1 Introduction
The investigation of allornothing transforms dates back to [6], in which Rivest suggested using it as a preprocessing for block ciphers in the setting of computational security. However, little attention was attracted to this topic until Stinson proposed unconditionally secure allornothing transforms in [7]. Later D’Arco et al. [1] introduced more general types of unconditionally secure allornothing transforms.
We begin with the following definition.
Definition 1.1
Let be a finite set known as an alphabet. Let be a positive integer and consider a map . For an input tuple, say , maps it to an output tuple, say , where for . The map is an unconditionally secure tallornothing transform provided that the following properties are satisfied:
is a bijection.
If any out of the output values are fixed, then any of the input values are completely undetermined, in an informationtheoretic sense.
We will call such a map as a AONT, where . And when and are clear or not relevant, we just call it a AONT.
The theory of AONTs is at a rudimentary stage. The study of Rivest [6] and Stinson [7] concentrated on the case . The AONTs can provide a preprocessing called “package transform” for block ciphers. The idea is to use a AONT to encrypt plaintexts to . Due to the property of AONT, a partial decryption cannot provide any information about each symbol among the plaintexts. In [1], the authors mainly concerned the case and introduced “approximations” to AONT. In this case, more theoretical results could be found in [8] and additional computational results could be found in [4]. Recently, Nasr Esfahani et al. [5] concentrated on the case over arbitrary alphabets and suggested interesting open problems.
An AONT with alphabet is linear if each , is an linear function of . Then, we can write
(1.1) 
where is an invertible by matrix with entries from . Subsequently, when we refer to a “linear AONT”, we mean the matrix that transforms to , as specified in (1.1).
D’Arco et al. characterized linear allornothing transforms in terms of submatrices of the matrix as follows.
Lemma 1.2 ([1, Lemma 1])
Suppose that is a prime power and is an invertible by matrix with entries from . Then defines a linear AONT if and only if every by submatrix of is invertible.
Next, we review some known results on linear AONTs.
Theorem 1.3 ([7, Theorem 2])
Suppose is a prime power and
. Then there is a linear transform that is simultaneously a
AONT for all such that .Theorem 1.4 ([5, Theorem 14])
There is no linear AONT for any prime power .
Given a prime power , define
In this paper, we continue the study of AONTs. Our main contributions are as follows:

We give a negative answer to the open problem (4) in [5].
[5, Open Problem 4]: As mentioned in Section 2.2, we performed exhaustive searches for linear AONT in type standard form, for all primes and prime powers , and found that no such AONT exists. We ask if there exists any linear AONT in type standard form.

By establishing a connection between linear AONTs constructed and cyclic codes, we give positive answers to the open problems (1) and (2) in [5].
[5, Open Problem 1]: Are there infinitely many primes for which there exist linear AONT?
[5, Open Problem 2]: Are there infinitely many primes
for which there exist cyclic skewsymmetric
AONT?As a consequence, when is a prime, is completely determined.

When is a prime power, we construct a AONT and improve the lower bound on in general. To the best of our knowledge, this is the first infinite class of AONTs which is better than the AONTs defined by Cauchy matrices.

