Efficient class-group realizations of large cyclic groups

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For an integral domain $\mathcal O$ with fraction field $F$, an invertible fractional ideal is an invertible $\mathcal O$-submodule of $F$ that becomes an ideal after multiplication by some $f\in F^\times$. The Picard group $\operatorname{Pic}(\mathcal O)$ consists of such ideals modulo principal fractional ideals, with ideal multiplication as composition. Compact group encodings here have $O(\log\ell)$ bits, as in the source's computational-group convention.

Statement recorded in the supplied source.

Construct suitable rings $\mathcal O$ for large primes $\ell$ such that $\mathbb Z/\ell\mathbb Z$ embeds into $\operatorname{Pic}(\mathcal O)$, elements of $\operatorname{Pic}(\mathcal O)$ have compact encodings, and composition of encoded classes requires $O(\log\ell)$ operations. [S2]

Short English statement

Find class groups containing large prime-order cyclic groups with compact representations and logarithmic composition cost.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8800003 (AMR-087-0003). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Gerhard Frey, Duality in Arithmetic Geometry and Applications to Discrete Logarithms, IPAM Number Theory and Cryptography: Open Problems workshop (9--13 October 2006), Problem 2 and definition of Pic, PDF pages 13--14. https://helper.ipam.ucla.edu/publications/SCWS1/SCWS1_6219.pdf.