Abstract
We present a technique to recover f ∈ ℚ(ζ p ) where ζ p is a primitive p th root of unity for a prime p, given its norm \(g = f * \bar{f}\) in the totally real field \(\mathbb{Q}(\zeta_{p}+\zeta_{p}^{-1})\). The classical method of solving this problem involves finding generators of principal ideals by enumerating the whole class group associated with ℚ(ζ p ), but this approach quickly becomes infeasible as p increases. The apparent hardness of this problem has led several authors to suggest the problem as one suitable for cryptography. We describe a technique which avoids enumerating the class group, and instead recovers f by factoring N f , the absolute norm of f, (for example with a subexponential sieve algorithm), and then running the Gentry-Szydlo polynomial time algorithm for a number of candidates. The algorithm has been tested with an implementation in PARI.
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Howgrave-Graham, N., Szydlo, M. (2004). A Method to Solve Cyclotomic Norm Equations \(f * \bar{f}\) . In: Buell, D. (eds) Algorithmic Number Theory. ANTS 2004. Lecture Notes in Computer Science, vol 3076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24847-7_20
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DOI: https://doi.org/10.1007/978-3-540-24847-7_20
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