UDC 621.391.15 : 519.7
1 Borys Grinchenko Kyiv University and Institute of Physics and Technology of the National Technical University “Igor Sikorsky Kyiv Polytechnic Institute”
 bessalov@ukr.net
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2 Institute of Physics and Technology of the National Technical University “Igor Sikorsky Kyiv Polytechnic Institute”
lusi.kovalchuk@gmail.com
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SUPERSINGULAR TWISTED EDWARDS CURVES OVER A SIMPLE FIELD.
ІI. SUPERSINGULAR TWISTED EDWARDS CURVES WITH AN j-INVARIANT EQUAL TO 663
Abstract. Theorems on the existence conditions for Edwards super singular curves over a simple field
with an j-invariant equal to 663 and with other values of the j-invariants were formulated and proved.
A generalization of the previously obtained results using the isomorphism of curves in the Legendre and Edwards forms is given.
Keywords: supersingular curve, complete Edwards curve, twisted Edwards curve, quadratic Edwards curve, torsion pair, point order, Legendre symbol, quadratic residue, quadratic non-deduction.
FULL TEXT
REFERENCES
- Bessalov A.V., Kovalchuk L.V. Supersingular twisted edwards curves over prime fields. I. supersingular twisted Edwards curves with ј -invariants 0 and 123. Kibernetika i sistemnyj analiz. 2019. Vol. 55, N 3. P. 3–10.
- Bessalov A.V. Edwards-shaped elliptic curves and cryptography (in Russian). Kiev: Igor Sikorsky KPI. Izd-vo «Politekhnika», 2017. 272 с.
- Bernstein D.J., Lange T. Faster addition and doubling on elliptic curves. In: Advances in Cryptology—ASIACRYPT’2007 (Proc. 13th Int. Conf. on the Theory and Application of Cryptology and Information Security. Kuching, Malaysia. December 2–6, 2007). Lect. Notes Comp. Sci. Vol. 4833. Berlin: Springer, 2007. P. 29–50.
- Menezes A.J, Okamoto T., Vanstone S.A. Reducing elliptic curve logarithms to logarithms in a finite field. IEEE Transactions on Information Theory. 1993. Vol. 39, Iss. 5. P. 1639–1646.
- Washington L.C. Elliptic curvres. Number theory and cryptography. Second Edition. CRC Press, 2008. 513 p.
- De Feo L., Jao D., Plut J. Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies. J. Mathematical Cryptology. 2014. Vol. 8, N 3. P. 209–247.
- Unruh D. Non-interactive zero-knowledge proofs in the quantum random oracle model. Berlin; Heidelberg: Springer, 2015. P. 755–784.
- Yoo Y., Azarderakhsh R., Jalali A., Jao D., Soukharev V. A post-quantum digital signature scheme based on supersingular isogenies. Cryptology ePrint Archive, Report 2017/186, 2017. URL: http://eprint.iacr.org/2017/186. 18 p.
- Bernstein D.J., Birkner P., Joye M., Lange T., Peters Ch. Twisted Edwards curves. IST Programme under Contract IST–2002–507932 ECRYPT and in Part by the National Science Foundation under Grant ITR–0716498, 2008. Р. 1–17.
- Bessalov A.V.,Tsygankova O.V. Interrelation of families of points of high order on the Edwards curve over a prime field. Probl. Peredachi Inf. 2015. Vol. 51, Iss. 4. P. 92–98.
- Bessalov А.V., Tsygankova О.V. Classification of curves in the Edwards form over a prime field. Applied Radio Electronics. 2015. Vol. 14, N 4. P. 197–203.
- Bessalov A.V., Tsygankova O.V. The number of curves in the generalized Edwards form with a minimal even cofactor of the order of the curve. Probl. Peredachi Inf. 2017. Vol. 53, Iss. 1. P. 101–111.
- Bessalov A.V., Telizhenko A.B. Cryptosystems on elliptic curves (in Russian). Kiev: IVC "Politehnika", 2004. 224 p
- Morain F. Edwards curves and CM curves. ArXiv 0904/2243v1 [Math.NT] Apr.15, 2009. 15 p.
- Bessalov A.V., Kovalchuk L.V. The exact number of elliptic curves in the canonical form, which are isomorphic to Edwards curves over the prime field. Kibernetika i sistemnyj analiz. 2015. Vol. 51, N 2. P. 3–12.
- Bespalov O. Generalization of Gauss's lemma on the characteristics of pairs of elements of a simple finite field. In: Proc.V.M. Glushkov Institute of Cybernetics of the National Academy of Sciences of Ukraine and Kamenets-Podolsky Ivan Ogienko National University. 2017. Iss. 15. P. 26–31.
