SUPERSINGULAR EDWARDS CURVES AND EDWARDS CURVE POINTS COUNTING METHOD OVER FINITE FIELD
DOI:
https://doi.org/10.17721/2706-9699.2020.1.06Keywords:
finite field, elliptic curve, Edwards curve, group of points of an elliptic curveAbstract
We consider problem of order counting of algebraic affine and projective curves of Edwards [2, 8] over the finite field $F_{p^n}$. The complexity of the discrete logarithm problem in the group of points of an elliptic curve depends on the order of this curve (ECDLP) [4, 20] depends on the order of this curve [10]. We research Edwards algebraic curves over a finite field, which are one of the most promising supports of sets of points which are used for fast group operations [1]. We construct a new method for counting the order of an Edwards curve over a finite field. It should be noted that this method can
be applied to the order of elliptic curves due to the birational equivalence between elliptic curves and Edwards curves. We not only find a specific set of coefficients with corresponding field characteristics
for which these curves are supersingular, but we additionally find a general formula by which one can determine whether a curve $E_d [F_p]$ is supersingular over this field or not. The embedding degree of the supersingular curve of Edwards over $F_{p^n}$ in a finite field is investigated and the field characteristic, where this degree is minimal, is found. A birational isomorphism between the Montgomery curve and the Edwards curve is also constructed. A one-to-one correspondence between the Edwards supersingular curves and Montgomery supersingular curves is established. The criterion of supersingularity for Edwards curves is found over $F_{p^n}$.
References
Daniel J. Bernstein, Peter Birkner, Marc Joye, Tanja Lange, and Christiane Peters. Twisted edwards curves. In: Serge Vaudenay (ed.) Progress in Cryptology – AFRICACRYPT 2008, Berlin, Heidelberg, 2008. Springer. P. 389–405.
Edwards H. A normal form for elliptic curves. Bulletin of the American mathematical society. 2007. 44(3). P. 393–422.
Fulton W. Algebraic curves. An Introduction to Algebraic Geometry. AddisonWesley, 3 edition, 2008.
Koblitz N. Elliptic curve cryptosystems. Mathematics of computation. 1987. 48(177). P. 203–209.
Lidl R., Niederreiter H. Introduction to Finite Fields and their Applications. Cambridge University Press, 1994.
Montgomery P. L. Speeding the pollard and elliptic curve methods of factorization. Mathematics of computation. 1987. 48(177). P. 243–264.
Schoof R. Counting points on elliptic curves over finite fields. Journal de th´eorie des nombres de Bordeaux. 1995. 7(1). P. 219–254.
Skuratovskii R. V. The order of projective edwards curve over F_{p^n} and embedding degree of this curve in finite field. In: Cait 2018, Proceedings of Conferences. 2018. P. 75–80.
Skuratovskii R. V. Supersingularity of elliptic curves over F_{p^n}. Research in Mathematics and Mechanics. 2018. 31(1). P. 17–26. (in Ukrainian)
Skuratovskii R. V. Employment of minimal generating sets and structure of sylow 2-subgroups alternating groups in block ciphers. In: Advances in Computer Communication and Computational Sciences. Springer. 2019. P. 351–364.
Stepanov S. A. Arifmetika algebraicheskikh krivykh. Nauka. Glav. red. fizikomatematicheskoi lit-ry. 1991. (in Russian)
Vinogradov I. M. Elements of number theory. Courier Dover Publications. 2016.
Barreto P. S. L. M., Naehrig M. Pairing-friendly elliptic curves of prime order. In: Bart Preneel and Stafford Tavares (eds.) Selected Areas in Cryptography. Berlin, Heidelberg, 2006. Springer. P. 319–331.
Glazunov N. M., Skobelev S. P. Manifolds over the rings. IAMM National Academy of Sciences of Ukraine. Donetsk. 2011. P. 323.
Varbanec P. D., Zarzycki P. Divisors of the Gaussian integers in an arithmetic progression. Journal of Number Theory. 1989. Vol. 33. Iss. 2. P. 152–169
Silverman J. H. The Arithmetic of Elliptic Curves. Graduate Texts in Mathematics. Vol. 106. Springer-Verlag. 1986.
Skuratovskii R. V., Williams A. A solution of the inverse problem to doubling of twisted Edwards curve point over finite field. Processing, transmission and security of information. 2019. vol. 2. Wydawnictwo Naukowe Akademii TechnicznoHumanistycznej w Bielsku-Bialej.
Deligne P. La conjecture de Weil. Publications Mathematiques de l’IHES. 1974. Vol. 43. P. 273–307.
Ren´e Schoof. Counting points on elliptic curves over finite fields. Journal de th´eorie des nombres de Bordeaux, 7(1):219–254, 1995.
R. Skuratovskii. The Derived Subgroups of Sylow 2-Subgroups of the Alternating Group and Commutator Width of Wreath Product of Groups. Mathematics, Basel, Switzerland, 2020, 8 (4), pp. 3–22.
