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9

This is true of any group of prime order, over elliptic curves or not. This is due to Lagrange's Theorem which states that the order of a subgroup $H$ of group $G$ divides the order of $G$. Since orders are elements of the ring of integers and since this is a principal ideal domain, unique factorization exists and primes make sense. Or put another way, ...


5

Yes, the attack you sketched out would work - in theory. In practice, it's an efficient (computable in polynomial time) mapping $\psi:E\rightarrow\mathbb Z_n$ we're lacking. As for the unefficient mappings, $\psi:x\cdot P\mapsto x$ would be perfectly fine theoretically, but we don't know how to calculate it (it's the elliptic curve discrete logarithm ...


2

The fact is that the discrete logarithm problem (DLP) is solved using different algorithms in the cases of multiplicative groups (where normal DH applies) and elliptic curves (where ECDH applies). The behavior of these algorithms is quite different. For multiplicative groups, where the NFS for logarithm is used, a huge part of the computation depends only ...



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