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Is the following a correct formulation for the assumed worst-case hardness of $SIVP_\gamma$?

  • For every PPT algorithm $A$
  • for every $n\in\mathbb{N}$ there exists a basis $B_{n,A}=\{v_1,\dots,v_n\} \in \mathbb{Z}^n$ such that
  • Pr[A produces an answer for $SIVP_\gamma$ on $B_{n,A}$]$\to 0$ faster than any 1/poly(n).

Specifically, is that what one assumes when proving that Ajtai's family of functions is a one-way family of functions?

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