Showing papers in "Designs, Codes and Cryptography in 1994"

Universal hashing and authentication codes

Abstract: In this paper, we study the application of universal hashing to the construction of unconditionally secure authentication codes without secrecy. This idea is most useful when the number of authenticators is exponentially small compared to the number of possible source states (plaintext messages). We formally define some new classes of hash functions and then prove some new bounds and give some general constructions for these classes of hash functions. Then we discuss the implications to authentication codes.

A survey of partial difference sets

Abstract: LetG be a finite group of order ν. Ak-element subsetD ofG is called a (ν,k, λ, μ)-partial difference set if the expressionsgh −1, forg andh inD withg≠h, represent each nonidentity element inD exactly λ times and each nonidentity element not inD exactly μ times. Ife∉D andg∈D iffg −1∈D, thenD is essentially the same as a strongly regular Cayley graph. In this survey, we try to list all important existence and nonexistence results concerning partial difference sets. In particular, various construction methods are studied, e.g., constructions using partial congruence partitions, quadratic forms, cyclotomic classes and finite local rings. Also, the relations with Schur rings, two-weight codes, projective sets, difference sets, divisible difference sets and partial geometries are discussed in detail.

Geometric secret sharing schemes and their duals

Abstract: Given a set of participants we wish to distribute information relating to a secret in such a way that only specified groups of participants can reconstruct the secret. We consider here a special class of such schemes that can be described in terms of finite geometries as first proposed by Simmons. We formalize the Simmons model and show that given a geometric scheme for a particular access structure it is possible to find another geometric scheme whose access structure is the dual of the original scheme, and which has the same average and worst-case information rates as the original scheme. In particular this shows that if an ideal geometric scheme exists then an ideal geometric scheme exists for the dual access structure.

Some new 2-resolvable Steiner quadruple systems

Abstract: Zaicev, Zinoviev and Semakov [12] and, independently, Baker [1], constructed 2-resolvableS(3, 4, 4n) for all ℕ. However, no 2-resolvableS(3, 4,v),v≥4, were known for any other value ofv. In this paper, we construct infinite classes of 2-resolvableS(3, 4,v) for values ofv that are not a power of 4. In particular, we construct a 2-resolvableS(3, 4, 100).

Hash functions and Cayley graphs

Abstract: We introduce cryptographic hash functions that are in correspondence with directed Cayley graphs, and for which finding collisions is essentially equivalent to finding short factorisations in groups. We show why having a large girth and a small diameter are properties that are relevant to hashing, and illustrate those ideas by proposing actual easily computable hash functions that meet those requirements.

On designs and formally self-dual codes

Abstract: Binary formally self-dual (f.s.d.) even codes are the one type of divisible [2n, n] codes which need not be self-dual. We examine such codes in this paper. On occasion a f.s.d. even [2n, n] code can have a larger minimum distance than a [2n, n] self-dual code. We give many examples of interesting f.s.d even codes. We also obtain a strengthening of the Assmus-Mattson theore. IfC is a f.s.d. extremal code of lengthn≡2 (mol 8) [n ≡6 (mod 8)], then the words of a fixed weight inC ∪C ⊥ hold a 3-design [1-design]. Finally, we show that the extremal f.s.d. codes of lengths 10 and 18 are unique.

The weight hierarchies of the projective codes from nondegenerate quadrics

Abstract: The weight hierarchies of the projective codes from nondegenerate quadrics in projective spaces over finite fields are calculated. These codes satisfy also the chain conditions.

A shift register construction of unconditionally secure authentication codes

Abstract: We consider the authentication problem, using the model described by Simmons. Several codes have been constructed using combinatorial designs and finite geometries. We introduce a new way of constructing authentication codes using LFSR-sequences. A central part of the construction is an encoding matrix derived from these LFSR-sequences. Necessary criteria for this matrix in order to give authentication codes that provides protection aginst impersonation and substitution attacks will be given. These codes also provide perfect secrecy if the source states have a uniform distribution. Moreover, the codes give a natural splitting of the key into two parts, one part used aginst impersonation attacks and a second part used against substitution attacks and for secrecy simultaneously. Since the construction is based on the theory of LFSR-sequences it is very suitable for implementation and a simple implementation of the construction is given.

Some constructions for key distribution patterns

Abstract: Key distribution patterns (KDPs) are finite incidence structures satisfying a certain property which enables them to be applied to a problem in network key distribution. Very few useful examples of KDPs are known. We discuss some ideas which can lead to constructions for KDPs and apply these ideas to construct infinite families of KDPs from conics in finite projective and affine planes.

