Original JLS alternating phase-Hadamard PRU conjecture

Determine whether the Ji–Liu–Song alternating phase-Hadamard ensemble $U=F_lH\cdots F_1H$, with a constant number of independently keyed phase functions $f_j:[N]\to[N]$, forms a pseudorandom-unitary ensemble.

Background

The Ji–Liu–Song construction alternates Hadamard transforms with independently keyed random phase unitaries whose phase functions use the full alphabet of size NN. The paper rules out a broad weaker variant in which the phase-function codomain is superpolynomially smaller than the domain, but explicitly states that this does not refute the original conjecture.

Thus, the security of the original full-alphabet alternating phase-Hadamard construction remains unresolved.

References

This does not refute the original JLS conjecture, which uses the full alphabet $K=N$.

— Distinctness threshold for pseudorandom unitaries  (2609.03065 - Raza et al., 2 Sep 2026) in Conjecture JLS, Section “Constraints on the JLS conjecture”