[Resource Topic] 2024/191: A Simpler and More Efficient Reduction of DLog to CDH for Abelian Group Actions

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A Simpler and More Efficient Reduction of DLog to CDH for Abelian Group Actions

Authors: Steven Galbraith, Yi-Fu Lai, Hart Montgomery


Abelian group actions appear in several areas of cryptography, especially isogeny-based post-quantum cryptography. A natural problem is to relate the analogues of the computational Diffie-Hellman (CDH) and discrete logarithm (DLog) problems for abelian group actions.
Galbraith, Panny, Smith and Vercauteren (Mathematical Cryptology '21) gave a quantum reduction of DLog to CDH, assuming a CDH oracle with perfect correctness. Montgomery and Zhandry (Asiacrypt '22, best paper award) showed how to convert an unreliable CDH oracle into one that is correct with overwhelming probability. However, while a theoretical breakthrough, their reduction is quite inefficient: if the CDH oracle is correct with probability \epsilon then their algorithm to amplify the success requires on the order of 1/\epsilon^{21} calls to the CDH oracle.

We revisit this line of work and give a much simpler and tighter algorithm. Our method only takes on the order of 1/\epsilon^{4} CDH oracle calls and is conceptually simpler than the Montgomery-Zhandry reduction. Our algorithm is also fully black-box, whereas the Montgomery-Zhandry algorithm is slightly non-black-box. Our main tool is a thresholding technique that replaces the comparison of distributions in Montgomery-Zhandry with testing equality of thresholded sets.

ePrint: https://eprint.iacr.org/2024/191

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