An optimal ternary [69, 5, 45] code and related codes

Abstract: A ternary [69, 5, 45] code is constructed, thus solving the problem of finding the minimum length of a ternary code of dimension 5 and minimum distance 45. Furthermore, this code is shown to be a unique two-weight code with weight enumerator 1+210Z45+32Z54. It is also shown that a ternary [70, 6, 45] code, which would have been a projective two-weight code giving rise to a new strongly regular graph, does not exist. In order to prove the main results, the uniqueness of some other optimal ternary codes with specified weight enumerators is also established.

On sequences with zero autocorrelation

Abstract: Normal sequences of lengthsn=18, 19 are constructed. It is proved through an exhaustive search that normal sequences do not exist forn=17, 21, 22, 23. Marc Gysin has shown that normal sequences do not exist forn=24. So the first unsettled case isn=27.

Quaternary constructions for the binary single-error-correcting codes of Julin, Best and others

Abstract: Certain nonlinear binary single-error-correcting codes found by Julin, Best and others have simple descriptions as codes over the ring of integers modulo 4.

Maximal three-independent subsets of {0, 1, 2} n

Abstract: We consider a variant of the classical problem of finding the size of the largest cap in ther-dimensional projective geometry PG(r, 3) over the field IF3 with 3 elements. We study the maximum sizef(n) of a subsetS of IF 3 n with the property that the only solution to the equationx 1+x2+x3=0 isx 1=x2=x3. Letc n=f(n)1/n andc=sup{c 1, c2, ...}. We prove thatc>2.21, improving the previous lower bound of 2.1955 ...

Decoding perfect maps

Abstract: Perfect Maps are two-dimensional arrays in which every possible sub-array of a certain size occurs exactly once. They are a generalisation of the de Bruijn sequences to two dimensions and are of practical significance in certain position location applications. In such applications the decoding problem, i.e. resolving the position of a particular sub-array within a specified Perfect Map, is of great significance. In this paper new constructions for (binary) Perfect Maps and 2 k -ary de Bruijn sequences are presented. These construction methods, although not yielding Perfect Maps for new sets of parameters, are significant because the Maps they yield can be efficiently decoded.

An exponent bound on skew Hadamard abelian difference sets

Abstract: A difference setD in a groupG is called a skew Hadamard difference set (or an antisymmetric difference set) if and only ifG is the disjoint union ofD, D(−1), and {1}, whereD(−1)={d−1|d∈D}. In this note, we obtain an exponent bound for non-elementary abelian groupG which admits a skew Hadamard difference set. This improves the bound obtained previously by Johnsen, Camion and Mann.

Translation planes of order 27

Abstract: Translation planes of order 27 are classified. Various invariants play an important role in a computer search.

On the [28, 7, 12] binary self-complementary codes and their residuals

Abstract: Recently Jungnickel and Tonchev have shown that there exist at least four inequivalent binary selfcomplementary [28, 7, 12] codes and have asked if there are other [28, 7] codes with weight distributionA0=A28=1,A12=A16=63. In the present paper we give a negative answer: these four codes are, up to equivalence, the only codes with the given parameters. Their residuals are also classified.

A coding theoretic solution to the 36 officer problem

Abstract: Using the tools of algebraic coding theory, we give a new proof of the nonexistence of two mutually orthogonal Latin squares of order 6.

A binary perfect code of length 15 and codimension 0

Abstract: We construct three new binary perfect codesC 1,C 2 andC 3 of length 15. We show that dim(ker(C i))=i fori=1, 2 and 3. It follows that the codimension ofC 1 equals 0.

On the rigidity of spherical t -designs that are orbits of finite reflection groups

Abstract: The concept of rigid sphericalt-designs was introduced by Bannai. He conjectured that there is a functionf(t, d) such that ifX is a sphericalt design in thed-dimensional Euclidean space so that |X|>f(t, d), theX is non-rigid. Furthermore, he asked to find examples of rigid but not tight sperical designs. In the present article we shall investigate the case whenX is an orbit of a finite reflection group and prove thatX is rigid iff tight for the groupsAn,Bn,Cn,Dn,E6,E7,F4,I3.

