F5 Algorithm Research Articles

Axioms for a theory of signature bases

Twenty years after the discovery of the F5 algorithm, Gröbner bases with signatures are still challenging to understand and to adapt to different settings. This contrasts with Buchberger's algorithm, which we can bend in many directions keeping correctness and termination obvious. I propose an axiomatic approach to Gröbner bases with signatures with the purpose of uncoupling the theory and the algorithms, giving general results applicable in many different settings (e.g. Gröbner for submodules, F4-style reduction, noncommutative rings, non-Noetherian settings, etc.), and extending the reach of signature algorithms.

Lightweight and efficient approach for multi-secret steganography

This paper compares an efficiency of two approaches of multi-secret steganography in lightweight systems: interlacing and multi-level. The study was conducted for two and three secrets with use of F5 algorithm for both approaches. The embedding times were measured with and without I/O operations. At the end the application of these techniques in lightweight solutions is discussed.

On affine tropical F5 algorithms

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gröbner bases taking into account the valuation of K. Because of the use of the valuation, the theory of tropical Gröbner bases has proved to provide settings for computations over polynomial rings over a p-adic field that are more stable than that of classical Gröbner bases.Beforehand, these strategies were only available for homogeneous polynomials. In this article, we extend the F5 strategy to a new definition of tropical Gröbner bases in an affine setting. We also provide a competitor with an adaptation of the F4 strategy to tropical Gröbner bases computations.We provide numerical examples to illustrate time-complexity and p-adic stability of this tropical F5 algorithm. We also illustrate its merits as a first step before an FGLM algorithm to compute (classical) lex bases over p-adics.

Steganographic schemes from perfect codes on Cayley graphs

The main purpose of this paper is to give a new and general way to obtain steganographic schemes from perfect codes on Cayley graphs, motivated by the F5 algorithm based on binary Hamming codes. We obtain the steganography based on perfect Hamming codes as a special case and also give various equivalent conditions for the existence of a perfect code on a k-regular Abelian Cayley graph. Then we show that a perfect code on an Abelian Cayley graph produces a proper steganographic scheme. We further compute the various parameters for the steganographic scheme of type [n, k] over a finite field $$\mathbb {F}_q$$ arising from a linear [n, n − k, d] code over $$\mathbb {F}_q$$ and find also parameters for some steganographic schemes from perfect codes in k-regular Abelian Cayley graphs.

An Enhanced Steganographic Code and its Application in Voice-Over-IP Steganography

Voice-over IP (VoIP) technology is a kind of digital transmission technology based on IP network. It is one of the important methods to use voice service in VoIP as steganographic carrier to ensure secure transmission. However, the traditional steganographic code has some problems, such as low embedding efficiency and weak concealment, which cannot meet the requirements of VoIP streaming media information hiding for the security of secret information. Therefore, a steganographic algorithm combining F5 and simplified wet paper code (SWPC) algorithm is proposed. The main idea is to embed secret messages in each row of the carrier matrix using the F5 algorithm, and then the SWPC algorithm is used to embed the columns according to the wet and dry characteristics of the wet paper code without affecting the results before row embedding. We use the VoIP streams encoded by the ITU-T G.729a codec as a carrier to verify the proposed scheme. The experimental results demonstrate that the proposed scheme can achieve relatively better IP speech data steganographic transparency and that it can outperform F5-WPC and SWPC approaches.

Analysis of F5 Algorithm and Improvised F5 Algorithm in Image Hiding with quality safeguarding and Video Hiding by Improvised F5 Algorithm

Analysis of F5 Algorithm and Improvised F5 Algorithm in Image Hiding with quality safeguarding and Video Hiding by Improvised F5 Algorithm

IMPLEMENTATION OF CRYPTOGRAPHY AND STEGANOGRAPHY FOR TEXT ON COVER IMAGE USING AES AND F5 ALGORITHM

This journal explains about implementation that combine both cryptography and steganography method for texton cover image to increase the security level. Text will be encrypted with AES algorithm, and then it will be embedded to the cover image using F5 algorithm. The implemented AES algorithm has a good performance, with Avalanche Effect value ranges from 0.43 � 0.59. The resulting image, or stego image, has a very similar histogram with the original image, so there is no significant difference between the two of them. However, the file size change about 1.25 � 3.25 times larger than theoriginal image. If noise or disruption is given to stego image, the information can not be extracted.Keywords: cryptography, steganography, AES, F5

