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An analytic approach to smooth polynomials over finite fields
 in Algorithmic Number Theory: Third Intern. Symp., ANTSIII
, 1998
"... Abstract. We consider the largest degrees that occur in the decomposition of polynomials over finite fields into irreducible factors. We expand the range of applicability of the Dickman function as an approximation for the number of smooth polynomials, which provides precise estimates for the discr ..."
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Cited by 11 (2 self)
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Abstract. We consider the largest degrees that occur in the decomposition of polynomials over finite fields into irreducible factors. We expand the range of applicability of the Dickman function as an approximation for the number of smooth polynomials, which provides precise estimates for the discrete logarithm problem. In addition, we characterize the distribution of the two largest degrees of irreducible factors, a problem relevant to polynomial factorization. As opposed to most earlier treatments, our methods are based on a combination of exact descriptions by generating functions and a specific complex asymptotic method. 1
Computing in groups of Lie type
 Math. Comp
, 2001
"... Abstract. We describe two methods for computing with the elements of untwisted groups of Lie type: using the Steinberg presentation and using highest weight representations. We give algorithms for element arithmetic within the Steinberg presentation. Conversion between this presentation and linear r ..."
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Cited by 8 (2 self)
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Abstract. We describe two methods for computing with the elements of untwisted groups of Lie type: using the Steinberg presentation and using highest weight representations. We give algorithms for element arithmetic within the Steinberg presentation. Conversion between this presentation and linear representations is achieved using a new generalisation of row and column reduction. 1.
A Multilevel Blocking Distinctdegree Factorization Algorithm
 CONTEMPORARY MATHEMATICS
, 2008
"... We give a new algorithm for performing the distinctdegree factorization of a polynomial P(x) over GF(2), using a multilevel blocking strategy. The coarsest level of blocking replaces GCD computations by multiplications, as suggested by Pollard (1975), von zur Gathen and Shoup (1992), and others. ..."
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Cited by 5 (4 self)
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We give a new algorithm for performing the distinctdegree factorization of a polynomial P(x) over GF(2), using a multilevel blocking strategy. The coarsest level of blocking replaces GCD computations by multiplications, as suggested by Pollard (1975), von zur Gathen and Shoup (1992), and others. The novelty of our approach is that a finer level of blocking replaces multiplications by squarings, which speeds up the computation in GF(2)[x]/P(x) of certain interval polynomials when P(x) is sparse. As an application we give a fast algorithm to search for all irreducible trinomials x r + x s + 1 of degree r over GF(2), while producing a certificate that can be checked in less time than the full search. Naive algorithms cost O(r 2) per trinomial, thus O(r 3) to search over all trinomials of given degree r. Under a plausible assumption about the distribution of factors of trinomials, the new algorithm has complexity O(r 2 (log r) 3/2 (log log r) 1/2) for the search over all trinomials of degree r. Our implementation achieves a speedup of greater than a factor of 560 over the naive algorithm in the case r = 24036583 (a Mersenne exponent). Using our program, we have found two new primitive trinomials of degree 24036583 over GF(2) (the previous record degree was 6972593).
The average lengths of the factors of the standard factorization of Lyndon words
"... A nonempty word w of {a, b}* is a Lyndon word if and only if it is strictly smaller for the lexicographical order than any of its proper suffixes. Such a word w is either a letter or admits a standard factorization uv where v is its smallest proper suffix. For any Lyndon word v, we show that the se ..."
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Cited by 2 (1 self)
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A nonempty word w of {a, b}* is a Lyndon word if and only if it is strictly smaller for the lexicographical order than any of its proper suffixes. Such a word w is either a letter or admits a standard factorization uv where v is its smallest proper suffix. For any Lyndon word v, we show that the set of Lyndon words having v as right factor of the standard factorization is rational and compute explicitly the associated generating function. Next we establish that, for the uniform distribution over the Lyndon words of length n, the average length of the right factor v of the standard factorization is asymptotically 3n/4.
Analysis Of Rabin's Irreducibility Test For Polynomials Over Finite Fields
, 2000
"... We give a precise averagecase analysis of Rabin's algorithm for testing the irreducibility of polynomials over finite fields. The main technical contribution of the paper is the study of the probability that a random polynomial of degree n contains an irreducible factor of degree dividing sever ..."
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We give a precise averagecase analysis of Rabin's algorithm for testing the irreducibility of polynomials over finite fields. The main technical contribution of the paper is the study of the probability that a random polynomial of degree n contains an irreducible factor of degree dividing several maximal divisors of the degree n. We then study the expected value and the variance of the number of operations performed by the algorithm. We present an exact analysis when n = Pl and n = PiP2 for Pl,P2 prime numbers, and an asymptotic analysis for the general case.
A Rigorous Proof Of The Waterloo Algorithm For The Discrete Logarithm Problem
"... In this paper we are concerned with the Waterloo variant of the index calculus method for the discrete logarithm problem in F 2 n . We provide a rigorous proof for the heuristic arguments for the running time of the Waterloo algorithm. This implies in studying the behavior of pairs of coprime smooth ..."
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Cited by 1 (0 self)
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In this paper we are concerned with the Waterloo variant of the index calculus method for the discrete logarithm problem in F 2 n . We provide a rigorous proof for the heuristic arguments for the running time of the Waterloo algorithm. This implies in studying the behavior of pairs of coprime smooth polynomials over finite fields. Our proof involves a double saddle point method, and it is in nature similar to the one of Odlyzko for the rigorous analysis of the basic index calculus.
On the Number of Permutations
"... Let m be a positive integer, and p n (m) the proportion of permutations of the symmetric group # n that admit an mth root. Calculating the exponential generating function of these permutations, we show the following asymptotic formula where # is the Euler function and #m an explicit cons ..."
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Let m be a positive integer, and p n (m) the proportion of permutations of the symmetric group # n that admit an mth root. Calculating the exponential generating function of these permutations, we show the following asymptotic formula where # is the Euler function and #m an explicit constant.
In
"... Abstract. We count the number of irreducible polynomials in several variables of a given degree over a finite field. The results are expressed in terms of a generating series, an exact formula and an asymptotic approximation. We also consider the case of the multidegree and the case of indecomposab ..."
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Abstract. We count the number of irreducible polynomials in several variables of a given degree over a finite field. The results are expressed in terms of a generating series, an exact formula and an asymptotic approximation. We also consider the case of the multidegree and the case of indecomposable polynomials. 1.