# Article

Full entry | PDF   (0.2 MB)
Keywords:
neofield; loop; orthomorphism; complete mapping; orthogonal latin squares; cryptography; balanced round robin tournament
Summary:
We give a short account of the construction and properties of left neofields. Most useful in practice seem to be neofields based on the cyclic group and particularly those having an additional divisibility property, called {\it D-neofields}. We shall give examples of applications to the construction of orthogonal latin squares, to the design of tournaments balanced for residual effects and to cryptography.
References:
[A1] Anderson I.: Balancing carry-over effects in tournaments. in Combinatorial Designs and their Applications, Eds. F.C. Holroyd, K.A.S. Quinn, C.Rowley, B.S. Webb, Chapman and Hall/CRC Research Notes in Mathematics, CRC Press, 1999, pp.1-16. MR 1678585 | Zbl 0958.05016
[A2] Artzy R.: On loops with a special property. Proc. Amer. Math. Soc. 6 (1955), 448-453. MR 0069804 | Zbl 0066.27101
[B1] Bruck R.H.: Loops with transitive automorphism groups. Pacific J. Math. 1 (1951), 481-483. MR 0045705 | Zbl 0044.01101
[D1] Dénes J., Keedwell A.D.: Latin Squares and their Applications. Akadémiai Kiadó, Budapest; English Universities Press, London; Academic Press, New York, 1974. MR 0351850
[D2] Dénes J., Keedwell A.D.: Some applications of non-associative algebraic systems in cryptology. submitted.
[D3] Doner J.R.: CIP-neofields and Combinatorial Designs. Ph.D. Thesis, University of Michigan, U.S.A., 1972.
[D4] Dulmage A.L., Mendelsohn N.S., Johnson D.M.: Orthomorphisms of groups and orthogonal latin squares I. Canad. J. Math. 13 (1961), 356-372. MR 0124229 | Zbl 0097.25102
[E1] ElGamal T.: A public key cryptosystem and a signature scheme based on discrete logarithms. IEEE Trans. Information Theory IT-31, 1985, pp.469-472. MR 0798552 | Zbl 0571.94014
[H1] Hsu D.F.: Cyclic neofields and combinatorial designs. Lecture Notes in Mathematics No. 824, Springer, Berlin, 1980. MR 0616639 | Zbl 0443.16022
[H2] Hsu D.F., Keedwell A.D.: Generalized complete mappings, neofields, sequenceable groups and block designs I, II. Pacific J. Math. 111 (1984), 317-332 and 117 (1985), 291-311. MR 0734858
[K1] Keedwell A.D.: On orthogonal latin squares and a class of neofields. Rend. Mat. e Appl. (5) 25 (1966), 519-561. MR 0220611 | Zbl 0153.32902
[K2] Keedwell A.D.: On property $D$ neofields. Rend. Mat. e Appl. (5) 26 (1967), 383-402. MR 0229538 | Zbl 0153.33001
[K3] Keedwell A.D.: The existence of pathological left neofields. Ars Combinatoria B16 (1983), 161-170. MR 0737119 | Zbl 0529.16030
[K4] Keedwell A.D.: Designing Tournaments with the aid of Latin Squares: a presentation of old and new results. Utilitas Math., to appear. MR 1801302 | Zbl 0971.05031
[K5] Keedwell A.D.: A characterization of the Jacobi logarithms of a finite field. submitted. Zbl 0986.12001
[M1] MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North Holland, Amsterdam, 1977. Zbl 0657.94010
[M2] Mann H.B.: The construction of orthogonal latin squares. Ann. Math. Statist. 13 (1942), 418-423. MR 0007736 | Zbl 0060.02706
[O1] Odlyzko A.M.: Discrete logarithms in finite fields and their cryptographic significance. in Lecture Notes in Computer Science No. 209; Advances in Cryptology, Proc. Eurocrypt 84, Eds. T. Beth, N. Cot, I. Ingemarsson, Springer, Berlin, 1955, pp.224-314. MR 0825593 | Zbl 0594.94016
[P1] Paige L.J.: Neofields. Duke Math. J. 16 (1949), 39-60. MR 0028326 | Zbl 0040.30501
[R1] Russell K.G.: Balancing carry-over effects in round robin tournaments. Biometrika 67 (1980), 127-131. MR 0570514
[T1] Tripke A.: Algebraische and Kombinatorische Structuren von Spielplänen mit Anwendung auf ausgewogen Spielpläne. Diploma Thesis, Ruhr-Universität in Bochum, 1983.

Partner of