Integer Coordinate Families and Threshold RSA Signatures for Any Prime Exponent

What it is

Paper by Shivansh Jaiswal and Somitra Sanadhya (IACR ePrint 2026/2601, preprint, received 22 October 2026). It builds on the CT-RSA 2026 work of Ng and Lau, which introduced (t,n)-integer coordinate families for threshold RSA signatures with e=3 and found families by heuristic search for 2 ≤ t ≤ n ≤ 9. Ng and Lau left open whether such families always exist and whether they can be built efficiently. This paper answers both questions positively, with limits on the construction.

What the paper establishes

• A (t,n)-integer coordinate family is also a (t+1,n)-integer coordinate family.
• Families can be constructed for t=2 and 2 ≤ n < 19, by an algorithm polynomial in n and in the output bit length. The abstract names no construction for larger t or n=19 and above.
• Pairwise families have super-polynomial lower bounds on their coordinates, which answers another question Ng and Lau raised.
• The minimum families are found for n ≤ 9.
• The Ng-Lau signature scheme admits trivial forgeries and has a flawed proof. The abstract does not describe the forgery mechanism.
• PiIC is the repaired scheme built on integer coordinate families.
• PiZ is an integer secret-sharing scheme for every prime e and every (t,n). It does not depend on any special coordinate family.
• Both PiIC and PiZ are proven TS-UF-CMA secure under the standard RSA assumption.

What to do

Anyone implementing the Ng-Lau threshold scheme should not deploy it as published. For a new design with an exponent other than 3, or a threshold outside the small searched range, PiZ is the option the abstract presents as general. PiIC depends on a coordinate family, and the paper constructs those only for t=2, n < 19, plus the earlier searched cases.

Caveats

Only the abstract was available, so the forgery details, the concrete PiIC and PiZ constructions, and any size or performance costs are not covered here. The paper is a preprint and has not been peer reviewed.