Cryptographic Smooth Neighbors
We revisit the problem of finding two consecutive $B$-smooth integers by giving an optimised implementation of the Conrey-Holm\-strom-McLaughlin ``smooth neighbors'' algorithm. While this algorithm is not guaranteed to return the complete set of $B$-smooth neighbors, in practice it returns a very close approximation to the complete set but does so in a tiny fraction of the time of its exhaustive counterparts. We exploit this algorithm to find record-sized solutions to the pure twin smooth problem, and subsequently to produce instances of cryptographic parameters whose corresponding isogeny degrees are significantly smoother than prior works. Our methods seem well-suited to finding parameters for the SQISign signature scheme, especially for instantiations looking to minimize the cost of signature generation. We give a number of examples, among which are the first parameter sets geared towards efficient SQISign instantiations at NIST's security levels III and V.