A survey on signature-based algorithms for computing Gröbner bases

In 1965 Buchberger introduced an algorithmic approach to compute Gröbner bases. Later on, he and many others presented various attempts to improve the computation by removing useless elements a priori. One approach, initiated by Gebauer, Möller, Mora and Traverso in the 1990s, is to keep track of the corresponding syzygies which is related to the topic of this survey: signature-based algorithms for Gröbner bases. This area was initiated by Faugère's F5 algorithm in 2002. The general idea of signatures is to keep track of the history of the computation with a minimal overhead and to exploit this information to detect redundant elements. Here we give a summary of the literature on signature-based algorithms and show how to classify known algorithms by 3 different orderings. For this we give translations between different notations and show the relationships (differences and similarities) among many approaches. Moreover, we give a general description of how the idea of signatures is quite natural when performing the reduction process using linear algebra. We hope that this survey would help to outline this field of active research.

Relating the Embedding Efficiency of LSB Steganography Techniques in Spatial and Transform Domains

This paper is based on image steganography that is Least Significant Bits (LSB) techniques on images to enhance the security of the communication. The LSB-based technique is the most challenging one because it is difficult to differentiate between the cover-object and stego-object, if few LSB bits of the cover object are replaced. The LSB approach combined with F5 algorithm and matrix embedding which is applied on both spatial and frequency domain of an image. The Mean squared error (MSE) and Peak signal to noise ratio (PSNR) was used as the performance measure to compare the different LSB techniques. This paper shows that MSE and PSNR of LSB techniques with Matrix embedding yields better results.

A Secure Steganographic Method for Efficient Data Sharing in Public Clouds

Data relaying process through internet has become more faster which makes it easier to send the data precisely to the destination. Security and transmission time are the most important factors of data transmission. The proposed system improves these factors using the concepts of cryptography and steganography. In this paper, the data hiding technique is implemented in encryption-decryption domain, by joining the concepts of steganography and public key cryptography using the sequence encryption-compression then decompression-decryption. The proposed system uses the mediated certificate less encryption (mCL-PKE) scheme which is combined with the data hiding scheme for providing more confidentiality in public key cryptography. The main idea is to implement the separable reversible data hiding scheme into the mCL-PKE scheme where the cover media decryption and hidden data freeing are separated based on the availability of keys. Among the various steganographic algorithms like Least Significant Bit (LSB) algorithm, Jsteg and F5 algorithms, the proposed system uses F5 algorithm which possess the matrix encoding technique that provides high steganographic capacity and can also prevent visual and statistical attacks.

On implementing signature-based Gröbner basis algorithms using linear algebraic routines from M4RI

On implementing signature-based Gröbner basis algorithms using linear algebraic routines from M4RI

A survey on signature-based Gröbner basis computations

No abstract available.

Sparse Gröbner bases

Toric (or sparse) elimination theory is a framework developed during the last decades to exploit monomial structures in systems of Laurent polynomials. We describe sparse variants of the F5 and FGLM algorithms that allow us to compute efficiently sparse Grobner bases of sparse systems. In the case where the generating subset of monomials corresponds to the points with integer coordinates in a normal lattice polytope and under some regularity assumptions, we prove complexity bounds which depend only on the combinatorial properties of the polytope. These bounds yield new estimates on the complexity of solving zerodimensional systems where all polynomials share the same Newton polytope. For instance, we generalize the bound min(n1, n2)+2 on the maximal degree in a Grobner basis of a zero-dim. bilinear system with blocks of variables of sizes (n1, n2) to the multi-homogeneous case: for polynomials of multi-degree (d1,d2, . . .) w.r.t. a partition of the variables in blocks of sizes (n1, n2, . . .) the maximal degree occurring in a Grobner basis computation is bounded by 2+ ∑ ni−max( ⌈ ni+1 di ⌉ ). From a practical point of view, a proof-of-concept implementation shows large speed-ups compared to optimized Grobner bases implementation.

