Designs, Codes and Cryptography最新文献

In 2020, Cao et al. proved that any repeated-root constacyclic code is monomially equivalent to a matrix product code of simple-root constacyclic codes. In this paper, we study a family of matrix product codes with wonderful properties, which is a generalization of linear codes obtained from the ([u+v|u-v])-construction and ([u+v|lambda ^{-1}u-lambda ^{-1}v])-construction. Then we show that any (lambda )-constacyclic code (not necessary repeated-root (lambda )-constacyclic code) of length N over the finite field (mathbb {F}_q) with (textrm{gcd}(frac{q-1}{textrm{ord}(lambda )},N)ge 2), where (textrm{ord}(lambda )) is the order of (lambda ) in the cyclic group (mathbb {F}^*_q=mathbb {F}_qbackslash {0}), is a matrix product code of some constacyclic codes. It is a highly interesting question that the existence of sequences ({C_1,C_2,C_3,...}) of Euclidean (or Hermitian) self-dual codes with square-root-like minimum Hamming distances, i.e., (C_i) is an ([n(C_i),k(C_i),d(C_i)]_q)-linear code such that

Based on the ([u+v|lambda ^{-1}u-lambda ^{-1}v])-construction, we construct several families of Euclidean (or Hermitian) self-dual codes with square-root-like minimum Hamming distances by using Reed-Muller codes, projective Reed-Muller codes. And we construct some new Euclidean isodual (lambda )-constacyclic codes with square-root-like minimum Hamming distances from Euclidean self-dual cyclic codes and Euclidean self-dual negacyclic codes by monomial equivalences.

BCH codes as a subclass of constacyclic BCH codes have been widely studied, while the results on the parameters of BCH codes over finite fields are still very limited. In this paper, we investigate some q-ary BCH codes and (lambda )-constacyclic BCH codes of length (q^{m}+1), where q is a prime power and (textrm{ord}(lambda )mid q-1). We determine the dimensions of these codes with some large designed distances, and give good lower bounds on the minimum distance. The code examples presented in this paper indicate that these codes contain many distance-optimal codes and codes with best known parameters.

In this paper, we survey on the recent results and methods in the study of compositional inverses of permutation polynomials over finite fields. In particular, we describe a framework in terms of a commutative diagram which unifies several recent methods in finding the inverses of permutation polynomials.

Deterministic random bit generators (DRBGs) are essential tools in modern cryptography for generating secure and unpredictable random numbers. The ISO DRBG standards provide guidelines for designing and implementing DRBGs, including four algorithms: (textsf{HASH}text {-}textsf{DRBG}), (textsf{HMAC}text {-}textsf{DRBG}), (textsf{CTR}text {-}textsf{DRBG}), and (textsf{OFB}text {-}textsf{DRBG}). While security analyses have been conducted for the former three algorithms, there is a lack of specific security analysis for the (textsf{OFB})-(textsf{DRBG}) algorithm. We prove its security in the robustness security framework that has been used to analyze (mathsf {CTRtext {-}DRBG}) by Hoang and Shen at Crypto 2020. More precisely, we prove that (textsf{OFB})-(textsf{DRBG}) provides (O(min left{ frac{lambda }{3}, frac{n}{2} right} ))-bit security, including ideal cipher queries, where (lambda ) and n denote the lower bound of min-entropy and the size of the underlying block cipher, respectively. The proof strategy is to transform the robustness game of (textsf{OFB})-(textsf{DRBG}) into an indistinguishability game and then apply the H-coefficient technique to upper bound the distinguishing advantage.

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.