For general AONTs, we present a recursive construction for general AONTs and a new relationship between AONTs and orthogonal arrays, and get a general construction for nonlinear AONT, except for .
We summarize upper and lower bounds on in Table 1 and the main contributions in this paper are in bold form.
2 Linear AONT
In this section, we describe some theoretical results for linear AONTs and answer three open problems proposed in [5]. The first one about standard form is reported in Section 2.1. Then we observe a general construction of linear AONT for all primes and give a positive answer to the existence results which attain the theoretical upper bounds of Theorem 1.4, see Section 2.2. Finally, we construct an infinite class of AONTs which is better than the AONTs defined by Cauchy matrices over , where is a prime power.
2.1 Linear AONT in type standard form
First we define a ‘standard form’ for a linear AONT.
Definition 2.1
Suppose is a matrix for a linear AONT. Then we can permute the rows and columns so that the ’s comprise the first entries on the main diagonal of . If , then we can multiply rows and columns by nonzero field elements so that all the entries in the first rows and first columns consist of ’s. If , we can multiply rows and columns by nonzero field elements so that all the entries in the first row and first column consist of ’s, except for the entry in the top left corner, which is a . Such a matrix is said to be of type standard form.
In [5], the authors obtained the structural conditions for a linear AONT and suggested an open problem as follows.
Lemma 2.2 ([5, Lemma 16])
Suppose is a matrix for a linear AONT in standard form. Then is of type or type .
[5, Open Problem 4]: As mentioned in Section 2.2, we performed exhaustive searches for linear AONT in type standard form, for all primes and prime powers , and found that no such AONT exists. We ask if there exists any linear AONT in type standard form.
In this subsection we give a negative answer to this open problem.
Theorem 2.3
For any prime power , there does not exist a linear AONT in type standard form.
Proof Suppose, on the contrary, that is a matrix for a linear AONT in type standard form. Then
where . Since every 2 by 2 submatrix of is invertible, for any . Let
Clearly, is also invertible and every 2 by 2 submatrix of is invertible. Consider 2 by 2 submatrices from the last row and and one row out of the first row. It is easy to see that the first rows each have distinct entries. It follows that the sum of each row of is zero and the matrix is singular, a contradiction.
Corollary 2.4
Suppose is a matrix for a linear AONT in standard form, then is of type .
2.2 Construction for linear AONT for all primes
In [5], the authors did an exhaustive search for a special subclass of linear AONTs and found that there exists a linear AONT for each value and proposed the following question: are there infinitely many primes for which there exist linear AONT. We will give a construction to answer this problem.
Construction 2.5
Let be a prime and be a matrix over , where denotes the entry in the th row and th column of . Let for , for and for , , .
Example 2.6
When , the matrix would be
When , the matrix would be
Lemma 2.7
For any prime , any submatrix of the matrix in Construction 2.5 is invertible.
Proof Consider a submatrix defined by rows and columns , where and . We consider the following cases:

If (or ), then .

If and , then .

If , and , then , so if and only if . This condition is equivalent to , which happens if and only if or . We assumed that and , so is invertible.
It is enough to show that any submatrix of the matrix in Construction 2.5 is invertible.
In order to show the construction above yields a linear AONT for , it remains to show that is invertible. The trick of the proof is to introduce an auxiliary matrix which is closely related to the matrix we construct in Construction 2.5.
Construction 2.8
Let be a prime and be a matrix over , where denotes the entry in the th row and th column of . Let for and for , , .
Remark 2.9
Since the last columns of each contain every element of exactly once, is invertible if and only if the lower right by submatrix of which is the same as the lower right by submatrix of is invertible. Since each column and each row of contain every element of exactly once, rank , thereby, to prove the lower right by submatrix of is invertible is equivalent to show the fact that rank().
Next, we will regard the matrix as a generator matrix of a cyclic code in coding theory. Before proceeding further, let us introduce some basic facts about cyclic codes.
A linear code of length over
is cyclic provided that for each vector
in the vector , obtained from by the cyclic shift of coordinates , is also in . When examining cyclic codes over , it is convenient to represent the codewords in polynomial form. There is a bijective correspondence between the vectors in and the polynomials in of degree at most . Notice that if , then if is set equal to . Thus cyclic codes are ideals of and ideals of are cyclic codes. It is well known that there is a relation between the dimension of and , where is generated by the polynomial and the polynomial is called a generator polynomial.Theorem 2.10 ([3, Theorem 4.2.1])
Let be a cyclic codes of length over with a generator polynomial . Then the dimension of is equal to , where is the degree of .
Lemma 2.11
rank.
Proof First we observe that the matrix is cyclic and . It is not difficult to verify that , and over , then . By Theorem 2.10, rank follows.
Theorem 2.12
There exists a linear AONT for all primes .
2.3 Existence results for linear AONT for prime power
In this subsection, we discuss linear AONTs for prime powers . As a consequence, we improve the lower bound on in general and answer an open problem proposed in [5]. To the best of our knowledge, there are only two systematic results in this topic. In [7], Cauchy matrices were mentioned as a possible method of constructing AONT, see Theorem 1.3. Recently, Nasr Esfahani et al. [5] pointed out that Vandermonde matrices can be treated as a method of constructing AONTs and get the following theorem.
Theorem 2.13 ([5, Theorem 2.1])
Suppose , is prime and . Then there exists a linear AONT over .
Remark 2.14
Since the parameter can attain , by Theorem 1.4, the result above yields a good construction for linear AONT. However, it actually requires that is a Mersenne prime, up to now, only such primes are known. Thus, the construction above cannot provide a infinite class of linear AONT.
Next, we will provide a general construction of linear AONT for all prime powers . We divide our proof in two steps. First, we construct a by matrix over such that any by submatrix is invertible. Then, we find a by submatrix which is invertible.
Construction 2.15
Let be a prime power, be a primitive element of and be a by matrix over , where denotes the entry in the th row and th column of . Let for and for , , .
Example 2.16
When , let be a primitive element of defined by , the matrix defined over would be
When , let be a primitive element of defined by , the matrix defined over would be
Lemma 2.17
For any prime power , any submatrix of the matrix in Construction 2.15 is invertible.
Proof Consider a submatrix defined by rows and columns , where and . We consider the following cases:

