## An analytic approach to smooth polynomials over finite fields (1998)

Venue: | in Algorithmic Number Theory: Third Intern. Symp., ANTS-III |

Citations: | 15 - 2 self |

### BibTeX

@INPROCEEDINGS{Panario98ananalytic,

author = {Daniel Panario and Xavier Gourdon},

title = {An analytic approach to smooth polynomials over finite fields},

booktitle = {in Algorithmic Number Theory: Third Intern. Symp., ANTS-III},

year = {1998},

pages = {226--236},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract. We consider the largest degrees that occur in the decomposi-tion 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 dis-tribution of the two largest degrees of irreducible factors, a problem relevant to polynomial factorization. As opposed to most earlier treat-ments, our methods are based on a combination of exact descriptions by generating functions and a specific complex asymptotic method. 1

### Citations

2959 | New directions in cryptography
- Diffie, Hellman
- 1976
(Show Context)
Citation Context ...ecurity of many applications in public-key cryptography relies on the computational intractability of finding discrete logarithms in finite fields. Examples are the Diffie-Hellman key exchange scheme =-=[7]-=-, E1 Gamal's cryptosystem [8], and pseudorandom bit generators [3,10]. On the other hand, algorithms for computing discrete logarithms in finite fields depend on finding polynomials with all of their ... |

2449 |
Handbook of mathematical functions
- Abramowitz, Stegun
- 1967
(Show Context)
Citation Context ... e -h/n) - n n -h" n(1--e-h/n )-- h § +0 . Nq(n,m)=qn 1 ] e_E(t~h) q- q-O ehdh. 2"~i J l_in ~ n We treat separatedly the three integrals. The fact that e -E(z) is bounded in the domain ~(z) _> 0 (see =-=[1]-=-, w 5.1) entails that the contribution of the big-Oh term in the integral is O(logn/m). Then, an integration by parts gives also a small contribution of order O(log n/n) for the term containing r Fina... |

1229 | A public key cryptosystem and a signature scheme based on the discrete logarithm
- ElGamal
- 1985
(Show Context)
Citation Context ...in public-key cryptography relies on the computational intractability of finding discrete logarithms in finite fields. Examples are the Diffie-Hellman key exchange scheme [7], E1 Gamal's cryptosystem =-=[8]-=-, and pseudorandom bit generators [3,10]. On the other hand, algorithms for computing discrete logarithms in finite fields depend on finding polynomials with all of their irreducible factors with degr... |

623 |
How to generate cryptographically strong sequences of pseudo-random bits
- Blum, Micali
- 1984
(Show Context)
Citation Context ...the computational intractability of finding discrete logarithms in finite fields. Examples are the Diffie-Hellman key exchange scheme [7], E1 Gamal's cryptosystem [8], and pseudorandom bit generators =-=[3,10]-=-. On the other hand, algorithms for computing discrete logarithms in finite fields depend on finding polynomials with all of their irreducible factors with degree not greater than certain bound m -- s... |

49 |
On the frequency of numbers containing prime factors of a certain relative magnitude
- Dickman
- 1930
(Show Context)
Citation Context ...ble factors. The results are expressed in terms of a familiar number-theoretic function, the Dickman function, that is already known to underlie the study of numbers with no primes larger than m; see =-=[5, 6]-=-. Our approach starts with an exact representation of enumeration problems by meanssSmooth Polynomials over Finite Fields 227 of combinatorial generating functions. From there, we develop dedicated co... |

48 |
Integers without large prime factors
- Hildebrand, Tenenbaum
- 1993
(Show Context)
Citation Context ...oth polynomials. This classical number-theoretic function describes the distribution of the largest prime divisor of a random integer [5, 6]. A survey on this topic is due to Hildebrand and Tenenbanm =-=[12]-=-. Our general reference for this paper is Tenenbaum's book [18].s228 Panario, Gourdon, and Flajolet Definition 1. The Dickman function, p(u), is the unique continuous solution of the difference-differ... |

