International Association for Cryptologic Research

International Association
for Cryptologic Research


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

Steven Galbraith , The University of Auckland
Yi-Fu Lai , Ruhr-Universit├Ąt Bochum
Hart Montgomery , Linux Foundation
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Presentation: Slides
Conference: PKC 2024
Abstract: 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 circuit 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 $q$ then their algorithm to amplify the success requires on the order of $1/q^{21}$ 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 $1/q^{4}$ CDH oracle calls and is much conceptually simpler than the Montgonery-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. We also give evidence that $1/q^{2}$ calls to the CDH oracle (or perhaps even more) is necessary, showing that it will be potentially difficult to substantially improve our method further.
  title={A Simpler and More Efficient Reduction of DLOG to CDH for Abelian Group Actions},
  author={Steven Galbraith and Yi-Fu Lai and Hart Montgomery},