Formulary for elliptic divisibility sequences and elliptic nets KATHERINE E STANGE Abstract Just the formulas No warranty is expressed or implied May cause side effects Not to be taken internally Remove label before using Not to be used as a flotation device May contain nuts Please report any errors you may find Let E be the elliptic curve defined over the rationals with eierstrass equation As usual, let y + a 1 xy + a 3 y = x 3 + a x + a 4 x + a 6 b = a 1 + 4a, b 4 = a 4 + a 1 a 3, b 6 = a 3 + 4a 6, b 8 = a 1a 6 + 4a a 6 a 1 a 3 a 4 + a a 3 a 4 1 Recurrence relation These formulas hold for elliptic divisibility sequences and elliptic nets, according to whether the indices are considered in Z or a larger free abelian group See [5] 11 Definition Definition 11 in [5] 1) p + q + s) p q) r + s) r) + q + r + s) q r) p + s) p) + r + p + s) r p) q + s) q) = 0 1 Stephens form Due to Nelson Stephens Obtained from 1) by s a, r b a, p c a, q d a a + b) a b) c + d) c d) + a + c) a c) d + b) d b) 13 Brown form Due to Dan Brown; equation 3) in [1] + a + d) a d) b + c) b c) = 0 p) q) r) s) ) ) ) ) p + q + r + s p q + r + s p + q r + s p + q + r s ) ) ) ) p + q + r + s p + q r s p q + r s p q r + s = 0 Date: May, 01, Draft #1 1
14 ard s elliptic divisibility sequences recurrence relation Not sufficient for generating a net of higher rank Equation 411) in [9] Obtained from 1) by p n, q m, s 0, r 1 n + m) n m) 1) = n + 1) n 1) m) m + 1) m 1) n) 15 Miscellaneous special cases n) ) 1) = n) n + ) n 1) n ) n + 1) ), n + 1) 1) 3 = n + ) n) 3 n 1) n + 1) 3, nm) ) 1) = nm nm) n) 1) = ) nm + ) nm 1) nm ) nm ) ) ) nm+1) 1 nm 1) nm+1) + 1 nm 1) + 1 ) 16 Special cases for rank two nets Theorem 5 in [5] nm 1) 1 ) 1, 1) 1, 1) 3 = 0, 1) 3, 1) 1, 0) 3 1, ) nm+1) + 1) ), The following formulas assume some terms near the origin are equal to one: 1, 0) = 0, 1) = 1, 1) = 1 Equations 1)-17) in [6] i 1, 0) = i + 1, 0) i 1, 0) 3 i, 0) i, 0) 3, i, 0), 0) = i, 0) i +, 0) i 1, 0) i, 0) i, 0) i + 1, 0), k 1, 1) 1, 1) = k + 1, 1) k 1, 1) k 1, 0) k, 0) k, 0) k, 1), k, 1) = k 1, 1) k + 1, 1) k, 0) k 1, 0) k + 1, 0) k, 1), k + 1, 1) 1, 1) = k 1, 1) k + 1, 1) k + 1, 0) k, 0) k +, 0) k, 1), k +, 1), 1) = k + 1, 0) k + 3, 0) k, 1) k 1, 1) k + 1, 1) k +, 0), Complex Function Formulas 1 eierstrass σ-function definition of net polynomials See Definition 31 in [5] For n = 1 sequences), see Theorem 11 in [9] 1) n = 1: ) n = : 3) general n: Ω u,v z, w; Λ) = Ω v z; Λ) = Ω v z; Λ) = σvz; Λ) σz; Λ) v σuz + vw; Λ) σz; Λ) u uv σz + w; Λ) uv σw; Λ) v uv σv 1 z 1 + + v n z n ; Λ) n σz i ; Λ) v i n j=1 v iv j σz i + z j ; Λ) v iv j i=1 1 i,j n i j )
