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A New Efficient Algorithm for Computing Gröbner Bases (F4)
 IN: ISSAC ’02: PROCEEDINGS OF THE 2002 INTERNATIONAL SYMPOSIUM ON SYMBOLIC AND ALGEBRAIC COMPUTATION
, 2002
"... This paper introduces a new efficient algorithm for computing Gröbner bases. To avoid as much as possible intermediate computation, the algorithm computes successive truncated Gröbner bases and it replaces the classical polynomial reduction found in the Buchberger algorithm by the simultaneous reduc ..."
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Cited by 365 (57 self)
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This paper introduces a new efficient algorithm for computing Gröbner bases. To avoid as much as possible intermediate computation, the algorithm computes successive truncated Gröbner bases and it replaces the classical polynomial reduction found in the Buchberger algorithm by the simultaneous reduction of several polynomials. This powerful reduction mechanism is achieved by means of a symbolic precomputation and by extensive use of sparse linear algebra methods. Current techniques in linear algebra used in Computer Algebra are reviewed together with other methods coming from the numerical field. Some previously untractable problems (Cyclic 9) are presented as well as an empirical comparison of a first implementation of this algorithm with other well known programs. This comparison pays careful attention to methodology issues. All the benchmarks and CPU times used in this paper are frequently updated and available on a Web page. Even though the new algorithm does not improve the worst case complexity it is several times faster than previous implementations both for integers and modulo computations.
An algorithm for solving the discrete log problem on hyperelliptic curves
, 2000
"... Abstract. We present an indexcalculus algorithm for the computation of discrete logarithms in the Jacobian of hyperelliptic curves defined over finite fields. The complexity predicts that it is faster than the Rho method for genus greater than 4. To demonstrate the efficiency of our approach, we de ..."
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Cited by 92 (7 self)
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Abstract. We present an indexcalculus algorithm for the computation of discrete logarithms in the Jacobian of hyperelliptic curves defined over finite fields. The complexity predicts that it is faster than the Rho method for genus greater than 4. To demonstrate the efficiency of our approach, we describe our breaking of a cryptosystem based on a curve of genus 6 recently proposed by Koblitz. 1
Writing on Wet Paper
"... In this paper, we show that the communication channel known as writing in memory with defective cells [1][2] is a relevant informationtheoretical model for a specific case of passive warden steganography when the sender embeds a secret message into a subset C of the cover object X without sharing ..."
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Cited by 42 (10 self)
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In this paper, we show that the communication channel known as writing in memory with defective cells [1][2] is a relevant informationtheoretical model for a specific case of passive warden steganography when the sender embeds a secret message into a subset C of the cover object X without sharing C with the recipient. The set C, also called the selection channel, could be arbitrary, determined by the sender from the cover object using a deterministic, pseudorandom, or a truly random process. We call this steganography “writing on wet paper ” and realize it using a simple variablerate random linear code that gives the sender a convenient flexibility and control over the embedding process and is thus suitable for practical implementation. The importance of the wet paper scenario for covert communication is discussed within the context of adaptive steganography and perturbed quantization steganography [3]. Heuristic arguments supported by tests using blind steganalysis [4] indicate that the wet paper steganography provides improved steganographic security and is less vulnerable to steganalytic attacks compared to existing methods with shared selection channels.
Discrete Logarithms: the Effectiveness of the Index Calculus Method
, 1996
"... . In this article we survey recent developments concerning the discrete logarithm problem. Both theoretical and practical results are discussed. We emphasize the case of finite fields, and in particular, recent modifications of the index calculus method, including the number field sieve and the func ..."
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Cited by 34 (1 self)
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. In this article we survey recent developments concerning the discrete logarithm problem. Both theoretical and practical results are discussed. We emphasize the case of finite fields, and in particular, recent modifications of the index calculus method, including the number field sieve and the function field sieve. We also provide a sketch of the some of the cryptographic schemes whose security depends on the intractibility of the discrete logarithm problem. 1 Introduction Let G be a cyclic group generated by an element t. The discrete logarithm problem in G is to compute for any b 2 G the least nonnegative integer e such that t e = b. In this case, we write log t b = e. Our purpose, in this paper, is to survey recent work on the discrete logarithm problem. Our approach is twofold. On the one hand, we consider the problem from a purely theoretical perspective. Indeed, the algorithms that have been developed to solve it not only explore the fundamental nature of one of the basic s...
