Lightweight sorting in approximate homomorphic encryption

Sorting encrypted values is an open research problem that plays a crucial role in the broader objective of providing efficient and practical privacy-preserving online services. The current state of the art work by Mazzone, Everts, Hahn and Peter [USENIX Security ’25] proposes efficient algorithms for indexing and sorting based on the CKKS scheme, which deviates from the compare-and-swap paradigm, typically used by sorting networks, using a permutation-based approach. In this work, we follow up their work and explore different approaches to approximate the nonlinear functions required by the circuit. We propose simpler and concrete solutions that allow for faster computations, smaller memory requirements and higher precision. For example, our framework allows to sort 128 real elements in roughly 22 seconds, while maintaining a precision of 0.001 and requiring 3 GB of memory. Furthermore, we propose an implementation of a swap-based bitonic network that is not based on approximations of the sgn(x) function, which scales linearly with the number of values, useful when the number of available slots is small.