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-   -   Linear recurrence on elliptic curve (https://www.mersenneforum.org/showthread.php?t=6982)

 Unregistered 2007-01-18 15:31

Linear recurrence on elliptic curve

Let Po be a point on elliptic curve E over Zn, n=p*q composite.
Let Fib(k) be k-th fibonacci number or any other linear
recurrence.

Is it possible to efficiently compute Fib(big) * Point?
Or is it known to be a hard problem?

The usual doubling computing of Fib() gives integer explosion
and the group order of E is unknown, so direct reduction is
impossible.

Basic idea:
Consider E over Fp. The group order may be smooth * r, r not
large prime.
Compute Po2 = smooth2 * Po.
The period of k*Po2 is r.
Fib( m*(r-1)) mod r = 0 or Fib( m*(r+1) ) mod r = 0 so
Fib( m*(r +/-1)) * Po2 is the point at infinity.
r +/-1 may be smooth.

Hakmem ITEM 14 (Gosper & Salamin):
[url]http://www.inwap.com/pdp10/hbaker/hakmem/recurrence.html[/url]
Mentions "rate doubling formula".

 R.D. Silverman 2007-01-18 16:20

[QUOTE=Unregistered;96442]Let Po be a point on elliptic curve E over Zn, n=p*q composite.
Let Fib(k) be k-th fibonacci number or any other linear
recurrence.

Is it possible to efficiently compute Fib(big) * Point?
Or is it known to be a hard problem?

The usual doubling computing of Fib() gives integer explosion
and the group order of E is unknown, so direct reduction is
impossible.

[/QUOTE]

I am not sure I know what you mean by "direct reduction"?

I don't follow you. Given a large integer k, one can compute
Fib(k) in O(log(k)) multiplications. If we are computing over Z,
the numbers get exponentially large, so the entire computation
takes exponentially many bit operations (even with FFT's to do
the multiplication).

However, computing Fib(k) mod N (or Fib(k)*P where P is an EC point
over Z/NZ) has *bounded* intermediate values. Computing
M*P on an elliptic curve over Z/NZ is a polynomial time computation
because the intermediate values are bounded in size and only polynomially
many multiplications are required.

OTOH, computing Fib(k)*P over Q is a purely exponential problem
because the heights of the points explode exponentially.

 Unregistered 2007-01-18 17:13

[QUOTE=R.D. Silverman;96447]I am not sure I know what you mean by

OTOH, computing Fib(k)*P over Q is a purely exponential problem
because the heights of the points explode exponentially.

The point P is bounded mod N.
All computations are mod N and they involve just point additions.
P + P = 2*P
P + 2*P = 3*P
2*P + 3*P = 5*P
...
FIB(K-2)*P + FIB(K-1)*P= FIB(K)*P

so if the group order is r, r prime, FIB(K) should be taken mod r and FIB(K)*P hits the point at infinity for K multiples of r-1 or r+1.

The question is may FIB(K)*P (this bounded mod N) be computed efficiently for large K product of small primes. The value of FIB(K) is not needed.

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