On supersingular primes of the Elkies' elliptic curve

2019 ◽  
Vol 60 (1) ◽  
pp. 41-59
Author(s):  
Naoki Murabayashi
Keyword(s):  
Elliptic Curve ◽  
Supersingular Primes

On the Distribution of Supersingular Primes

1996 ◽  
Vol 48 (1) ◽  
pp. 81-104 ◽  
Author(s):  
Etienne Fouvry ◽  
M. Ram Murty
Keyword(s):  
Elliptic Curve ◽  
Rational Numbers ◽  
Supersingular Primes ◽  
Image Position

AbstractLet E be a fixed elliptic curve defined over the rational numbers. We prove that the number of primes p ≤ x such that E has supersingular reduction mod p is greater than for any positive δ and x sufficiently large. Here logkx is defined recursively as log(logk-1 x) and log1x = logx. We also establish several results related to the Lang-Trotter conjecture.

scholarly journals THE PLUS/MINUS SELMER GROUPS FOR SUPERSINGULAR PRIMES

2013 ◽  
Vol 95 (2) ◽  
pp. 189-200 ◽  
Author(s):  
BYOUNG DU KIM
Keyword(s):  
Elliptic Curve ◽  
Galois Group ◽  
Finite Index ◽  
Selmer Groups ◽  
Selmer Group ◽  
Supersingular Primes ◽  
Reduction Case ◽  
Ordinary Reduction ◽  
Iwasawa Algebra

AbstractSuppose that an elliptic curve $E$ over $ \mathbb{Q} $ has good supersingular reduction at $p$. We prove that Kobayashi’s plus/minus Selmer group of $E$ over a ${ \mathbb{Z} }_{p} $-extension has no proper $\Lambda $-submodule of finite index under some suitable conditions, where $\Lambda $ is the Iwasawa algebra of the Galois group of the ${ \mathbb{Z} }_{p} $-extension. This work is analogous to Greenberg’s result in the ordinary reduction case.

scholarly journals Tate–Shafarevich groups in anticyclotomic ℤp-extensions at supersingular primes

2009 ◽  
Vol 145 (2) ◽  
pp. 293-308 ◽  
Author(s):  
Mirela Çiperiani
Keyword(s):  
Elliptic Curve ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Supersingular Primes

AbstractLet E/ℚ be an elliptic curve and p a prime of supersingular reduction for E. Denote by $\mathrm {K}_\infty $ the anticyclotomic ℤp-extension of an imaginary quadratic field K which satisfies the Heegner hypothesis. Assuming that p splits in K/ℚ, we prove that ${\mbox {\textcyr {Sh}}} (\mathrm {K}_\infty , \mathrm {E})_{p^\infty }$ has trivial Λ-corank and, in the process, also show that $\mathrm {H^1_{Sel}}(\mathrm {K}_\infty , \mathrm {E}_{p^\infty })$ and $\mathrm {E}(\mathrm {K}_\infty )\otimes \mathbb {Q}_p/\mathbb {Z}_p$ both have Λ-corank two.

scholarly journals NON-COMMUTATIVE p-ADIC L-FUNCTIONS FOR SUPERSINGULAR PRIMES

2012 ◽  
Vol 08 (08) ◽  
pp. 1813-1830
Author(s):  
ANTONIO LEI
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Extension ◽  
Abelian Extension ◽  
Supersingular Primes ◽  
Cyclotomic Extension

Let E/ℚ be an elliptic curve with good supersingular reduction at p with ap(E) = 0. We give a conjecture on the existence of analytic plus and minus p-adic L-functions of E over the ℤp-cyclotomic extension of a finite Galois extension of ℚ where p is unramified. Under some technical conditions, we adopt the method of Bouganis and Venjakob for p-ordinary CM elliptic curves to construct such functions for a particular non-abelian extension.

Study of Safe Elliptic Curve Cryptography over Gaussian Integer

2020 ◽  
Vol E103.A (12) ◽  
pp. 1624-1628
Author(s):  
Kazuki NAGANUMA ◽  
Takashi SUZUKI ◽  
Hiroyuki TSUJI ◽  
Tomoaki KIMURA
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
Gaussian Integer

Enhanced Security Authentication of WiMAX using EC3A

2017 ◽  
Vol 7 (8) ◽  
pp. 219
Author(s):  
Mohd Javed ◽  
Khaleel Ahmad ◽  
Ahmad Talha Siddiqui
Keyword(s):  
Cellular Automata ◽  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
Information Rate ◽  
Security Authentication ◽  
Encryption And Decryption ◽  
High Information

WiMAX is the innovation and upgradation of 802.16 benchmarks given by IEEE. It has numerous remarkable qualities, for example, high information rate, the nature of the service, versatility, security and portability putting it heads and shoulder over the current advancements like broadband link, DSL and remote systems. Though like its competitors the concern for security remains mandatory. Since the remote medium is accessible to call, the assailants can undoubtedly get into the system, making the powerless against the client. Many modern confirmations and encryption methods have been installed into WiMAX; however, regardless it opens with up different dangers. In this paper, we proposed Elliptic curve Cryptography based on Cellular Automata (EC3A) for encryption and decryption the message for improving the WiMAX security

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