Title: **Tightness Subtleties for Multi-user PKE Notions** Publication: https://link.springer.com/chapter/10.1007/978-3-030-92641-0_5 Eprint: https://eprint.iacr.org/2022/855.pdf Presentation: virtual talk at [IMACC 2021](https://ima.org.uk/16366/online-event-18th-ima-international-conference-on-cryptography-and-coding/), 21/12/14 ([slides](https://drive.google.com/file/d/16IL2OZUO13ZEBq-jtPrk7th7kk8PGn-U/view?usp=sharing)) Authors: Hans Heum, Martijn Stam Abstract: *Public key encryption schemes are increasingly being studied concretely, with an emphasis on tight bounds even in a multi-user setting. Here, two types of formalization have emerged, one with a single challenge bit and one with multiple challenge bits. Another modelling choice is whether to allow key corruptions or not. How tightly the various notions relate to each other has hitherto not been studied in detail. We show that in the absence of corruptions, single-bit left-or-right indistinguishability is the preferred notion, as it tightly implies the other (corruption-less) notions. However, in the presence of corruptions, this implication no longer holds; we suggest the use of a more general notion that tightly implies both existing options. Furthermore, for completeness we study how the relationship between left-or-right versus real-or-random evolves in the multi-user PKE setting.* Contained in [[Cryptology in the Crowd]].