Secure Video Transmission

Nowadays, as the technology has developed, methods need to be found out that not only hide the information but also hide the existence of that information. The art of hiding information is known as steganography. Till now many algorithms have been proposed for steganography but they suffer few or more drawbacks. This paper uses F5 algorithm for information hiding. F5 offers huge capacity for steganography. F5 is based on matrix encoding. The proposed system is evaluated on the basis of its ability to hide and retrieve the information correctly. In recent days video steganography is considered as a boon for secure and secret transmission of data. The main aim of this paper is to provide new ways of improving the existing methodologies to hide information. In continuation with this, we start by first describing the previously used algorithms and then enhancing them by describing the use of F5. General Terms Digital image processing, Network security, Algorithms.

A new signature-based algorithms for computing Gröbner bases

Gao, Volny and Wang (2010) gave a simple criterion for signature-based algorithms to compute Grobner bases. It gives a unified frame work for computing Grobner bases for both ideals and syzygies, the latter is very important in free resolutions in homological algebra. Sun and Wang (2011) later generalized the GVW criterion to a more general situation (to include the F5 Algorithm). Signature-based algorithms have become increasingly popular for computing Grobner bases. The current paper introduces a concept of factor pairs that can be used to detect more useless J-pairs than the generalized GVW criterion, thus improving signature-based algorithms.

Pixel Steganalysis - A Novel Approach in ImageSteganography Using F5 Algorithm

The Faugere F5 algorithm first calculates the Grobner basis of a pair of generator polynomials of the ideal. And it uses to reduce the sizes of initial matrices of created messages which had to be embedded in image steganography. The stenographical systems are weak against statistically and visually weak which offers only a relatively small capacity of stenographic message but the newly developed f5 algorithm help to embedded messages in images large capacity of steganography. It implement matrix encoding and permutation straddling to embedded message with large capacity and reduces the embedded-image.

On the complexity of the [formula omitted] Gröbner basis algorithm

We study the complexity of Gröbner bases computation, in particular in the generic situation where the variables are in simultaneous Noether position with respect to the system.We give a bound on the number of polynomials of degree d in a Gröbner basis computed by Faugère's F5 algorithm (2002) in this generic case for the grevlex ordering (which is also a bound on the number of polynomials for a reduced Gröbner basis, independently of the algorithm used). Next, we analyse more precisely the structure of the polynomials in the Gröbner bases with signatures that F5 computes and use it to bound the complexity of the algorithm.Our estimates show that the version of F5 we analyse, which uses only standard Gaussian elimination techniques, outperforms row reduction of the Macaulay matrix with the best known algorithms for moderate degrees, and even for degrees up to the thousands if Strassen's multiplication is used. The degree being fixed, the factor of improvement grows exponentially with the number of variables.

Termination of the F5 algorithm

The F5 algorithm, which calculates the Grobner basis of an ideal generated by homogeneous polynomials, was proposed by Faugere in 2002; simultaneously, the correctness of this algorithm was proved under the condition of termination. However, termination itself was demonstrated only for a regular sequence of polynomials. In this paper, it is proved that the algorithm terminates for any input data. First, it is shown that if the algorithm does not terminate, it eventually generates two polynomials where the first is a reductor for the second. However, it is not argued that such a reduction is permitted by all the criteria introduced in F5. Next, it is shown that if such a pair exists, then there exists another pair for which the reduction is permitted by all the criteria, which is impossible.

Effect of morphing on embedding capacity and embedding efficiency

This paper makes use a morphing technique to increase embedding efficiency and embedding capacity of a cover image. It has modified F5 algorithm (Westfeld, 2001) [1] which provides higher security and embedding capacity in transform domain such that in the same cover image larger amount of data can be embedded. In this paper, the embedding capacity of the cover image has been improved by increasing the non-zero DCT coefficients. Experimental results show that for the test images under consideration, it can achieve the improvement of embedding capacity by about 53% and has better embedding efficiency in transform domain as compared to the original F5 algorithm for the same cover image.

A non-commutative [formula omitted] algorithm with an application to the computation of Loewy layers

We provide a non-commutative version of the F5 algorithm, namely for right-modules over path algebra quotients. It terminates, if the path algebra quotient is a basic algebra. We show that the signatures used in the F5 algorithm allow to read off a basis for each Loewy layer, provided that a negative degree monomial ordering is used. As a byproduct, Gröbner bases in this setting can be computed more efficiently with the F5 algorithm than with Buchberger's algorithm.