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Old 2007-01-18, 16:20   #2
R.D. Silverman
 
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"Bob Silverman"
Nov 2003
North of Boston

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Quote:
Originally Posted by Unregistered View Post
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.
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.

Please clarify your question.
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