[seqfan] Re: Crypto EllipticK & A000984
Sven Simon
sven-h.simon at gmx.de
Sun Jul 23 14:29:04 CEST 2017
Hello together,
that's an interesting session. Concerning crypting I have a tendency to crypt things which are already published... That's another aspect to those crypting sessions of CCC - do they have anything worth to crypt as secure as a number bank account in Switzerland ? - this is not the case as far as I can see. It is just to keep privacy and that's OK.
Sven
-----Ursprüngliche Nachricht-----
Von: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] Im Auftrag von Brad Klee
Gesendet: Mittwoch, 5. Juli 2017 00:26
An: Sequence Fanatics Discussion list
Betreff: [seqfan] Crypto EllipticK & A000984
Hi Seqfans,
Recently I found the following presentation "ECCHacks" from Bernstein and Lange at Chaos Congress 2014:
https://media.ccc.de/v/31c3_-_6369_-_en_-_saal_1_-_201412272145_-_ecchacks_-_djb_-_tanja_lange
Very nice, and easy to understand! By discussing the small-d limit, also the wall-clock limit, Bernstein and Lange hint an analogy between the Edward's curve and integrable Hamiltonian oscillations along one dimension.
To expand upon this possible connection, it's useful to enumerate "time-dependence" of the addition formula.
Let {x1,y1} be arbitrary along the genus one Edward's curve
x^2 + y^2 - (1 + d * x^2 * y^2) = 0
and {x2,y2} a small displacement from the fixed point. We add these points together using the addition rule:
{x1,y1} + {x2,y2} |---> {
-(y1*y2 - x1*x2) / (1 - d*x1*x2*y1*y2),
(x1*y2 + y1*x2) / ( 1 + d*x1*x2*y1*y2 ) }
Assuming that small displacement {x2,y2} sets a time-scale "dt", we can apply physical calculus. In polar coordinates where tan(phi) = y/x, we construct the integrand for the addition rule time-dependence:
dt = dphi / sqrt( 1 - 4*d*x^2 ) with x = cos(phi) * sin(phi)
This is essentially the generating function for A000984. The extensive encyclopedia entry, one of the "core", does not appear to include reference to the Edward's curve.
In polar-coordinate formulation, integration is easy as the integrand expands in a Fourier cosine series of the 2Pi-periodic phi. Integrating over one complete period yields a function proportional to EllipticK[d] ( cf. A038534 ), yet again! Probably this is all obvious to the experts???
Personally, I did not expect to get EllipticK out with so little hassle.
Ultimately re-discovery of EllipticK draws a quantitative connection between ciphers using Edward's Curve and physical systems such as the simple pendulum.
Cheers,
Brad
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