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Hey I got a pointer a while ago to hidden order groups and I found papers like https://eprint.iacr.org/2006/178.pdf dating way back using this, but I couldn't find any elementary read on what can and can't I do with hidden order groups.

Does anyone have a small pointer for me on where to pick up reading?

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  • Cryptographic assumptions in hidden order groups.

    • Alin Tomescu: Cryptographic Assumptions in Hidden-Order Groups. This is a great overview of cryptographic assumptions in hidden-order groups. Link to the blog post.
    • Dankrad Feist: RSA assumptions. I like this website a lot as it gives a nice visualization of known reduction between cryptographic assumptions in RSA groups. There are several open questions about novel RSA assumptions, e.g., a classic and well-known is whether the RSA assumption and Factoring are (non)-equivalent. Link to the website. This page also has links to several classic papers.
    • Aron van Baarsen and Marc Stevens: On Time-Lock Cryptographic Assumptions in Abelian Hidden-Order Groups. This is the latest and maybe most comprehensive academic work on cryptographic assumptions in groups of unknown order. They work in the algebraic group model and show several new reductions. Link to the paper.
  • Instantiations of groups of unknown order.

    • RSA group. You can learn about RSA groups in any of your favorite undergrad number theory textbook. Though, their cryptographic importance diminishes as they require a trusted setup, unlike the other two candidates for hidden order groups.
    • Class groups of imaginary quadratic fields. This is an accessible blog post as an introduction to this topic. Michael Straka: Class Groups for Cryptographic Accumulators. Link to the blog post.
    • Jacobians of hyperelliptic curves: an excellent academic work on trustless unknown order groups, especially on the Jacobians of hyperelliptic curves by Samuel Dobson, Steven Galbraith, and Benjamin Smith. Link to the paper.
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  • $\begingroup$ Thank you so much! This is really helpful :) $\endgroup$ Commented Dec 29, 2022 at 22:04

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