Philip S. Hirschhorn

Choosing NTRUEncrypt Parameters in Light of Combined Lattice Reduction and MITM Approaches

We present the new NTRUEncrypt parameter generation algorithm, which is designed to be secure in light of recent attacks that combine lattice reduction and meet-in-the-middle (MITM) techniques. The parameters generated from our algorithm have been submitted to several standard bodies and are presented at the end of the paper.

Free simplicial groups and the second relative homotopy group of an adjunction space

(If (YX) is a pair of spaces, then 7rnz(YX) is a ;rrtX-crossed module.) Suppose that X is a connected CW-complex, and that Y is obtained from X by attaching 2-cells. Whitehead [9] showed that in this case, x2(xX) is a free x,X-crossed module, i.e., if {c,} are the elements of ;71#‘,X) corresponding to the 2-cells of Y-X, and if (g,} are elements of another n,X-crossed module G with ag,=ac,, then there is a unique homomorphism of crossed modules h : 7t2(Y,X) + G for which h(c,) =g,. Whitehead’s proof of this theorem is rather difficult and geometric (see [l] for a more modern exposition of Whitehead’s proof). Brown and Higgins [2] have shown that this theorem follows from their 2-dimensional generalization of the van Kampen theorem, in the context of ‘double groupoids’. Ratcliffe [7], using his study of free and projective crossed modules, was able to give a more algebraic proof. The purpose of the present paper is to give a completely algebraic proof of this theorem. We will obtain the theorem through a study offree simplicial groups, which are models for loop spaces (and are therefore useful for computing (absolute) homotopy groups).