If (or , , ), then .

Otherwise, , so if and only if . This condition happens if and only if or . We assumed that and , so is invertible.
It is enough to show that any submatrix of the matrix in Construction 2.15 is invertible.
Next, we will prove that the rank of is .
Lemma 2.18
rank.
Proof We observe that the matrix is cyclic and . Since in , by Theorem 2.10, in order to prove this lemma, we need to show has exactly distinct roots in .
Claim 2.19
We consider the following three cases:

If , then .

If , where , then . We expand as follows:
Then . Thus , , are the roots of .

If , where and , then .
This completes the proof of Claim 2.19, and Lemma 2.18 follows.
Theorem 2.20
There exists a linear AONT for all primes power .
In [5], the authors did an exhaustive search and observed a very interesting structure for linear AONTs in type standard form, which the authors call it ‘skewsymmetric’. Suppose is a matrix for a linear AONT in type standard form. We say that the matrix is skewsymmetric if for any pair of cells and of , where and , it holds that . Furthermore, we say that is cyclic if (the lower right by submatrix of ) is a cyclic matrix. Up to equivalence, the authors in [5] actually found there was exactly one skewsymmetric AONT for each value and proposed the following open problem:
[5, Open Problem 2]: Are there infinitely many primes for which there exist (cyclic) skewsymmetric AONT?
For the matrix in Construction 2.15, when is a prime, we construct a linear AONT. Let be a by matrix in type standard form and choose the matrix as . Then the matrix is as following.
Construction 2.21
Let be a prime and be a primitive element of . Define a by matrix over as follows, where denotes the entry in the th row and th column of . Let for , and for and for , and .
By the property of the matrix in Construction 2.15, it is easy to see that is cyclic and skewsymmetric ( for , ). By the same argument of Remark 2.9, the matrix is invertible and every submatrix of the matrix is also invertible. For , when we replace the lower right by submatrix of with , the resultant by matrix is skewsymmetric. The corollary is a direct consequence of what we observed, which answers the open problem mentioned above.
Corollary 2.22
There exists a cyclic skewsymmetric AONT for any prime .
We should note that Construction 2.21 gives a theoretical explanation of the exhaustive searches in [5, Section III.B]. When , is a primitive element of . Suppose , we choose the lower right by submatrix as in Construction 2.21, then the matrix would be
which is coincident with [5, Example 35].
2.4 Dual property of linear AONTs
In this subsection, we will show the dual property of linear AONT.
Let be an by array whose entries are elements chosen from an alphabet of size . We will refer to as an array. Suppose the columns of are labelled by the elements in the set . Let and define to be the subarray obtained from by deleting all the columns . We say that is unbiased with respect to if the rows of contain every tuple of elements of exactly times. The following result characterizes AONT in terms of arrays that are unbiased with respect to certain subsets of columns.
Theorem 2.23
([1, Theorem 34]) A AONT is equivalent to a array that is unbiased with respect to the following subsets of columns:
,
, and
, for all with and all with .
It is easy to see that if a array satisfies the conditions in Theorem 2.23, then the array obtained by interchanging the first columns and the last columns of is unbiased with respect to the following subsets of columns:
,
, and
, for all with and all with .
Therefore, is an AONT.
Theorem 2.24
If there exists a AONT, then there exists an AONT.
By the definition of linear AONT, i.e., the relationship between the first columns and the last columns as in (1.1), interchanging the first columns and the last columns of a linear AONT yields a linear AONT by Theorem 2.24. So, the matrix must satisfy that every by submatrix is invertible.
