Designs, Codes and Cryptography最新文献

An affine spread is a set of subspaces of (textrm{AG}(n, q)) of the same dimension that partitions the points of (textrm{AG}(n, q)). Equivalently, an affine spread is a set of projective subspaces of (textrm{PG}(n, q)) of the same dimension which partitions the points of (textrm{PG}(n, q) setminus H_{infty }); here (H_{infty }) denotes the hyperplane at infinity of the projective closure of (textrm{AG}(n, q)). Let (mathcal {Q}) be a non-degenerate quadric of (H_infty ) and let (Pi ) be a generator of (mathcal {Q}), where (Pi ) is a t-dimensional projective subspace. An affine spread (mathcal {P}) consisting of ((t+1))-dimensional projective subspaces of (textrm{PG}(n, q)) is called hyperbolic, parabolic or elliptic (according as (mathcal {Q}) is hyperbolic, parabolic or elliptic) if the following hold:

• Each member of (mathcal {P}) meets (H_infty ) in a distinct generator of (mathcal {Q}) disjoint from (Pi );

• Elements of (mathcal {P}) have at most one point in common;

• If (S, T in mathcal {P}), (|S cap T| = 1), then (langle S, T rangle cap mathcal {Q}) is a hyperbolic quadric of (mathcal {Q}).

In this note it is shown that a hyperbolic, parabolic or elliptic affine spread of (textrm{PG}(n, q)) is equivalent to a spread of (mathcal {Q}^+(n+1, q)), (mathcal {Q}(n+1, q)) or (mathcal {Q}^-(n+1, q)), respectively.

The purpose of the present paper is to introduce recursive methods for constructing simple t-designs, s-resolvable t-designs, and large sets of t-designs. The results turn out to be very effective for finding these objects. In particular, they reveal a fundamental property of the considered designs. Consequently, many new infinite series of simple t-designs, t-designs with s-resolutions and large sets of t-designs can be derived from the new constructions. For example, by starting with an important result of Teirlinck stating that for every natural number t and for all (N > 1) there is a large set (LS[N](t, t+1, t+Ncdot ell (t))), where (ell (t)=prod _{i=1}^t lambda (i)cdot lambda ^*(i)), (lambda (t)=mathop {textrm{lcm}}(left( {begin{array}{c}t mend{array}}right) ,vert , m=1,2,ldots , t)) and (lambda ^*(t)=mathop {textrm{lcm}}(1,2, ldots , t+1)), we obtain the following statement. If ((t+2)) is composite, then there is a large set (LS[N](t, t+2, t+1+Ncdot ell (t))) for all (N > 1). If ((t+2)) is prime, then there is an (LS[N](t, t+2, t+1+Ncdot ell (t))) for any N with (gcd (t+2,N)=1).

An oriented supersingular elliptic curve is a curve which is enhanced with the information of an endomorphism. Computing the full endomorphism ring of a supersingular elliptic curve is a known hard problem, so one might consider how hard it is to find one such orientation. We prove that access to an oracle which tells if an elliptic curve is (mathfrak {O})-orientable for a fixed imaginary quadratic order (mathfrak {O}) provides non-trivial information towards computing an endomorphism corresponding to the (mathfrak {O})-orientation. We provide explicit algorithms and in-depth complexity analysis. We also consider the question in terms of quaternion algebras. We provide algorithms which compute an embedding of a fixed imaginary quadratic order into a maximal order of the quaternion algebra ramified at p and (infty ). We provide code implementations in Sagemath (in Stein et al. Sage Mathematics Software (Version 10.0), The Sage Development Team, http://www.sagemath.org, 2023) which is efficient for finding embeddings of imaginary quadratic orders of discriminants up to O(p), even for cryptographically sized p.

