A handful of recent cryptographic proposals rely on the conjectured hardness of the following problem in the ring of integers of a cyclotomic number field: given a basis of an ideal that is guaranteed to have a “rather short” generator, find such a generator. In the past year, Bernstein and Campbell-Groves-Shepherd have sketched potential attacks against this problem. Most notably, the latter authors claimed a quantum polynomial-time attack; alternatively, replacing the quantum component with an algorithm of Biasse and Fieker would yield a classical subexponential-time attack. A key claim of Campbell et al. is that one step of the algorithm—namely, decoding the log-unit lattice of the ring to recover a short generator from an arbitrary one—is efficient (whereas the standard approach takes exponential time). However, very few convincing details were provided to substantiate this claim, and as a result it has met with some skepticism. In this work, we remedy the situation by giving a rigorous theoretical and practical confirmation that the log-unit lattice in indeed efficiently decodable, in cyclotomics of prime-power index. The proof consists of two main technical contributions: the first is a geometrical analysis, using tools from analytic number theory, of the canonical generators of the group of cyclotomic units. The second shows that for a wide class of typical distributions of the short generator, a standard lattice-decoding algorithm can recover it, given any generator.
Preprint in Cryptology ePrint Archive: Report 2015/313 - Recovering Short Generators of Principal Ideals in Cyclotomic Rings
Published on 6 April 2015
The first preprint full text on candidate highly-versatile crypto systems being less secure appeared on 23 March 2015 on the CWI website.
Ronald Cramer (Cryptology Group, CWI & Mathematical Institute, Leiden University)
Leo Ducas (Cryptology Group, CWI)
Chris Peikert (School of Computer Science, College of Computing, Georgia Institute of Technology)
Oded Regev (Courant Institute of Mathematical Sciences, New York University)