On Bounded Distance Decoding, Unique Shortest Vectors, and the Minimum Distance Problem

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Provided by: University of Calgary
Topic: Security
Format: PDF
The authors prove the equivalence, up to a small polynomial approximation factor p n= log n, of the lattice problems uSVP (unique Shortest Vector Problem), BDD (Bounded Distance Decoding) and GapSVP (the decision version of the Shortest Vector Problem). This resolves a long-standing open problem about the relationship between uSVP and the more standard GapSVP, as well the BDD problem commonly used in coding theory. The main cryptographic application of this paper is the proof that the Ajtai-D work ([AD97]) and the Regev ([Reg04a]) cryptosystems, which were previously only known to be based on the hardness of uSVP, can be equivalently based on the hardness of worst-case GapSVPO(n2:5) and GapSVPO(n2), respectively.
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