Integer Coordinate Families and Threshold RSA Signatures for Any Prime Exponent
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Resumen
At CT-RSA 2026, Ng and Lau introduced $(t,n)$-integer coordinate families for threshold RSA signatures with $e=3$, and found some families by heuristic search for $2\le t \le n \le 9$. They asked whether such families always exist and, if so, can they be constructed efficiently. We answer both of these questions positively. We prove that a $(t,n)$-integer coordinate family is also a $(t+1,n)$-integer coordinate family. We construct such families for $t=2$, and $2\le n < 19$, using a polynomial time algorithm in $n$ and its output bit length. Answering another question raised in the Ng-Lau work, we prove super-polynomial lower bounds on coordinates of pairwise families. We also find the minimum families for $n \le 9$. Ng and Lau proposed a threshold signature scheme based on integer coordinate families. We show that the Ng-Lau scheme admits trivial forgeries, and has a flawed proof. We give a repaired scheme $\mathrm{PiIC}$ which does not suffer from these issues. Finally, we propose an integer secret-sharing scheme $\mathrm{PiZ}$ that works for every prime $e$ and every $(t,n)$, without relying on any special coordinate family. We prove that both the schemes $\mathrm{PiIC}$ and $\mathrm{PiZ}$ are TS-UF-CMA secure under the standard RSA assumption.