Shapes of Elliptic Curves
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1 Shapes of Elliptic Curves Daniel J. Bernstein Tanja Lange University of Illinois at Chicago and Technische Universiteit Eindhoven D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 1
2 Starring... D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 2
3 Weierstrass curve y 2 = x 3 0.4x D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 3
4 Weierstrass curve y 2 = x 3 0.4x D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 4
5 Jacobi quartic x 2 = y 4 1.9y D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 5
6 Jacobi quartic x 2 = y 4 1.9y D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 6
7 Hessian curve x 3 y = 0.3xy D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 7
8 Hessian curve x 3 y = 0.3xy D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 8
9 Edwards curve x 2 + y 2 = 1 300x 2 y 2 D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 9
10 Edwards curve x 2 + y 2 = 1 300x 2 y 2 D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 10
11 The race zoom on Weierstrass and Edwards D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 11
12 Weierstrass vs. Edwards I D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 12
13 Weierstrass vs. Edwards II D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 13
14 Weierstrass vs. Edwards III D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 14
15 Weierstrass vs. Edwards IV D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 15
16 Weierstrass vs. Edwards V D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 16
17 all competitors... D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 17
18 All competitors I D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 18
19 All competitors II D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 19
20 All competitors III D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 20
21 All competitors IV D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 21
22 All competitors V D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 22
23 Read the full story at: hyperelliptic.org/efd D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 23
24 One year passes I feel so odd... D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 24
25 Exceptions, How can there be an incomplete set of complete curves??? D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 25
26 After extensive (finite) field studies... (joint work with Reza Rezaeian Farashahi) Assume d 1, d 2 IF 2 n, d 1 0, Tr(d 2 ) = 1. Then d 1 (x + y) + d 2 (x + y) 2 = xy + xy(x + y) + x 2 y 2 describes an elliptic curve. (Curve shape is symmetric and has highest term x 2 y 2 like classic Edwards curves x 2 + y 2 = 1 + dx 2 y 2.) Neutral element is (0, 0). Negative of (x, y) is (y, x). The addition law on this curve is complete! It works for adding arbitrary points doubling, adding negatives, adding the neutral element,... Every ordinary elliptic curve over IF 2 n is birationally equivalent to a complete binary Edwards curve. D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 26
27 Timeline (after some early aborts) February 13th, 2008: Binary Edwards curves born February 15th, 2008: Binary Edwards takes first steps February 15th, 2008: Binary Edwards curves are complete (!) for d 2 = 1 and n odd. February 16th, 2008: Binary Edwards plays with d 1, d 2 February 20th, 2008: Binary Edwards adds differentially February 29th, 2008: Binary Edwards reaches all ordinary curves March 31st, 2008: Intel announces support for binary Edwards curves (PCMULQDQ in Westmere) April 16th, 2008: Sun announces support for binary Edwards curves in Rock. D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 27
28 Operation counts These curves are the first binary curves to offer complete addition laws. They are also surprisingly fast: ADD on binary Edwards curves takes 21M+1S+4D, madd takes 13M+3S+3D. Latest results ADD in 18M+2S+7D. Differential addition (P + Q given P, Q, and Q P ) takes 8M+1S+2D; mixed version takes 6M+1S+2D. Differential addition+doubling (typical step in Montgomery ladder) takes 8M+4S+2D; mixed version takes 6M+4S+2D. See our preprint (eprint 2008/171) or cr.yp.to/papers.html#edwards2 for full details, speedups for d 1 = d 2, how to choose small coefficients, affine formulas,... D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 28
29 Comparison with other doubling formulas Assume curves are chosen with small coefficients. System Projective Jacobian Lopez-Dahab Edwards Lopez-Dahab a 2 = 1 Cost of doubling 7M+4S; see HEHCC 4M+5S; see HEHCC 3M+5S; Lopez-Dahab 2M+6S; new, complete 2M+5S; Kim-Kim Explicit-Formulas Database for characteristic 2 is in preparation; our paper already has some speed-ups for Lopez-Dahab coordinates. D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 29
30 Happy End! D. J. Bernstein & T. Lange cr.yp.to/papers.html p. 30
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