On row-cyclic codes with algebraic structure

Abstract: In this article, some row-cyclic error-correcting codes are shown to be ideals in group rings in which the underlying group is metacyclic. For a given underlying group, several nonequivalent codes with this structure may be generated. Each is related to a cyclic code generated in response, to the metrics associated with the underlying metacyclic group. Such codes in the same group ring are isomorphic as vector spaces but may vary greatly in weight distributions and so are nonequivalent. If the associated cyclic code is irreducible, examining the structure of its isomorphic finite field yields all nonequivalent codes with the desired structure. Several such codes have been found to have minimum distances equalling those of the best known linear codes of the same length and dimension.

Constructing c -ary perfect factors

Abstract: Ac-ary Perfect Factor is a set of uniformly long cycles whose elements are drawn from a set of sizec, in which every possiblev-tuple of elements occurs exactly once. In the binary case, i.e. wherec=2, these perfect factors have previously been studied by Etzion [2], who showed that the obvious necessary conditions for their existence are in fact sufficient. This result has recently been extended by Paterson [4], who has shown that the necessary existence conditions are sufficient wheneverc is a prime power. In this paper we conjecture that the same is true for arbitrary values ofc, and exhibit a number of constructions. We also construct a family of related combinatorial objects, which we callPerfect Multi-factors.

On evaluating the linear complexity of a sequence of least period 2 n

Abstract: The linear complexity of a periodic binary sequence is the length of the shortest linear feedback shift register that can be used to generate that sequence. When the sequence has least period 2 n ,n≥0, there is a fast algorithm due to Games and Chan that evaluates this linear complexity. In this paper a related algorithm is presented that obtains the linear complexity of the sequence requiring, on average for sequences of period 2 n ,n≥0, no more than 2 parity checks sums.

A counter-example to a recent result on the q -ary image of a q s -ary cyclic code

Abstract: We show, by means of a counter-example, that the necessary and sufficient conditions given in a recent paper [3] in order for theq-ary image of aqs-ary cyclic code to be cyclic are incorrect.

On completely free elements in finite fields

Abstract: Let q > 1 be a prime power, m > 1 an integer, GF(q^m) and GF(q) the Galois fields of order q^m and q, respectively. We show that the different module structures of (GF(q^m), +) arising from the intermediate fields of the field extension GF(q^m) over GF(q) can be studied simultaneously with the help of some basic properties of cyclotomic polynomials. The results can be generalized to finite cyclic Galois extensions over arbitrary fields. In 1986, D. Blessenohl and K. Johnsen proved that there exist elements in GF(q^m) which generate normal bases in GF(q^m) over any intermediate field GF(q^d) of GF(q^m) over GF(q). Such elements are called completely free in GF(q^m) over GF(q). Using our ideas, we give a detailed and constructive proof of the most difficult part of that theorem, i.e., the existence of completely free elements in GF(q^m) over GF(q) provided that m is a prime power. The general existence problem of completely free elements is easily reduced to this special case. Furthermore, we develop a recursive formula for the number of completely free elements in GF(q^m) over GF(q) in the case where m is a prime power.

A new class of symmetric ( v, k , l)-designs

Abstract: We construct new families of symmetric (v, k, λ)-designs with parameters $$\begin{gathered} v = p^s \cdot (q^{2m} - 1)/(q - 1). \hfill \\ k = q^{2m - 1} \cdot p^{s - 1} , \hfill \\ \lambda = p^{s - 1} \cdot q^{2m - 2} \cdot (p^{s - 1} - 1)(p - 1) \hfill \\ \end{gathered} $$ wherep is a prime andq is a prime power with $$q = (p^{s - 1} - 1)/(p - 1).$$ The orders of our designs aren=p 2s−2 ·q 2m−2 .

A Bruen chain for q =19

Abstract: The one-to-one correspondence between the class of two-dimensional translation planes of orderq2 and the collection of spreads ofPG(3,q) has long provided a natural context for describing new planes. The method often used for constructing “interesting” spreads is to start with a regular spread, corresponding to a desarguesian plane, and then replace some “nice” subset of lines by another partial spread covering the same set of points. Indeed the first approach was replacing the lines of a regulus by the lines of its opposite regulus, or doing this process for a set of disjoint reguli. Nontrivial generalizations of this idea include thechains of Bruen and thenests of Baker and Ebert. In this paper we construct a replaceable subset of a regular spread ofPG (3, 19) which is the union of 11 reguli double covering the lines in their union, hence is a chain in the terminology of Bruen or a 11-nest in the Baker-Ebert terminology.