Sealing the Leak on Classical NTRU Signatures

Initial attempts to obtain lattice based signatures were closely related to reducing a vector modulo the fundamental parallelepiped of a secret basis (like GGH [9], or

NTRUSign

[12]). This approach leaked some information on the secret, namely the shape of the parallelepiped, which has been exploited on practical attacks [24].

NTRUSign

was an extremely efficient scheme, and thus there has been a noticeable interest on developing countermeasures to the attacks, but with little success [6].

In [8] Gentry, Peikert and Vaikuntanathan proposed a randomized version of Babai’s nearest plane algorithm such that the distribution of a reduced vector modulo a secret parallelepiped only depended on the size of the base used. Using this algorithm and generating large, close to uniform, public keys they managed to get provably secure GGH-like lattice-based signatures. Recently, Stehlé and Steinfeld obtained a provably secure scheme very close to

NTRUSign

[26] (from a theoretical point of view).

In this paper we present an alternative approach to seal the leak of

NTRUSign

. Instead of modifying the lattices and algorithms used, we do a classic leaky

NTRUSign

signature and hide it with gaussian noise using techniques present in Lyubashevky’s signatures. Our main contributions are thus a set of strong

NTRUSign

parameters, obtained by taking into account latest known attacks against the scheme, a statistical way to hide the leaky

NTRU

signature so that this particular instantiation of CVP-based signature scheme becomes zero-knowledge and secure against forgeries, based on the worst-case hardness of the

$\mathcal{\tilde{O}}(N^{1.5})$

-Shortest Independent Vector Problem over

NTRU

lattices. Finally, we give a set of concrete parameters to gauge the efficiency of the obtained signature scheme.