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The closest that I can think of is a "symmetric bilinear group" (a.k.a. Type-I pairing group) that was popular when bilinear groups were first introduced. This is actually a pair of groups $(G, G_T)$ together with an efficient non-degenerate bilinear map $\otimes: G \times G \to G_T$. Obviously DDH is easy in $G$, since one can on input $(g, U, V, W) \in ...


5

This is a reduction showing that if you can compute $g^{a^2}$ given $g^a$, then you can solve the computational Diffie Hellman problem. Here is the reduction. Let $A$ be an adversary that given $g^a$ for a random $a$, outputs $g^{a^2}$ with probability $\epsilon$. We construct $A'$ who receives $u=g^a$ and $v=g^b$ and works as follows. $A'$ runs $A$ three ...


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You probably know that cyclic groups of any order exist, so it is perfectly possible to have a cyclic group of order $pq$, and to consider a generator thereof.


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There is no standard "multiply two group elements" operation in an additive group. So you first need to define what you mean by $P*Q$. From the comments I gather that you want $P*Q = q P = p Q = (p \cdot q) G$. The computational Diffie-Hellman (CDH) problem is: Given $P=pG$ and $Q=qG$ compute $(p\cdot q)G$. which is clearly equivalent to your problem. ...


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Well the equation $R = P * Q$ simply isn't possible on an elliptic curve. The group of points on the EC is an additive group. Meaning it is only possible to compute $P + Q$ or $[m] P$ for some integer m. Taken $P=p \cdot G$ and $Q=q \cdot G$ you already got the answer yourself: $R=(p \cdot q)G$. Simply add the point $G$ to itself $(p \cdot q)$-times.


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When using any cryptosystem that relies on the Decisional Diffie-Hellman assumption (e.g., ElGamal, ECIES, Diffie-Hellman key exchange, etc.) then you need a group of prime order. Note that $\mathbb Z_p^*$ where $p$ is prime has order $p-1$. However, if $p=rq+1$ where $q$ is also a large prime, then you have a subgroup of $\mathbb Z_p^*$ of order $q$. If you ...



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