Elliptic Nets With Applications to Cryptography
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1 Elliptic Nets With Applications to Cryptography Katherine Stange Brown University
2 Elliptic Divisibility Sequences: Seen In Their Natural Habitat
3 Example
4 Elliptic Divisibility Sequences: Two Good Definitions Definition A
5 Elliptic Divisibility Sequences: Two Good Definitions Definition A Definition B
6 Theorem (M Ward, 1948): A and B are equivalent. From the initial conditions in Definition B, one can explicitly calculate the curve and point needed for Definition A. Definition A Definition B
7 Reflects the structure of a cyclic subgroup of the Mordell-Weil group
8 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
9 From Sequences to Nets It is natural to look for a generalisation that reflects the structure of the entire Mordell-Weil group:
10 In this talk, we work with a rank 2 example Nearly everything can be done for general rank
11 Definition A Elliptic Nets: Rank 2 Case
12 Definition B Elliptic Nets: Rank 2 Case
13 Example Q P
14 Example Q P
15 Example Q P
16 Example Q P
17 Example Q P
18 Example Q P
19 Equivalence of Definitions
20 For any given n, one can compute the explicit bijection.
21 Proof of Equivalence satisfy recurrence (check divisors & value) The axes of a net are elliptic divisibility sequences, from which we determine curve and points A proof using the recurrence relation shows that the axes determine a net
22 Nets are Integral
23 Proof of Integrality By clever choice of recurrence relations, you can control the divisions necessary to calculate each term Very messy & long multivariable induction!
24 Reduction Mod p
25 Example Q P
26 Example Q P
27 Proof of Reduction Theorem Relies on integrality Requires understanding how nets behave under endomorphisms of E n, to reduce to the rank 1 case
28 Divisibility Property
29 Periodicity of Sequences
30 Periodicity of Sequences: Restatement
31 Periodicity of Nets
32 Proof of Periodicity The vanishing condition gives a relation on the points generating the net Prove identity on elliptic functions By reduction theorem, this applies mod p There are a great many more periodicity results!
33 The Tate Pairing
34 Tate Pairing from Elliptic Nets
35 Proof of Tate Pairing Relation Show that the formula is independent of equivalence Choose an appropriate equivalent net so that the quotient of functions is exactly f P (D Q ).
36 Choosing a Nice Net
37 Calculating the Net (Rank 2) Based on an algorithm by Rachel Shipsey Double DoubleAdd
38 Calculating the Tate Pairing Find the initial values of the net associated to E, P, Q (there are simple formulae) Use a Double & Add algorithm to calculate the block centred on m Use the terms in this block to calculate
39 Embedding Degree k
40 Efficiency
41 Possible Research Directions Extend this to Jacobians of higher genus curves? Use periodicity relations to find integer points? (M. Ayad does this for sequences) Other computational applications: counting points on elliptic curves over finite fields? Other cryptographic applications of Tate pairing relationship?
42 References Morgan Ward. Memoir on Elliptic Divisibility Sequences. American Journal of Mathematics, 70:13-74, Christine S. Swart. Elliptic Curves and Related Sequences. PhD thesis, Royal Holloway and Bedford New College, University of London, Graham Everest, Alf van der Poorten, Igor Shparlinski, and Thomas Ward. Recurrence Sequences. Mathematical Surveys and Monographs, vol 104. American Mathematical Society, Elliptic net algorithm for Tate pairing implemented in the PBC Library, Slides, preprint, scripts at
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