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74
Discrete Logarithms in Finite Fields and Their Cryptographic Significance
, 1984
"... Given a primitive element g of a finite field GF(q), the discrete logarithm of a nonzero element u GF(q) is that integer k, 1 k q  1, for which u = g k . The wellknown problem of computing discrete logarithms in finite fields has acquired additional importance in recent years due to its appl ..."
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Cited by 87 (6 self)
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Given a primitive element g of a finite field GF(q), the discrete logarithm of a nonzero element u GF(q) is that integer k, 1 k q  1, for which u = g k . The wellknown problem of computing discrete logarithms in finite fields has acquired additional importance in recent years due to its applicability in cryptography. Several cryptographic systems would become insecure if an efficient discrete logarithm algorithm were discovered. This paper surveys and analyzes known algorithms in this area, with special attention devoted to algorithms for the fields GF(2 n ). It appears that in order to be safe from attacks using these algorithms, the value of n for which GF(2 n ) is used in a cryptosystem has to be very large and carefully chosen. Due in large part to recent discoveries, discrete logarithms in fields GF(2 n ) are much easier to compute than in fields GF(p) with p prime. Hence the fields GF(2 n ) ought to be avoided in all cryptographic applications. On the other hand, ...
Subquadratictime factoring of polynomials over finite fields
 Math. Comp
, 1998
"... Abstract. New probabilistic algorithms are presented for factoring univariate polynomials over finite fields. The algorithms factor a polynomial of degree n over a finite field of constant cardinality in time O(n 1.815). Previous algorithms required time Θ(n 2+o(1)). The new algorithms rely on fast ..."
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Cited by 68 (11 self)
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Abstract. New probabilistic algorithms are presented for factoring univariate polynomials over finite fields. The algorithms factor a polynomial of degree n over a finite field of constant cardinality in time O(n 1.815). Previous algorithms required time Θ(n 2+o(1)). The new algorithms rely on fast matrix multiplication techniques. More generally, to factor a polynomial of degree n over the finite field Fq with q elements, the algorithms use O(n 1.815 log q) arithmetic operations in Fq. The new “baby step/giant step ” techniques used in our algorithms also yield new fast practical algorithms at superquadratic asymptotic running time, and subquadratictime methods for manipulating normal bases of finite fields. 1.
A New Polynomial Factorization Algorithm and its Implementation
 Journal of Symbolic Computation
, 1996
"... We consider the problem of factoring univariate polynomials over a finite field. We demonstrate that the new baby step/giant step factoring method, recently developed by Kaltofen & Shoup, can be made into a very practical algorithm. We describe an implementation of this algorithm, and present the re ..."
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Cited by 64 (5 self)
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We consider the problem of factoring univariate polynomials over a finite field. We demonstrate that the new baby step/giant step factoring method, recently developed by Kaltofen & Shoup, can be made into a very practical algorithm. We describe an implementation of this algorithm, and present the results of empirical tests comparing this new algorithm with others. When factoring polynomials modulo large primes, the algorithm allows much larger polynomials to be factored using a reasonable amount of time and space than was previously possible. For example, this new software has been used to factor a "generic" polynomial of degree 2048 modulo a 2048bit prime in under 12 days on a Sun SPARCstation 10, using 68 MB of main memory. 1 Introduction We consider the problem of factoring a univariate polynomial of degree n over the field F p of p elements, where p is prime. This problem has been wellstudied, and many algorithms for its solution have been proposed. In general, the running tim...
Factoring Multivariate Polynomials via Partial Differential Equations
 Math. Comput
, 2000
"... A new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic. It is based on a simple partial differential equation that gives a system of linear equations. Like Berlekamp's and Niederreiter's algorithms for factorin ..."
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Cited by 51 (9 self)
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A new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic. It is based on a simple partial differential equation that gives a system of linear equations. Like Berlekamp's and Niederreiter's algorithms for factoring univariate polynomials, the dimension of the solution space of the linear system is equal to the number of absolutely irreducible factors of the polynomial to be factored and any basis for the solution space gives a complete factorization by computing gcd's and by factoring univariate polynomials over the ground field. The new method finds absolute and rational factorizations simultaneously and is easy to implement for finite fields, local fields, number fields, and the complex number field. The theory of the new method allows an effective Hilbert irreducibility theorem, thus an efficient reduction of polynomials from multivariate to bivariate.
Testing Modules for Irreducibility
 J. Austral. Math. Soc. Ser. A
, 1994
"... A practical method is described for deciding whether or not a finitedimensional module for a group over a finite field is reducible or not. In the reducible case, an explicit submodule is found. The method is a generalisation of the ParkerNorton `Meataxe' algorithm, but it does not depend for its e ..."
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Cited by 42 (2 self)
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A practical method is described for deciding whether or not a finitedimensional module for a group over a finite field is reducible or not. In the reducible case, an explicit submodule is found. The method is a generalisation of the ParkerNorton `Meataxe' algorithm, but it does not depend for its efficiency on the field being small. The principal tools involved are the calculation of the nullspace and the characteristic polynomial of a matrix over a finite field, and the factorisation of the latter. Related algorithms to determine absolute irreducibility and module isomorphism for irreducibles are also described. Details of an implementation in the GAP system, together with some performance analyses are included. 1991 Mathematics subject classification (Amer. Math. Soc.): 20C40, 2004. 1 Introduction The purpose of this paper is to describe a practical method for deciding whether or not a finite dimensional FGmodule M is irreducible, where F = GF (q) is a finite field and G is a fi...
A Recognition Algorithm for Classical Groups over Finite Fields
 Proc. London Math. Soc
, 1998
"... 2. Classical groups and primitive prime divisors...... 121 3. Generic and nongeneric parameters........ 123 4. Groups with two different primitive prime divisor elements... 126 ..."
