Score:0

# Diffie-Hellman: how to solve an alternative of Diffie-Hellman given an algorithm that solves Square Diffie-Hellman?

A simple question. How can I proof the next polynomial reduction? : $$DH’ ≤_{p} SQ$$

where DH': given $$g^{a}$$ and $$g^{b}$$, compute {$$g^{ab}$$,$$yg^{ab}$$} where $$y= g^{d/2}$$, d is the order of cyclic group G.

and SQ: Square Diffie-Hellman (SDH) given $$g^{a}$$ , compute $$(g^{a})^{2}$$

Your notation of SDH is improper - it's about, given \$g^a\$, finding \$g^{a^2} = g^{(a^2)}\$. Finding \$(g^a)^2\$ would be rather straight-forward. With that said, there exist previous questions on this site about the equivalence of SDH and CDH, such as [this](https://crypto.stackexchange.com/questions/27152/show-how-to-efficiently-solve-the-computational-diffie-hellman-assumption-given) or [this](https://crypto.stackexchange.com/questions/82041/diffie-hellman-difficulty-of-computing-gx2-given-gx/82042#82042).
\$(g^{a})^{2} = g^{a}g^{a} = g^{2a} = g^{(a^2)}\$
I dont want for the equivalence of SDH and CDH, DH' its a different variant.
The last equality of yours does not hold. \$a^2 \neq 2a\$ for most values of \$a\$. In which context have you encountered this DH' problem? It seems equivalent to CDH, but if it's part of an assignment then I'd rather not provide a full solution.