this paper, a design algorithm for quantitative robust optimal control of uncertain multivariable feedback systems is developed. By use of the frequency-domain approach, the two-degree-of-freedom Wiener–Hopf optimal control method and the quantitative feedback theory are employed to derive the linear quadratic optimal system design with quantitative performance robustness. Two frequency-dependent weighting matrices are shaped and two-degree-of-freedom compensators are designed to achieve quantitative robust optimal control. The proposed design algorithm is then applied to an AFTI/F-16 flight control system design to illustrate the design algorithm. © 1998 John Wiley & Sons, Ltd.

With the rapid development of quantum computing technology, traditional public key cryptographic systems face serious threats. Post-quantum cryptographic algorithms have become an important research direction in the field of information security. The Unbalanced Oil and Vinegar (UOV) signature is one of the important candidate schemes in post-quantum cryptography. It can effectively resist quantum computing attacks. However, the extensive matrix multiplication and addition operations during key generation and the intensive computations in the random number generation stage consume substantial CPU resources. This causes the UOV key generation phase to account for more than 90% of the total UOV computational workload. To address these problems, this paper proposes a GPU parallel computing-based optimization scheme for UOV key generation. The approach focuses on accelerating two key components. First, GPU threads are used to parallelize matrix multiplication and addition operations. Thread-level parallel reconstruction and triangular matrix storage optimization are adopted. This achieves a 2.34× acceleration effect. Second, the natural parallel characteristics of AES-CTR mode are fully utilized. GPU multi-level buffering and batch processing techniques are used to optimize the Pseudo-Random Number Generator (PRNG) generation process for partial public key matrices. This obtains a significant 9.88× acceleration. Experimental results show that compared to the current optimal CPU implementation, the proposed GPU parallelization scheme achieves an overall 1.64× acceleration for the UOV algorithm key generation phase.

A novel image encryption scheme comprising of one permutation process and one diffusion process is proposed. In the permutation process, the image sized is expanded to one sized by dividing the plain-image into two parts: one consisting of the higher 4bits and one consisting of the lower 4bits. The permutation operations are done row-by-row and column-by-column to increase the speed. The chaotic generalized Arnold map is utilized to generate chaotic sequence, which is quantized to shuffle the expanded image. The chaotic sequence for permutation process is dependent on plain-image and cipher keys, resulting in good key sensitivity and plain-image sensitivity. To achieve more avalanche effect and larger key space, Chinese Remainder Theorem is applied to diffuse the shuffled image. The key sensitivity and key space of the proposed image encryption have been analyzed as well. The experimental results suggest that the proposed image encryption scheme can be used for secure image and video communication applications.

As a response to serious transmission security problems in wireless communications, physical layer encryption (PLE) provides an effective security measure which is very different from upper layer cryptography technologies. PLE can take advantage of the effects of channel and noise and the processing objects are complex vector signals, which are essentially different from Boolean algebra based traditional cryptography. This paper establishes mathematical models, design frameworks and cryptographic primitives for PLE. Two design frameworks are proposed: stream PLE and block PLE. For stream PLE, a new 3D security constellation mapping is derived. For block PLE, two types of sub-transforms are defined: isometry transformations and stochastic transformations. The proposed PLE framework has a large cipher signal space and key space; it provides more freedom in design and can resist known plaintext attacks and chosen-plaintext attacks.

The Internet advances in the past decade have been upping at a tremendous pace; especially in the fields like medicine, military, and technology, where lot of internet threats have also been raising, at least at the same rate, if not higher. The communication takes place in the form of text, image, voice, and video signals etc. And we deal with more image-based applications. There are many encryption techniques existing-DES, AES, RSA, etc. but they provide less security and are vulnerable to brute force attacks due to factors like small key space, to avoid which, chaos-based encryption was introduced. Chaotic systems are very sensitive to initial conditions and distort the image to beyond reversible state, except with a protection key with which it is encrypted, which is hard to find in such a large key space. In this paper we will discuss image encryption using combined chaos by combining various chaotic maps like logistics map, singer map, tent map, sine map, piecewise map etc. and attempt to compare the results for better combination for encryption.

This paper proposes a digital image compression-encryption scheme based on the theory of compressive sensing and cyclic shift, which use random Gauss matrix and sparse transform to compress the digital image, and then cyclic shift and diffusion operation are developed subsequently to the compressive sensing (CS) phase. The algorithm has three advantages: First, the measurement matrix used in the algorithm is generated by Chebyshev mapping, which increases the key space and reduces the burden of key transmission. Second, The Sigmoid function is used to transform the range of compressed data to 0~255, which are stored as 8-bit binary data, thus further reducing the amount of data transmission and avoiding the expansion of encrypted data. Third, the generation of key stream in encryption phase is related to ciphertext, there are different key streams for different plain images. Thus, our algorithm can resist against the chosen-plaintext and known-plaintext attacks effectively. In addition, the implementation of cyclic shift and diffusion operation further enhances the security of the system. Each pixel of the encrypted image is output in the form of 8-bit integer to facilitate data storage, display and transmission. The experimental results and security analysis show that the algorithm has the advantages of large key space, no obvious statistical characteristics of ciphertext, sensitive to plaintext and key, and able to resist chosen-plaintext attack.

ABSTRACTIn this paper, an asymmetric hybrid encryption scheme, using coherent superposition and modified equal modulus decomposition in a hybrid multi-resolution wavelet, is proposed. The hybrid multi-resolution wavelet is generated using fractional Fourier transform of multiple orders and Walsh transform. The fractional orders of the fractional Fourier transform increase the key space and hence provide additional strength to the cryptosystem. The designed scheme has a large key space to avoid brute-force attack and is non-linear in nature. The scheme is validated on grey-scale images. Computer-based simulations have been performed to verify the validity and performance of the proposed scheme against various attacks. Scheme's robustness to the special attack is also checked. Results show that single equal modulus decomposition with fractional Fourier transform and hybrid transform are vulnerable to the special attack, whereas the proposed scheme endures the special attack.

In open network, it is important to keep sensitive information secure from becoming vulnerable to unauthorized access. Encryption is used to ensure high security for Images. Chaos has been widely used for image encryption for its different features. There are many chaos based encryption techniques. Most of the proposed discrete chaotic cryptographic approaches are based on stream or block cipher schemes. If these two schemes are combined the security level is improved. Novel image encryption is proposed based on combination of pixel shuffling. Chaos is used for expand diffusion & confusion in image. Chaotic maps gives advantages of large key space and high level security. We survey exiting work which uses different techniques for image encryption and analyse them with respect to various parameters. To evaluate security & performance, key space analysis, correlation coefficient, histogram, information entropy, NPCR, UACI etc. will be studied.

Encryption of a digital image is very important especially in applications of body area networks (BANs) since the image may include a number of privacy. Past encryption methods have disadvantages of the small key space and low ability of resistance to attack. In this paper, we propose a new encryption algorithm based on discrete wavelet transform (DWT) and multichaos which has characteristics of the deterministic, pseudorandomness, and sensitivity of initial values. The image is first decomposed and spatial reconstructed by two-dimensional DWT and then is performed by multichaos matrices for space encryption. The experimental results indicate that the proposed algorithm has a large key space, high key sensitivity, and excellent ability of resistance to attack.