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Cited by 29 (0 self)
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2. Classical groups and primitive prime divisors...... 121 3. Generic and nongeneric parameters........ 123 4. Groups with two different primitive prime divisor elements... 126
Computing Frobenius Maps And Factoring Polynomials
 Comput. Complexity
, 1992
"... . A new probabilistic algorithm for factoring univariate polynomials over finite fields is presented. To factor a polynomial of degree n over F q , the number of arithmetic operations in F q is O((n 2 +n log q) \Delta (log n) 2 loglog n). The main technical innovation is a new way to compute F ..."
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Cited by 28 (2 self)
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. A new probabilistic algorithm for factoring univariate polynomials over finite fields is presented. To factor a polynomial of degree n over F q , the number of arithmetic operations in F q is O((n 2 +n log q) \Delta (log n) 2 loglog n). The main technical innovation is a new way to compute Frobenius and trace maps in the ring of polynomials modulo the polynomial to be factored. Subject classifications. 68Q40; 11Y16, 12Y05. 1. Introduction We consider the problem of factoring a univariate polynomial over a finite field. This problem plays a central role in computational algebra. Indeed, many of the efficient algorithms for factoring univariate and multivariate polynomials over finite fields, the field of rational numbers, and finite extensions of the rationals solve as a subproblem the problem of factoring univariate polynomials over finite fields (Kaltofen 1990). This problem also has important applications in number theory (Buchmann 1990), coding theory (Berlekamp 1968), and ...
Short Presentations for Finite Groups
 JOURNAL OF ALGEBRA
, 1997
"... We conjecture that every finite group G has a short presentation (in terms of generators and relations) in the sense that the total length of the relations is (log jGj) O(1) . We show that it suffices to prove this conjecture for simple groups. Motivated by applications in computational complexity ..."
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Cited by 26 (11 self)
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We conjecture that every finite group G has a short presentation (in terms of generators and relations) in the sense that the total length of the relations is (log jGj) O(1) . We show that it suffices to prove this conjecture for simple groups. Motivated by applications in computational complexity theory, we conjecture that for finite simple groups, such a short presentation is computable in polynomial time from the standard name of G, assuming in the case of Lie type simple groups over GF (p m ) that an irreducible polynomial f of degree m over GF (p) and a primitive root of GF (p m ) are given. We verify this (stronger) conjecture for all finite simple groups except for the three families of rank 1 twisted groups: we do not handle the unitary groups PSU(3; q) = 2 A 2 (q), the Suzuki groups Sz(q) = 2 B 2 (q), and the Ree groups R(q) = 2 G 2 (q). In particular, all finite groups G without composition factors of these types have presentations of length O((log jGj) 3 ). For...
Open Problems in Number Theoretic Complexity, II
"... this paper contains a list of 36 open problems in numbertheoretic complexity. We expect that none of these problems are easy; we are sure that many of them are hard. This list of problems reflects our own interests and should not be viewed as definitive. As the field changes and becomes deeper, new ..."
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Cited by 26 (0 self)
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this paper contains a list of 36 open problems in numbertheoretic complexity. We expect that none of these problems are easy; we are sure that many of them are hard. This list of problems reflects our own interests and should not be viewed as definitive. As the field changes and becomes deeper, new problems will emerge and old problems will lose favor. Ideally there will be other `open problems' papers in future ANTS proceedings to help guide the field. It is likely that some of the problems presented here will remain open for the forseeable future. However, it is possible in some cases to make progress by solving subproblems, or by establishing reductions between problems, or by settling problems under the assumption of one or more well known hypotheses (e.g. the various extended Riemann hypotheses, NP 6= P; NP 6= coNP). For the sake of clarity we have often chosen to state a specific version of a problem rather than a general one. For example, questions about the integers modulo a prime often have natural generalizations to arbitrary finite fields, to arbitrary cyclic groups, or to problems with a composite modulus. Questions about the integers often have natural generalizations to the ring of integers in an algebraic number field, and questions about elliptic curves often generalize to arbitrary curves or abelian varieties. The problems presented here arose from many different places and times. To those whose research has generated these problems or has contributed to our present understanding of them but to whom inadequate acknowledgement is given here, we apologize. Our list of open problems is derived from an earlier `open problems' paper we wrote in 1986 [AM86]. When we wrote the first version of this paper, we feared that the problems presented were so difficult...
On the Deterministic Complexity of Factoring Polynomials over Finite Fields
 Inform. Process. Lett
, 1990
"... . We present a new deterministic algorithm for factoring polynomials over Z p of degree n. We show that the worstcase running time of our algorithm is O(p 1=2 (log p) 2 n 2+ffl ), which is faster than the running times of previous deterministic algorithms with respect to both n and p. We also ..."
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Cited by 21 (4 self)
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. We present a new deterministic algorithm for factoring polynomials over Z p of degree n. We show that the worstcase running time of our algorithm is O(p 1=2 (log p) 2 n 2+ffl ), which is faster than the running times of previous deterministic algorithms with respect to both n and p. We also show that our algorithm runs in polynomial time for all but at most an exponentially small fraction of the polynomials of degree n over Z p . Specifically, we prove that the fraction of polynomials of degree n over Z p for which our algorithm fails to halt in time O((log p) 2 n 2+ffl ) is O((n log p) 2 =p). Consequently, the averagecase running time of our algorithm is polynomial in n and log p. Keywords: factorization, finite fields, irreducible polynomials. This research was supported by NSF grants DCR8504485 and DCR8552596. Appeared in Information Processing Letters 33, pp. 261267, 1990. An preliminary version of this paper appeared as University of WisconsinMadison, Comput...