The first Gröbner basis algorithm was constructed by Buchberger in 1965; thus it bears his name to this day – Buchberger’s algorithm.[9] Though Buchberger’s algorithm looks relatively simple, it can take a very large amount of time. The step that creates 0 h via a normal form calculation is computationally very difficult. This is particularly frustrating (and wasteful) if the normal form calcul...

Many steganographic systems are weak against visual and statistical attacks. Systems without these weaknesses offer only a relatively small capacity for steganographic messages. The newly developed algorithm F5 withstands visual and statistical attacks, yet it still offers a large steganographic capacity. F5 implements matrix encoding to improve the efficiency of embedding. Thus it reduces the ...

Signature-based algorithms are a popular kind of algorithms for computing Gröbner basis, including the famous F5 algorithm, F5C, extended F5, GV and the GVW algorithm. In this paper, an efficient method is proposed to solve the detachability problem. The new method only uses the outputs of signature-based algorithms, and no extra Gröbner basis computations are needed. When a Gröbner basis is ob...

Article history: Received 2 June 2013 Accepted 15 July 2013 Available online xxxx

This paper introduces a new efficient algorithm for computing Gröbner bases. We replace the Buchberger criteria by an optimal criteria. We give a proof that the resulting algorithm (called F5) generates no useless critical pairs if the input is a regular sequence. This a new result by itself but a first implementation of the algorithm F5 shows that it is also very efficient in practice: for ins...

The F5 algorithm is presented by Faugère in 2002. However, Faugère have not provide the rigorous proofs so far. In this paper, we will give a new complete proof and hence reveal the essence of this algorithm. The proof consists of the correctness and termination of the algorithm and the correctness of two criteria in it.

This paper introduces a new efficient algorithm for computing Gröbner bases. We replace the Buchberger criteria by an optimal criteria. We give a proof that the resulting algorithm (called F5) generates no useless critical pairs if the input is a regular sequence. This a new result by itself but a first implementation of the algorithm F5 shows that it is also very efficient in practice: for ins...

In order to improve the steganographic robustness of the algorithm in binary images, suppose an new steganographic method which deals with the edge of the binary image using mathematical morphology and combines F5 encoding method to embedded information. Marginalization and reconstruction on binary image by dilation and erosion operations, and tag blocks that can embed information to. Finally e...

Buchberger’s algorithm for computing Gröbner bases was introduced in 1965, and subsequently there have been extensive efforts in improving its efficiency. Major algorithms include F4 (Faugère 1999), XL (Courtois et al. 2000) and F5 (Faugère 2002). F5 is believed to be the fastest algorithm known in the literature. Most recently, Gao, Guan and Volny (2010) introduced an incremental algorithm (G2...

The techniques for parametrizing nonsingular cubic surfaces have shown to be of great interest in recent years. This paper is devoted to the rational parametrization of nonsingular cubic blending surfaces. We claim that these nonsingular cubic blending surfaces can be parametrized using the symbolic computation due to their excellent geometric properties. Especially for the specific forms of th...