Complex function identities See Lemmas 35 and 36 in [5] For the first equation, see also [] or any book on elliptic functions σz + w)σz w) z) w) =, σz) σw) v z) w z) = Ω v+wz)ω v w z) Ω v z) Ω w z), ζx + a) ζa) ζx + b) + ζb) = ζx + a + b) ζx + a) ζx + b) + ζx) = 3 Division and Net polynomials σx + a + b)σx)σa b) σx + a)σx + b)σa)σb), σx + a + b)σa)σb) σx + a + b)σx + a)σx + b)σx) 31 Division polynomials See [], [3, p80], [4, Exercise 37] or many other resources Ψ 1 = 1, Ψ = y + a 1 x + a 3, Ψ 3 = 3x 4 + b x 3 + 3b 4 x + 3b 6 x + b 8, Ψ 4 = y + a 1 x + a 3 )x 6 + b x 5 + 5b 4 x 4 + 10b 6 x 3 + 10b 8 x + b b 8 b 4 b 6 )x + b 4 b 8 b 6); 3 Net polynomials See Proposition 38 in [5] 1) for n = : ) for n = 3: Ψ 1, 1) = x x 1, ) ) y y 1 y y 1 Ψ,1) = x 1 + x a 1 + a, x x 1 x x 1 Ψ, 1) = y 1 + y ) x 1 + x )x 1 x ) ; Ψ 1,1,1) = y 1x x 3 ) + y x 3 x 1 ) + y 3 x 1 x ), x 1 x )x 1 x 3 )x x 3 ) Ψ 1,1,1) = y 1x x 3 ) y x 3 x 1 ) y 3 x 1 x ) + a 1 x 1 + a 3, x x 3 ) Ψ 1, 1,1) = y 1x x 3 ) + y x 3 x 1 ) y 3 x 1 x ) + a 1 x + a 3, x 3 x 1 ) Ψ 1,1, 1) = y 1x x 3 ) y x 3 x 1 ) + y 3 x 1 x ) + a 1 x 3 + a 3 x 1 x ) 4 Formulas relating curves and nets 41 Points in terms of division polynomials See any of the resources in Section 31 Define φ m = xp )Ψ m Ψ m+1 Ψ m 1, 4yω m = Ψ m+ Ψ m 1 Ψ m Ψ m+1 3
Then [m]p = φm P ) Ψ m P ), ω ) mp ), Ψ m P ) 3 x[m]p ) x[n]p ) = Ψ m+np )Ψ m n P ) Ψ mp )Ψ np ) 4 Curves from sequences and nets, rank 1 For the case n = 1, the simplest formulas are given in Theorem 453 in [8] C : y + a 1 xy + a 3 y = x 3 + a x + a 4 x + a 6, P = 0, 0), a 1 = 4) + )5 ) 3) ) 3) a = ) 3) + 4) + ) 5 ) 3) ) 3 3) a 3 = ), a 4 = 1, a 6 = 0 Morgan ard had more complicated formulas for the usual g and g 4 giving an elliptic curve equations 136) and 137) of [9]): g = g 3 = u) = 1 1 8 4 3 0 + 4 15 4 16 1 3 3 + 6 10 4 8 7 3 3 4 + 4 5 3 4 + 16 4 3 6 + 8 3 3 4 + 4 4 ) 1 30 16 1 3 6 + 6 5 4 4 3 3 + 15 0 4 60 17 3 3 4 + 0 15 4 3 + 10 14 3 6 36 1 3 3 4 + 15 10 4 4 48 9 3 6 4 + 1 7 3 3 4 3 + 64 6 3 9 + 6 5 4 5 + 48 4 3 6 4 + 1 3 3 4 4 + 4 6 ) 1 1 4 3 u) = 4 + 5 4 + 4 3 3 + 10 ) For n =, see Proposition 64 and Remark 66 in [5] 1) in rank n = : C : y + a 1 xy + a 3 y = x 3 + a x + a 4 x + a 6, P 1 = 0, 0), P =, 1) 1, ), 0), a 1 =, 0) 0, ), 1) 1, ), a =, 1) 1, ), a 3 =, 0) a 4 =, 1) 1, )), 1), a 6 = 0 4
) alternative in rank n = and characteristic : C : y + a 1 xy + a 3 y = x 3 + a x + a 4 x + a 6, P 1 = v, 0), P = v, 0),, a 1 = v =, 1) 1, ),, 0) 0, ), 1) 1, ), a =, 1) + 1, ), a 3 =, 0) + 0, ) 4a 4 =, 1) 1, )), 8a 6 =, 1) 1, )), 1) + 1, )) 5 Change of basis for elliptic nets See Proposition 43 in [5] Let T be any n m matrix Let P E m, v Z n n E,P T tr v)) = E,T P) v) E,P T tr e i )) v i v i j i v j) E,P T tr e i + e j )) v iv j i=1 6 Partial periodicity 1 i<j n 61 Periodicity formulas for non-degenerate elliptic nets The rank n = 1 case is Theorem 81 in [9] For rank n =, see Theorem 5 in [7] 1) rank