{"title":"The Number of Rational Points on Elliptic Curves y2 = x3 + b2 Over Finite Fields","authors":"Bet\u00fcl Gezer, Hacer \u00d6zden, Ahmet Tekcan, Osman Bizim","volume":1,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":97,"pagesEnd":104,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/3721","abstract":"

Let p be a prime number, Fpbe a finite field and let Qpdenote the set of quadratic residues in Fp. In the first section we givesome notations and preliminaries from elliptic curves. In the secondsection, we consider some properties of rational points on ellipticcurves Ep,b: y2= x3+ b2 over Fp, where b ∈ F*p. Recall that theorder of Ep,bover Fpis p + 1 if p ≡ 5(mod 6). We generalize thisresult to any field Fnp for an integer n≥ 2. Further we obtain someresults concerning the sum Σ[x]Ep,b(Fp) and Σ[y]Ep,b(Fp), thesum of x- and y- coordinates of all points (x, y) on Ep,b, and alsothe the sum Σ(x,0)Ep,b(Fp), the sum of points (x, 0) on Ep,b.<\/p>\r\n","references":" G.E. Andrews. Number Theory. Dover Pub., 1971 A.O.L. Atkin and F. Moralin. Eliptic Curves and Primality Proving.Math. Comp. 61 (1993), 29?68. S. Goldwasser and J. Kilian. Almost all Primes can be Quickly Certified.In Proc. 18th STOC, Berkeley, May 28-30, 1986, ACM, New York(1986), 316-329. N. Koblitz. A Course in Number Theory and Cryptography. Springer-Verlag, 1994. H.W.Jr. Lenstra. Factoring Integers with Elliptic Curves. Annals ofMaths. 126(3) (1987), 649?673. V.S. Miller. Use of Elliptic Curves in Cryptography, in Advances inCryptology?CRYPTO?85. Lect. Notes in Comp. Sci. 218, Springer-Verlag, Berlin (1986), 417?426. R.A. Mollin. An Introduction to Cryptography. Chapman&Hall\/CRC,2001. L.J. Mordell. On the Rational Solutions of the Indeterminate Eqnarraysof the Third and Fourth Degrees. Proc. Cambridge Philos. Soc. 21(1922),179?192. R. Schoof. Counting Points on Elliptic Curves Over Finite Fields.Journal de Theorie des Nombres de Bordeaux 7(1995), 219?254. J.H. Silverman. The Arithmetic of Elliptic Curves. Springer-Verlag, 1986. A.Tekcan. Elliptic Curves y2 = x3?t2x over Fp. International Journalof Mathematics Sciences 1(3)(2007), 165-171. L.C. Washington. Elliptic Curves, Number Theory and Cryptography.Chapman&Hall \/CRC, Boca London, New York, Washington DC, 2003. A. Wiles. Modular Elliptic Curves and Fermat?s Last Theorem. Annalsof Maths. 141(3) (1995), 443?551.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 1, 2007"}