Score:4

Reducing exact SVP to exact SIVP

gf flag

In "Efficient reductions among lattice problems" by Micciancio (2007) it is said, that

SVP reduces to SIVP in their exact versions.

I did not found anything about this fact, is a reduction that trivial? Does the same hold for their approximation versions?

Score:5
cn flag

It is non-trivial, You need to see the Corollary 7 in Mic07, Micciancio proved that a series of problems (including CVP and SIVP) in the Euclidean norm are equivalent in their exact version under polynomial time rank-preserving reductions. And in GMSS99, Goldreich et al showed a reduction from SVP to CVP, So, combination of both, you got a reduction from SVP to SIVP. As for the approximation versions, you could see Noah Stephens-Davidowitz's paper NSD16.

user108492 avatar
gf flag
Thank you, that helps a lot!
user108492 avatar
gf flag
In Stephens-Davidowitz's paper, it is mentioned that SIVP_gamma reduces to SVP_gamma, but he said that he is not aware of any other source. He gave a proof, as far as I can see, but his paper is not been officially published. Is there anything behind? Does the relation hold?
constantine avatar
cn flag
Yes, But I don't know if it will be published later, Maybe this proof is too trivial to published, In fact, the reduction is from $SIVP_{\sqrt{n}\gamma}$ to $SVP_{\gamma}$. Maybe, If someone can do better than $\sqrt{n}$ between two approximate factors, there will be a publication
user108492 avatar
gf flag
Ah, I see, thanks again!
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