MaTRU: A New NTRU-Based Cryptosystem

Abstract

In this paper, we propose a new variant of the NTRU public key cryptosystem − the MaTRU cryptosystem. MaTRU works under the same general principles as the NTRU cryptosystem, except that it operates in a different ring with a different linear transformation for encryption and decryption. In particular, it operates in the ring of k by k matrices of polynomials in R = Z[X]/(X n − 1), whereas NTRU operates in the ring Z[X]/(X N − 1). Note that an instance of MaTRU has the same number of bits per message as an instance of NTRU when nk 2 = N . The improved efficiency of the linear transformation in MaTRU leads to respectable speed improvements by a factor of O(k) over NTRU at the cost of a somewhat larger public key.

NTRU over rings beyond $${\mathbb{Z}}$$

“…, e such that q = p e 1 1 p e 2 2 · · · p e . In particular, the ideal generated by q factorises as (q) = p e 1 1 p e 2 2 · · · p e . By the Chinese Remainder Theorem, we can write…”

“…Kouzmenko, in turn, discussed possible conditions on alternatives to A = Z or F 2 [x], and presented the example of GNTRU, based on A = Z[i ], the Gaussian integers. Other works have followed, including MaTRU [2] by Coglianese and Goi in 2005, based on matrix rings.…”

Generalisations of NTRU cryptosystem

“…In 2005, Coglianse et al proposed a variant, called MaTRU, of NTRU cryptosystem. One of its strongest features of NTRU is the speed of NTRU.…”

A non-commutative cryptosystem based on quaternion algebras

“…Generalizations of NTRU over the Dedekind domains including Z[i], Z[ √ −2], Z[ζ 3 ] and Z[ζ 5 ], where ζ 3 and ζ 5 are 3rd and 5th roots of unity, are presented in [23] and [33] . A non-commutative version of NTRU over the non-commutative ring of k × k matrices of polynomials in the ring R is also available in the literature [9]. Another non-commutative version of NTRU, called QTRU, is addressed by Malekian et al over quaternion algebras [28], [29].…”