J. von zur Gathen and I. E. Shparlinski, Order of Gauss periods in finite fields, Proc ISAAC '95, Cairns, Australia, (1995), to appear. Home/Search Document Details and Download
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Elements Of Provable High Orders In Finite Fields - Gao (1997) (1 citation) (Correct)
....of finite field arithmetic [6] A special class of Gauss periods generate optimal normal bases [11, 10] In [5] Gao and Vanstone find by computer experiments that type II optimal normal basis generators are often primitive and always have high orders. Later, von zur Gathen and Shparlinski [8] prove that type II optimal normal basis generators indeed have orders at least 2 2n 2 . This is the first result proving that certain elements in F 2 n Date: November 7, 1997. 1991 Mathematics Subject Classification. Primary 11T55; Secondary 11Y16, 68Q25, 11T06, 12Y05. Key words and ....
....7, 1997. 1991 Mathematics Subject Classification. Primary 11T55; Secondary 11Y16, 68Q25, 11T06, 12Y05. Key words and phrases. Finite fields, primitive elements, elements of provable high orders, compositions of polynomials. To appear in Proc. American Math. Soc. Von zur Gathen Shparlinski [8] prove only for a subclass of type II optimal normal basis generators, i.e. when 2 is primitive modulo 2n 1, but their argument can be easily modified to work for the general case. have high orders without factoring 2 1 for infinitely many n. But this does not work for all n since, by Gao ....
J. von zur Gathen and I. Shparlinski, "Orders of Gauss periods in finite fields," Proc. 6th International Symposium on Algorithms and Computation, Cairns, LNCS 1004, 1995, 208--215.
Gauss Periods: Orders and Cryptographical Applications - Gao, Gathen, Panario (1996) Self-citation (Gathen) (Correct)
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J. von zur Gathen and I. E. Shparlinski, Order of Gauss periods in finite fields, Proc ISAAC '95, Cairns, Australia, (1995), to appear.
Constructing Elements Of Large Order In Finite Fields - Gathen, Shparlinski (1999) Self-citation (Gathen Shparlinski) (Correct)
....On the other hand, for many applications instead of a primitive root just an element of high multiplicative order is su#cient. Such applications include but are not limited to cryptography, coding theory, pseudo random number generation and combinatorial designs. Following this idea, the authors [9] have proposed some algorithms to construct elements of exponentially large order for some su#ciently dense sequence of extensions of a prime field F p of small characteristic. The algorithms are based on studying multiplicative orders of Gauss periods in finite fields. As an additional bonus in ....
....the corresponding large order elements are generators of normal bases as well. On the other hand, for several applications it is desirable to provide a large order element either in some given field F q or at least in a not too large extension F q s of it. This work as well as its predecessor [9] provides yet another contribution to the theory of Gauss periods over finite fields and their generalizations and analogues, which have already proved to be useful for a number of various applications [1 9] Apparently Gauss periods are worth a further study of their properties and areas of ....
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J. von zur Gathen and I. E. Shparlinski, `Orders of Gauss periods in finite fields', Appl. Algebra in Engin., Commun. and Comp., 9 (1998), 15--24.
On the Multiplicative Orders of . . . - Shparlinski Self-citation (Shparlinski) (Correct)
....q does not belong to any proper subfield of IF q then at least one of the multiplicative orders of # and # # 1 exceeds c(p, #) ln q) 4 3 # , where c(p, #) 0 depends only on p and arbitrary # 0. Several more results about the multiplicative orders of # and # # 1 can be extracted from [2, 3]. Theorem. For any fixed # 0 and su#ciently large q, for any positive divisors n and m of q 1 with nm # q 3 2 # there exists # # IF # q with ord # = n and ord # # 1 = m. Proof. For a divisor d q 1 let X d denote the set of all multiplicative characters of orders dividing ....
....not rule out the possibility that for some special values of ord # one indeed can directly compute ord (# # 1 ) from ord # (for example, when ord # is very small or satisfies some additional arithmetic restrictions, see the remarks after Research Problem 5. 1 in [1] Moreover, the results of [2, 3] show that in several situations of this kind the multiplicative order of # # 1 is exponentially large. Nevertheless this does not seem to be plausible that under any reasonable assumptions on ord # one can evaluate ord (# # 1 ) precisely just from the value of ord #. Acknowledgments. The ....
J. von zur Gathen and I. Shparlinski, `Orders of Gauss periods in finite fields', Appl. Algebra in Engin., Commun. and Computing , 1998, v.9, 15-24.
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