International Association for Cryptologic Research

International Association
for Cryptologic Research


Joachim von zur Gathen


Finding Low Weight Polynomial Multiples Using Lattices
Laila El Aimani Joachim von zur Gathen
The low weight polynomial multiple problem arises in the context of stream ciphers cryptanalysis and of efficient finite field arithmetic, and is believed to be difficult. It can be formulated as follows: given a polynomial $f \in \F_2[X]$ of degree $d$, and a bound $n$, the task is to find a low weight multiple of $f$ of degree at most $n$. The best algorithm known so far to solve this problem is based on a time memory trade-off and runs in time ${\cal O}(n^{ \lceil {(w - 1)}/{2} \rceil})$ using ${\cal O}(n^{ \lceil {(w - 1)}/{4} \rceil})$ of memory, where $w$ is the estimated minimal weight. In this paper, we propose a new technique to find low weight multiples using lattice basis reduction. Our algorithm runs in time ${\cal O}(n(n-d)^5)$ and uses ${\cal O}(nd)$ of memory. This improves the space needed and gives a better theoretical time estimate when $w \geq 12$ or when the \textit{excess degree} $n-d$ is small, say, $(n-d)^5 < n^{\lceil {(w-3)}/{2} \rceil}$. The former situation is plausible when the bound $n$, which represents the available keystream, is small, whereas the latter one occurs in efficient finite field arithmetic. We also propose bounds for the minimal weight of such multiples, supplying in this sense the state-of-the art techniques with a method to check whether their estimated minimal weight is in the correct range. This provides a quantitative cryptographic quality criterion for such polynomials: the fewer low degree low weight multiples a polynomial has, the harder becomes this type of cryptanalysis of the corresponding stream cipher. As an example, the Bluetooth polynomial turns out to be of good quality in this sense. Moreover, we introduce the corresponding number problem and apply a similar strategy to find sparse multiples of a given number with respect to the Hamming weight of their 2-ary representation. Finally, we run our experiments using the NTL library on some known polynomials in cryptanalysis and we confirm our analysis.\\ \textbf{Keywords: } stream ciphers analysis, low weight polynomial multiples, lattices, shortest vector.


Laila El Aimani (1)
Michael Nöcker (1)