NTRU Primal Attacks

NTRU Primal Attacks#

We construct an (easy) example NTRU instance:

from estimator import *
params = NTRU.Parameters(n=200, q=7981, Xs=ND.UniformMod(3), Xe=ND.UniformMod(3))

The simplest (and quickest to estimate) model is solving via uSVP and assuming the Geometric Series Assumption (GSA) [Schnorr03]. The success condition was formulated in [USENIX:ADPS16] and studied/verified in [AC:AGVW17], [C:DDGR20], [PKC:PosVir21]. The treatment of small secrets is from [ACISP:BaiGal14]. Here, the NTRU instance is treated as a homoegeneous LWE instance:

NTRU.primal_usvp(params, red_shape_model="gsa")

We get a similar result if we use the GSA simulator. We do not get the identical result because we optimize β and d separately:

NTRU.primal_usvp(params, red_shape_model=Simulator.GSA)

To get a more precise answer we may use the CN11 simulator by Chen and Nguyen [AC:CheNgu11] (as implemented in FPyLLL):

NTRU.primal_usvp(params, red_shape_model=Simulator.CN11)

We can then improve on this result by first preprocessing the basis with block size β followed by a single SVP call in dimension η [RSA:LiuNgu13]. We call this the BDD approach since this is essentially the same strategy as preprocessing a basis and then running a CVP solver:

NTRU.primal_bdd(params, red_shape_model=Simulator.CN11)

We can improve these results further by exploiting the sparse secret in the hybrid attack [C:HowgraveGraham07] guessing ζ positions of the secret:

NTRU.primal_hybrid(params, red_shape_model=Simulator.CN11)

In addition to the primal secret key recovery attack, this module supports the dense sublattice attack as formulated in [EC:KirFou17], and refined/verified in [AC:DucWoe21]. The baseline dense sublattice attack uses a ‘z-shape’ variant of the Geometric Series Assumption, called the ZGSA:

NTRU.primal_dsd(params, red_shape_model=Simulator.ZGSA)

Of course we can also use the CN11 simulator for this attack as well:

NTRU.primal_dsd(params, red_shape_model=Simulator.CN11)

Note: Currently, dense sublattice attack estimation is only supported if the distributions of f and g are equal. NTRU.primal_dsd() will return a NotImplementedError if this is not the case.