34 |
On the number of positive integers ≤ x and free of prime factors > y
- Bruijn
- 1951
(Show Context)
Citation Context ...ble factors. The results are expressed in terms of a familiar number-theoretic function, the Dickman function, that is already known to underlie the study of numbers with no primes larger than m; see =-=[5, 6]-=-. Our approach starts with an exact representation of enumeration problems by meanssSmooth Polynomials over Finite Fields 227 of combinatorial generating functions. From there, we develop dedicated co... |

24 |
Discrete logarithms and their cryptographic significance
- Odlyzko
- 1985
(Show Context)
Citation Context ...res, like random mappings. The results on smooth polynomials are presented in Section 2. The number of m-smooth polynomials of degree n over lFq has already been considered in the literature. Odlyzko =-=[15]-=- provides an asymptotic estimate when n --+ oc for the case q = 2 and n 1/1~176 < m < n 99/l~176 using the saddle point method. This generalizes to any prime power q; see [13]. Car [4] has given an as... |

22 | Asymptotic semi-smoothness probabilities
- Bach, Peralta
- 1992
(Show Context)
Citation Context ...fields that are defined by constraints on the degrees of several of their largest irreducible factors. (These are the equivalent for polynomials of the semismooth integers defined by Bach and Peralta =-=[2]-=-.) We illustrate this fact by treating in some detail the joint distribution of the largest two irreducible factors, a problem that is again of relevance for polynomial factorization algorithms. Throu... |

22 | Rigorous discrete logarithm computations in finite fields via smooth polynomials - Bender, Pomerance - 1998 |

16 | The complete analysis of a polynomial factorization algorithm over finite fields
- Flajolet, Panario
(Show Context)
Citation Context ... distribution of the two largest degrees D~ ], Din 2] of the distinct factors of a random polynomial of degree n in lFq provides the halting condition for the distinct-degree factorization stage; see =-=[9]-=-. We first investigate the distribution of the largest degree D~ ] which is of independent interest. The same analysis techniques are then applied in order to produce the joint distribution of D~ ], D... |

10 |
subexponential algorithms for discrete logarithms over finite fields
- Lovorn, Rigorous
- 1992
(Show Context)
Citation Context ... the literature. Odlyzko [15] provides an asymptotic estimate when n --+ oc for the case q = 2 and n 1/1~176 < m < n 99/l~176 using the saddle point method. This generalizes to any prime power q; see =-=[13]-=-. Car [4] has given an asymptotic expression for this number in terms of the Dickman function, but Car's estimates only hold for m large with respect to n, typically m > c n log log n~ log n. Finally,... |

6 | Gauss Periods: Orders and Cryptographical Applications
- Gao, Gathen, et al.
- 1998
(Show Context)
Citation Context ...the computational intractability of finding discrete logarithms in finite fields. Examples are the Diffie-Hellman key exchange scheme [7], E1 Gamal's cryptosystem [8], and pseudorandom bit generators =-=[3,10]-=-. On the other hand, algorithms for computing discrete logarithms in finite fields depend on finding polynomials with all of their irreducible factors with degree not greater than certain bound m -- s... |

1 |
Th~or~mes de densit4 dans ]Fq[x
- CAR
- 1987
(Show Context)
Citation Context ...ature. Odlyzko [15] provides an asymptotic estimate when n --+ oc for the case q = 2 and n 1/1~176 < m < n 99/l~176 using the saddle point method. This generalizes to any prime power q; see [13]. Car =-=[4]-=- has given an asymptotic expression for this number in terms of the Dickman function, but Car's estimates only hold for m large with respect to n, typically m > c n log log n~ log n. Finally, Soundara... |

1 |
Combinatoire, algorithmique et gdomgtrie des polyn6mes. Th~se, l~cole Polytechnique
- GOURDON
- 1996
(Show Context)
Citation Context ... in the spirit of analytic number theory but have quite a different technical flavour since power series are used instead of Dirichlet series. Such an approach is of general applicability and Gourdon =-=[11]-=- introduced it in order to study the size of the largest cycle in random permutations (where nonconstructive Tauberian methods had been previously used), as well as largest components in several decom... |