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In the second example the curve has cardinality 8245498483844445086274997696534494324318871629323439587180432555305925857658038158971493931464052751511244450366220997579222992221013001791913941608268032 which is hard to factor. The computation of the order of a point will be trying to factor it.

How would you find the order of the point? Suppose the group order is n=a*b where a and b are primes, but you do not know this factorization, and I give you a point of order a. How could you find that order more easily than factoring n?

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