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Selmer groups of supersingular elliptic curves [PDF]
Journal of Soviet Mathematics, 1987 Translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 75, 16-21 (Russian) (1978; Zbl 0449.14009).
M. I. Bashmakov, A. S. Kurochkin
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Supersingular elliptic curves over ℤ𝑝-extensions
Journal für die Reine und Angewandte Mathematik, 2023Let E / Q \mathrm{E}/\mathbb{Q} be an elliptic curve and 𝑝 a prime of supersingular reduction for E \mathrm{E} . Consider a quadratic extension L / Q p L/\mathbb{Q}_{p} and the corresponding anticyclotomic Z p \mathbb{Z}_{p} -extension L ∞ / L L_{\infty}/
M. Çiperiani
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On the supersingular reduction of elliptic curves [PDF]
Proceedings of the Indian Academy of Sciences - Section A, 1987 Let E be an elliptic curve over \({\mathbb{Q}}\), and let p be a prime of supersingular reduction for E. The author shows that the 2-complement of \(E({\mathbb{F}}_ p)\) is cyclic. In particular, if \(E_ a\) is the curve \(y^ 2=(x^ 2+1)(x+a)\) (a\(\in {\mathbb{Q}})\) the author combines the above result with Elkies' theorem (that there are infinitely ...
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A supersingularity criterion of elliptic curves
Journal of Mathematical Sciences, 1997A well-known Belyi theorem states that an arbitrary algebraic curve defined over \(\overline \mathbb{Q}\) can be mapped onto the projective line \(\mathbb{P}^1\) so that the whole of ramification will be concentrated over three points of \(\mathbb{P}^1\) (we may assume that these points are \(\infty, 0,1)\).
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Iwasawa theory for elliptic curves at supersingular primes [PDF]
Inventiones Mathematicae, 2003 Let \(p\) be an odd prime, \({\mathbb Q}_{\infty} = \bigcup_{n}\;F_{n}\) the cyclotomic \({\mathbb Z}_{p}\)-extension of \({\mathbb Q},\) \(\wedge\) the usual Iwasawa algebra. In the Iwasawa theory of elliptic curves at good ordinary primes, the Main Conjecture states that the Selmer group over \({\mathbb Q}_{\infty}\) is \(\wedge\)-cotorsion and the ...
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Arithmetic on non supersingular elliptic curves
1991We discuss the different possibilities to choose elliptic curves over different finite fields with respect to application for public key cryptosystems.
Beth, Thomas, Schaefer, Frank
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Supersingular Elliptic Curves in Cryptography
2007I will survey the checkered history of supersingular elliptic curves in cryptography, from their first consideration in the seminal papers of Koblitz and Miller, to their rejection after the discovery of the Weil and Tate pairing attacks on the discrete logarithm problem for these curves, and concluding with their resurrection alongside the discovery ...
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An elementary proof for the number of supersingular elliptic curves
São Paulo Journal of Mathematical Sciences, 2020Building on Finotti in (Acta Arith 139(3):265–273, 2009), we give an elementary proof for the well known result that there exactly $$\lceil (p-1)/4 \rceil -\lfloor (p-1)/6 \rfloor$$ supersingular elliptic curves in characteristic p. We use a related polynomial instead of the supersingular polynomial itself to simplify the proof and this idea might ...
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Pseudorandom number generator based on supersingular elliptic curve isogenies
Science China Information Sciences, 2021Yan Huang +3 more
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On the Cost of Computing Isogenies Between Supersingular Elliptic Curves [PDF]
, 2019 The security of the Jao-De Feo Supersingular Isogeny Diffie-Hellman (SIDH) key agreement scheme is based on the intractability of the Computational Supersingular Isogeny (CSSI) problem—computing \({\mathbb F}_{p^2}\)-rational isogenies of degrees \(2^e\) and \(3^e\) between certain supersingular elliptic curves defined over \({\mathbb F}_{p^2}\).
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