S. A. Evdokimov. Factoring a solvable polynomial over a finite field and Generalized Riemann Hypothesis. Zapiski Nauchn. Semin. Leningr. Otdel. Matem. Inst. Acad. Sci. USSR, 176:104--117, 1989. In Russian.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that will construct a generating set for GF(p n ) in time p Delta n O(1) see [22] and [23] so for small p the problem of constructing nonresidues can be solved in deterministic polynomial time unconditionally. Assuming the ERH, the algorithm of Huang [12] as generalized by Evdokimov [11] allows us to deterministically construct a k th power nonresidue in GF(p n ) in time k A Delta (n log p) O(1) for some positive constant A, where currently the best known value for A appears to be at least 2. So for small k the problem of constructing nonresidues can be solved in ....

S. A. Evdokimov. Factoring a solvable polynomial over a finite field and Generalized Riemann Hypothesis. Zapiski Nauchn. Semin. Leningr. Otdel. Matem. Inst. Acad. Sci. USSR, 176:104--117, 1989. In Russian.

....of the dependence on p, the bound on the running time of our algorithm is better than the worstcase bounds on the running times of current algorithms in the literature for factoring arbitrary polynomials over F p . See [15] for an unconditional running time bound of p 1=2 (n log p) O(1) and [8, 14] for running time bounds (assuming ERH) of (n log p) O(1) for polynomials of a special form. The algorithms of von zur Gathen and Ronyai essentially reduce the problem of factoring a polynomial of degree n over F p to the following two problems in time (n log p) O(1) 1) computing the prime ....

S. A. Evdokimov. Factoring a solvable polynomial over a finite field and Generalized Riemann Hypothesis. Zapiski Nauchn. Semin. Leningr. Otdel. Matem. Inst. Acad. Sci. USSR, 176:104-- 117, 1989. In Russian.

....quadratic nonresidue is almost always small [Erd61] so C12 can be solved in deterministic polynomial time for almost all inputs. Rem12 94 On the problem of calculating kth power non residues in GF(p n ) the following is known. On ERH, the algorithm of Huang [Hua85] generalized by Evdokimov [Evd89], constructs a kth power nonresidue, in GF(p n ) in deterministic time (kn log p) O(1) Buchmann and Shoup [BS91] on ERH, construct a kth power non residue in GF(p n ) in deterministic time (log p) O(n) Bach [Bac90] on ERH, has given explicit bounds for estimations of the least kth ....

....that we did not mention the work of Schoof [Sch85] on this problem in our earlier manuscript. Schoof proved that for fixed a, there exists a deterministic algorithm with running time polynomial in log p. Ref14 Many additional references are given in [LN83, page 182] See also Ref16 and [Hua85] [Evd89], BS91] 15 Polynomial roots modulo a prime C15 Input p 2 N, f 2 (Z=pZ) x] Output a 2 Z with f(a) j 0 (mod p) if p 2 Primes and such an a exists. O15 Is C15 in P Rem15 86 See Rem14 86 . C15 is in R [Ber70] CZ81] Rab80b] If the extended Riemann hypothesis is assumed and f has abelian ....

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S. A. Evdokimov. Factoring a solvable polynomial over a finite field and generalized Riemann hypothesis. Zapiski Nauchn. Semin. Leningr. Otdel. Matem. Inst. Acad. Sci. USSR, 176:104--117, 1989. In Russian.

....of eliminating the need for randomness in algorithms. Recently, there has been much work on this problem in the area of number theoretic and algebraic algorithms. In this regard, we mention the deterministic algorithms for constructing irreducible polynomials of degree n over GF(p) in the papers [1, 11] and [8, 26, 28] The running times of the algorithms in [1, 11] are (n log p) O(1) assuming the ERH, whereas the running times of the algorithms in [8, 26, 28] are unconditionally (np) O(1) We also mention the result of Lenstra [20] that isomorphisms between two different models of a GF(p ....

....there has been much work on this problem in the area of number theoretic and algebraic algorithms. In this regard, we mention the deterministic algorithms for constructing irreducible polynomials of degree n over GF(p) in the papers [1, 11] and [8, 26, 28] The running times of the algorithms in [1, 11] are (n log p) O(1) assuming the ERH, whereas the running times of the algorithms in [8, 26, 28] are unconditionally (np) O(1) We also mention the result of Lenstra [20] that isomorphisms between two different models of a GF(p n ) can be computed in deterministic time (n log p) O(1) ....

S. A. Evdokimov. Factoring a solvable polynomial over a finite field and Generalized Riemann Hypothesis. Zapiski Nauchn. Semin. Leningr. Otdel. Matem. Inst. Acad. Sci. USSR, 176:104--117, 1989. In Russian.

....polynomial time a model for F p 2 together with a generating set for F p 2 . Thus, for n = 1 and n = 2, the problem of constructing a generating set can be solved in deterministic polynomial time under the ERH. With the ERH assumed, the algorithm of Huang [13] as generalized by Evdokimov [11] allows us to deterministically construct a kth power nonresidue in F p n in time k A Delta (n log p) O(1) for some positive constant A. The precise value of A is not worked out in [13] or [11] but is certainly at least 1. So for k = n log p) O(1) the problem of constructing kth power ....

....time under the ERH. With the ERH assumed, the algorithm of Huang [13] as generalized by Evdokimov [11] allows us to deterministically construct a kth power nonresidue in F p n in time k A Delta (n log p) O(1) for some positive constant A. The precise value of A is not worked out in [13] or [11], but is certainly at least 1. So for k = n log p) O(1) the problem of constructing kth power nonresidues can be solved in deterministic polynomial time under the ERH. Related to the problem of constructing a generating set is that of searching for a primitive root, i.e. a single element that ....

S. A. Evdokimov. Factoring a solvable polynomial over a finite field and Generalized Riemann Hypothesis. Zapiski Nauchn. Semin. Leningr. Otdel. Matem. Inst. Acad. Sci. USSR, 176:104-- 117, 1989. In Russian.

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S. A. Evdokimov. Factoring a solvable polynomial over a finite field and Generalized Riemann Hypothesis. Zapiski Nauchn. Semin. Leningr. Otdel. Matem. Inst. Acad. Sci. USSR, 176:104--117, 1989. In Russian.

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