Perturbed Quantization Steganography with Wet Paper Codes
 Proc. ACM Multimedia Workshop
"... In this paper, we introduce a new approach to passivewarden steganography in which the sender embeds the secret message into a certain subset of the cover object without having to share the selection channel with the recipient. An appropriate informationtheoretical model for this communication is w ..."
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Cited by 32 (3 self)
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In this paper, we introduce a new approach to passivewarden steganography in which the sender embeds the secret message into a certain subset of the cover object without having to share the selection channel with the recipient. An appropriate informationtheoretical model for this communication is writing in memory with (a large number of) defective cells [1]. We describe a simple variablerate random linear code for this channel (the “wet paper” code) and use it to develop a new steganographic methodology for digital media files – Perturbed Quantization. In Perturbed Quantization, the sender hides data while processing the cover object with an informationreducing operation, such as lossy compression, downsampling, A/D conversion, etc. The sender uses the cover object before processing as side information to confine the embedding changes to those elements of the processed cover object whose values are the most “uncertain”. This informedsender embedding and uninformedrecipient message extraction improves steganographic security because an attacker cannot easily determine from the processed stego object the location of embedding changes. Heuristic is presented and supported by blind steganalysis [2] that a specific case of Perturbed Quantization for JPEG images is significantly less detectable than current JPEG steganographic methods.
Massively parallel computation of discrete logarithms
, 1993
"... Numerous cryptosystems have been designed to be secure under the assumption that the computation of discrete logarithms is infeasible. This paper reports on an aggressive attempt to discover the size of fields of characteristic two for which the computation of discrete logarithms is feasible. We dis ..."
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Cited by 31 (0 self)
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Numerous cryptosystems have been designed to be secure under the assumption that the computation of discrete logarithms is infeasible. This paper reports on an aggressive attempt to discover the size of fields of characteristic two for which the computation of discrete logarithms is feasible. We discover several things that were previously overlooked in the implementation of Coppersmith’s algorithm, some positive, and some negative. As a result of this work we have shown that fields as large as GF(2^503) can definitely be attacked.
Distributed MatrixFree Solution of Large Sparse Linear Systems over Finite Fields
 Algorithmica
, 1996
"... We describe a coarsegrain parallel software system for the homogeneous solution of linear systems. Our solutions are symbolic, i.e., exact rather than numerical approximations. Our implementation can be run on a network cluster of SPARC20 computers and on an SP2 multiprocessor. Detailed timings a ..."
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Cited by 29 (6 self)
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We describe a coarsegrain parallel software system for the homogeneous solution of linear systems. Our solutions are symbolic, i.e., exact rather than numerical approximations. Our implementation can be run on a network cluster of SPARC20 computers and on an SP2 multiprocessor. Detailed timings are presented for experiments with systems that arise in RSA challenge integer factoring efforts. For example, we can solve a 252; 222 \Theta 252; 222 system with about 11.04 million nonzero entries over the Galois field with 2 elements using 4 processors of an SP2 multiprocessor, in about 26.5 hours CPU time. 1 Introduction The problem of solving large, unstructured, sparse linear systems using exact arithmetic arises in symbolic linear algebra and computational number theory. For example the sievebased factoring of large integers can lead to systems containing over 569,000 equations and variables and over 26.5 million nonzero entries, that need to be solved over the Galois field of two...
Improvements to the general number field sieve for discrete logarithms in prime fields. A comparison with . . .
 MATHEMATICS OF COMPUTATION
, 2003
"... In this paper, we describe many improvements to the number field sieve. Our main contribution consists of a new way to compute individual logarithms with the number field sieve without solving a very large linear system for each logarithm. We show that, with these improvements, the number field sie ..."
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Cited by 27 (2 self)
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In this paper, we describe many improvements to the number field sieve. Our main contribution consists of a new way to compute individual logarithms with the number field sieve without solving a very large linear system for each logarithm. We show that, with these improvements, the number field sieve outperforms the gaussian integer method in the hundred digit range. We also illustrate our results by successfully computing discrete logarithms with GNFS in a large prime field.