Non-overlapping codes are a set of codewords such that any nontrivial prefix of each codeword is not a nontrivial suffix of any codeword in the set, including itself. If the lengths of the codewords are variable, it is additionally required that every codeword is not contained in any other codeword as a subword. Let C(n, q) be the maximum size of a fixed-length non-overlapping code of length n over an alphabet of size q. The upper bound on C(n, q) has been well studied. However, the nontrivial upper bound on the maximum size of variable-length non-overlapping codes whose codewords have length at most n remains open. In this paper, by establishing a link between variable-length non-overlapping codes and fixed-length ones, we are able to show that the size of a q-ary variable-length non-overlapping code is upper bounded by C(n, q). Furthermore, we prove that the minimum average codeword length of a q-ary variable-length non-overlapping code with cardinality (tilde{C}), is asymptotically no shorter than (n-2) as q approaches (infty ), where n is the smallest integer such that (C(n-1, q) < tilde{C} le C(n,q)).

This paper shows that a ((mathbb {Z}_v,4,1))-disjoint difference family exists if and only if (vequiv 1pmod {12}) and (vne 25) by giving suitable translations of base blocks of a ((mathbb {Z}_v,4,1))-cyclic difference family. The Combinatorial Nullstellensatz finds its application in constructing disjoint difference families.

A quaternionic Hadamard matrix (QHM) of order n is an (ntimes n) matrix H with non-zero entries in the quaternions such that (HH^*=nI_n), where (I_n) and (H^*) denote the identity matrix and the conjugate-transpose of H, respectively. A QHM is dephased if all the entries in its first row and first column are 1, and it is non-commutative if its entries generate a non-commutative group. The aim of our work is to provide new constructions of infinitely many (non-commutative dephased) QHMs; such matrices are used by Farkas et al. (IEEE Trans Inform Theory 69(6):3814–3824, 2023) to produce mutually unbiased measurements.

Cryptanalysis of modern symmetric ciphers may be done by using linear equation systems with multiple right hand sides, which describe the encryption process. The tool was introduced by Raddum and Semaev (Des Codes Cryptogr 49(1):147–160, 2008) where several solving methods were developed. In this work, the probabilities are ascribed to the right hand sides and a statistical attack is then applied. The new approach is a multivariate generalisation of the correlation attack by Siegenthaler (IEEE Trans Comput C 49(1):81–85, 1985). A fast version of the attack is provided too. It may be viewed as an extension of the fast correlation attack in Meier and Staffelbach (J Cryptol 1(3):159–176, 1989), based on exploiting so called parity-checks for linear recurrences. Parity-checks are a particular case of the relations that we introduce in the present work. The notion of a relation is irrelevant to linear recurrences. We show how to apply the method to some LFSR-based stream ciphers including those from the Grain family. The new method generally requires a lower number of the keystream bits to recover the initial states than other techniques reported in the literature.

Several applications of function fields over finite fields, or equivalently, algebraic curves over finite fields, require computing bases for Riemann–Roch spaces. In this paper, we determine explicit bases for Riemann–Roch spaces of linearized function fields, and we give a lower bound for the minimum distance of generalized algebraic geometry codes. As a consequence, we construct generalized algebraic geometry codes with good parameters.

Locally repairable codes (LRCs) are linear codes with locality properties for code symbols, which have important applications in distributed storage systems. In this paper, we completely classify all the possible code parameters of optimal ternary LRCs achieving the Singleton-like bound proposed by Gopalan et al. Explicit constructions of optimal ternary LRCs are given for each group of possible code parameters. Moreover, it is also proved that optimal ternary LRCs with maximal minimum distance 6 are unique up to the equivalence of linear codes.

Let (n_k(s)) be the maximal length n such that a quaternary additive ([n,k,n-s]_4)-code exists. We solve a natural asymptotic problem by determining the lim sup (lambda _k) of (n_k(s)/s) for s going to infinity, and the smallest value of s such that (n_k(s)/s=lambda _k.) Our new family of quaternary additive codes has parameters ([4^k-1,k,4^k-4^{k-1}]_4=[2^{2k}-1,k,3cdot 2^{2k-2}]_4) (where (k=l/2) and l is an odd integer). These are constant-weight codes. The binary codes obtained by concatenation with inner code ([3,2,2]_2) meet the Griesmer bound with equality. The proof is in terms of multisets of lines in (PG(l-1,2)).