ISSN 0032-9460, Problems of Information Transmission, 2016, Vol. 52, No. 2, pp. 134–141.
Pleiades Publishing, Inc., 2016.
Original Russian Text
Y.L. Borissov, 2016, published in Problemy Peredachi Informatsii, 2016, Vol. 52, No. 2, pp. 37–45.
Some New Results on Hadamard
Modulo Prime Matrices
Y. L. Borissov
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Soﬁa, Bulgaria
Received April 9, 2015; in ﬁnal form, December 11, 2015
Abstract—First, some nonexistence and classiﬁcation results on Hadamard modulo prime
matrices whose size is relatively small with respect to their modulus, are presented. Second,
we show the existence of an inﬁnite class of matrices of that kind derived by ﬁnite projective
The Hadamard modulo prime (shortly HMP) matrices can be considered in the broader context
of modular Hadamard matrices introduced in . Also, their concept has recently resurfaced in the
engineering literature in the course of investigating the so-called jacket transforms .
In this paper our main concern, motivated by a remarkable cryptographic application, namely the
“All-Or-Nothing Transform” (AONT), is on the prime modular matrices. The AONT is proposed 
as a preprocessing step when encrypting data with a block cipher and aﬀords a certain amount of
additional security since it consolidates the data blocks into a single large block, so that a potential
adversary is forced to decrypt all blocks of a ciphertext (say, by an exhaustive key search) before
he/she could determine even one block of plaintext data. In  it is shown, among other things,
how to construct eﬃcient linear AONT schemes employing ordinary real Hadamard matrices in an
appropriate way. The recent work  proposes an extension of that construction applying instead
of them matrices of the HMP type, which enables sizes not restricted to 2 or multiples of 4.
The outline of the present article is as follows. After this introductory section, in Section 2
we recall necessary deﬁnitions and preliminary facts. In Section 3, we expose our results on HMP
matrices of relatively small size, and in Section 4, the results concerning matrices derived by ﬁnite
Deﬁnition 1. An HMP matrix H of size n modulo odd prime p is an n× n nonsingular over Z
matrix with entries ±1suchthat
= n (mod p) I
is the identity matrix of size n.
As usual, H
denotes the transpose matrix of a given matrix H. We shall also use the notation
HMP(n, p) for the set of HMP matrices of size n modulo p.
The work was presented in part at the 14th Int. Workshop on Algebraic and Combinatorial Coding Theory,
Svetlogorsk, Russia, Sept. 7–13, 2014.