n = 1 with E,P r) = 0: a = ) rank n = with E,P,Q r) = 0: a r = E,P sr + k) = E,P k)a sk b s E,P r + ) E,P r + 1) E,P ), b = E,P r + 1) E,P ) E,P r + ) E,P,Q lr + k) = E,P,Q k)a lk 1 E,P,Q r 1 +, r ) E,P,Q r 1 + 1, r ) E,P,Q, 0), b r = c r = E,P,Qr 1 + 1, r + 1) a r b r E,P,Q 1, 1) r b lk r c l r E,P,Q r 1, r + ) E,P,Q r 1, r + 1) E,P,Q 0, ), 6 Perfectly periodic elliptic divisibility sequence and elliptic net over F q See Theorem 6 in [7] ) 1 E,P q 1) ordp ) φp ) =, E,P q 1 + ordp )) n φv P) = E,P v) φp i ) v i v i j i v j) i=1 5 1 i<j n φp i + P j ) v iv j
7 Tate-Lichtenbaum and eil pairing formulas These are all from [6]; see Theorem 6 and Corollary 1 Special cases: mp + q + s)s) τ m P, Q) = mp + s)q + s), mp + q + s)p + s)mq + s) e m P, Q) = mp + s)q + s)p + mq + s) τ m P, P ) = P m + ) P 1) P m + 1) P ), τ m P, Q) = P,Qm + 1, 1) P,Q 1, 0) P,Q m + 1, 0) P,Q 1, 1) 8 Discrete log Type Equations Equations 9) and 11) in [7] Suppose [m]p = O and Q = [k]p ) k ) m E,P,Q m + 1, 0) E,P,Q, 0) E,P k 1) = ) E,P,Q1, m) E,P,Q, 0), E,P,Q m +, 0) E,P k) E,P,Q, m) E,P,Q 1, 1) m E,P m + 1) k+1 = ) mm+) E,P,Qm + 1, m + 1) E,P k + 1) E,P,Q 0, m + 1) E,P k) Acknowledgements Thank you to Dan Brown for corrections References [1] Daniel R L Brown Stange s elliptic nets and coxeter group f4 Cryptology eprint Archive, Report 010/161, 010 http://eprintiacrorg/ [] K Chandrasekharan Elliptic functions, volume 81 of Grundlehren der Mathematischen issenschaften [Fundamental Principles of Mathematical Sciences] Springer-Verlag, Berlin, 1985 [3] Gerhard Frey and Tanja Lange Background on curves and Jacobians In Handbook of elliptic and hyperelliptic curve cryptography, Discrete Math Appl Boca Raton), pages 45 85 Chapman & Hall/CRC, Boca Raton, FL, 006 [4] Joseph H Silverman The arithmetic of elliptic curves, volume 106 of Graduate Texts in Mathematics Springer, Dordrecht, second edition, 009 [5] Katherine Stange Elliptic nets and elliptic curves Algebra Number Theory, 5):197 9, 011 [6] Katherine E Stange The Tate pairing via elliptic nets In Pairing-Based Cryptography - PAIRING 007, volume 4575 of Lecture Notes in Comput Sci, pages 39 348 Springer, Berlin, 007 [7] Katherine E Stange The elliptic curve discrete logarithm problem and equivalent hard problems for elliptic divisibility sequences In Selected Areas in Cryptography 008, volume 5381 of Lecture Notes in Comput Sci, pages 309 37 Springer, Berlin, 009 [8] Christine Swart Elliptic curves and related sequences PhD thesis, Royal Holloway and Bedford New College, University of London, 003 [9] Morgan ard Memoir on elliptic divisibility sequences Amer J Math, 70:31 74, 1948 Department of Mathematics, Stanford University, 450 Serra Mall, Bldg 380, Stanford, CA 94305 E-mail address: stange@mathstanfordedu 6