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Research (Partial List) • Applications to Elliptic Curve Discrete Logarithm Problem in cryptography (R. Shipsey) • Finding integral points (M. Ayad) • Study of nonlinear recurrence sequences (Fibonacci numbers, Lucas numbers, and integers are special cases of EDS) • Appearance of primes (G. Everest, T. Ward, …) • EDS are a special case of Somos Sequences (A. van der Poorten, J. Propp, M. Somos, C. Swart, …) • p-adic & function field cases (J. Silverman) • Continued fractions & elliptic curve group law (W. Adams, A. van der Poorten, M. Razar) • Sigma function perspective (A. Hone, …) • Hyper-elliptic curves (A. Hone, A. van der Poorten, …) • More…
Part VI: Elliptic Nets: Jacking up the Dimension
Page 1 and 2: Elliptic Nets How To Catch an Ellip
Page 3 and 4: Part I: Elliptic Curves are Groups
Page 5 and 6: A Typical Elliptic Curve E E : Y 2
Page 7 and 8: Doubling a Point P on E Tangent Lin
Page 9 and 10: Part II: Elliptic Divisibility Sequ
Page 11 and 12: Example
Page 13 and 14: So What Happens to Point Multiples?
Page 15 and 16: Some Example Sequences 0, 1, 2, 3,
Page 17 and 18: Some Example Sequences 0, 1, 1, -1,
Page 19 and 20: Some more terms… 0, 1, 1, -3, 11,
Page 21 and 22: Take a Lattice Λ in the Complex Pl
Page 23 and 24: Elliptic Functions Zeroes at z = a
Page 25 and 26: Part IV: Elliptic Divisibility Sequ
Page 27 and 28: Elliptic Divisibility Sequences: De
Page 29 and 30: Part V: Reduction Mod p
Page 31 and 32: Reduction Mod p 0, 1, 1, -3, 11, 38
Page 33 and 34: Reduction Mod p 0, 1, 1, -3, 11, 38
Page 35: Periodicity Example 0, 1, 1, 8, 0,
Page 39 and 40: From Sequences to Nets It is natura
Page 41 and 42: Definition A Elliptic Nets: Rank 2
Page 43 and 44: ↑ Q Example 4335 5959 12016 -5528
Page 49 and 50: Equivalence of Definitions
Page 51 and 52: Nets are Integral
Page 53 and 54: Divisibility Property
Page 55 and 56: ↑ Q Example 0 4 1 3 1 2 4 4 4 4 4
Page 57 and 58: Periodicity of Nets
Page 59 and 60: Elliptic Curve Cryptography For cry
Page 61 and 62: Elliptic Curve Diffie-Hellman Key E
Page 63 and 64: Tate Pairing in Cryptography: Tripa
Page 65 and 66: Tate Pairing from Elliptic Nets
Page 67 and 68: Calculating the Net (Rank 2) Based
Page 69 and 70: Embedding Degree k
Page 71 and 72: Possible Research Directions • Ex

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