Theorem 2.25
If there exists a linear AONT with the corresponding matrix , then the linear AONT defined by is a linear AONT.
Corollary 2.26
There is no linear AONT for any prime power .
By Theorem 2.25 and the constructions in Theorems 2.12, 2.20 and [5, Theorem 11], we have the following existence results.
Corollary 2.27
Suppose is a prime. Then there exists a linear AONT over .
Corollary 2.28
Suppose is a prime power. Then there exists a linear AONT over .
Corollary 2.29
Suppose , is a prime. Then there exists a linear AONT over .
3 General AONT
In this section, we present a recursive construction for general AONTs and a new relationship between AONTs and orthogonal arrays. Based on the observations above, we get a general construction for nonlinear AONT, except for .
We begin with a direct product construction.
Lemma 3.1 (Product Construction)
If there is a AONT and a AONT, then there exists a AONT.
Proof Let be an array over corresponding to a AONT, and be an array over corresponding to a AONT. For and , denote
It is routine to check that the array consisting of all row vectors where and , is a array over corresponding to a AONT.
To obtain a relation between AONTs and orthogonal arrays, Stinson [7] completely determined the existence of AONTs.
Theorem 3.2
[7, Theorem 3.4] There exists a AONT if and only if .
For prime powers , the existence of AONT has been completely determined in [7].
Theorem 3.3
[7, Corollary 2.3] There exists a linear AONT for all prime powers and for all positive integers .
Applying Lemma 3.1 with the known linear AONT in Theorem 3.3, we could generalise the existence results of AONT for .
Corollary 3.4
There exists a AONT for any integer , and for all positive integers .
Next we will obtain a new relationship between AONTs and orthogonal arrays. A array over is called an orthogonal array and denoted by OA if it is unbiased with respect to every subset of columns . The following relationship between OA and AONT is immediate from Theorem 2.23.
Corollary 3.5 ([1, Corollary 35])
If there exists an OA, then there exists a AONT for all such that .
Theorem 3.6 ([5, Theorem 23])
Suppose there is a AONT. Then there is an OA.
Roughly speaking, a AONT is a combinatorial configuration between OA and OA. The main difference of our results is that we do not construct an AONT directly from the structure of OA, actually, we take an OA as the auxiliary array in our construction.
Theorem 3.7
If there exists an OA, then there exists a AONT.
Proof Let be an OA over , whose rows are indexed by , . Without loss of generality, we assume that . For , we construct row vectors as follows:
Let be a array consisting of the row vectors above. We claim that is a AONT. To prove this claim, we check the conditions , and of Theorem 2.23.
Case 1. The first two conditions follow immediately from the definition of OA.
Case 2. Suppose we choose two columns from and one column from , it suffices to establish the bijection between the row vectors chosen above and all tuples. By the symmetry of our construction, without loss of generality, we assume that and . For any tuple , let , and . By the definition of OA, we could uniquely determine the index from the first two equalities, then uniquely determine the index from the last one. The proof is completed.
Corollary 3.8
For any positive integer with , there is a AONT and a AONT.
4 Conclusions
In this paper, we continue the study of tallornothing transforms over alphabets of arbitrary size. First we solved three open problems proposed in [5] and showed that there exists linear AONT. Then for prime powers , we construct the first infinite class of linear AONTs over which is better than the linear AONTs defined by Cauchy matrices. Besides, we also presented a recursive construction for general AONTs and a new relationship between AONTs and orthogonal arrays.
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