Designs, Codes and Cryptography最新文献

Let (mathbb {F}_q) be the finite field with q elements, where q is a power of a prime p. Given a monic polynomial (f in mathbb {F}_q[x]) that is not divisible by x, there exists a positive integer (e=e(f)) such that f(x) divides the binomial (x^e-1) and e is minimal with this property. The integer e is commonly known as the order of f and we write (textrm{ord}(f)=e). Motivated by a recent work of the second author on primitive k-normal elements over finite fields, in this paper we introduce the concept of polynomials free of binomials. These are the polynomials (f in mathbb {F}_q[x]), not divisible by x, such that f(x) does not divide any binomial (x^d-delta in mathbb {F}_q[x]) with (1le d<textrm{ord}(f)). We obtain some general results on polynomials free of binomials and we focus on the problem of describing the set of degrees of the polynomials that are free of binomials and whose order is fixed. In particular, we completely describe such set when the order equals a positive integer (n>1) whose prime factors divide (p(q-1)). Moreover, we also provide a correspondence between the polynomials that are free of binomials and cyclic codes that cannot be submerged into smaller constacyclic codes.

The regular decoding problem asks for (the existence of) regular solutions to a syndrome decoding problem (SDP). This problem has increased applications in post-quantum cryptography and cryptanalysis. Recently, Esser and Santini explored in depth the connection between the regular (RSD) and classical syndrome decoding problems. They have observed that while RSD to SDP reductions are known (in any parametric regime), a similar generic reduction from SDP to RSD is not known. In our contribution, we examine two different generic polynomial reductions from a syndrome decoding problem to a regular decoding problem instance. The first reduction is based on constructing a special parity check matrix that encodes weight counter progression inside the parity check matrix, which is then the input of the regular decoding oracle. The target regular decoding problem has a significantly longer code length, that depends linearly on the weight parameter of the original SDP. The second reduction is based on translating the SDP to a non-linear system of equations in the Multiple Right-Hand Sides form, and then applying RSD oracle to solve this system. The second reduction has better code length. The ratio between RSD and SDP code length of the second reduction can be bounded by a constant (less than 8).

Let (n=2^k-1) and (m=2^{k-2}) for a certain (kge 3). Consider the point-line geometry of 2m-element subsets of an n-element set. Maximal singular subspaces of this geometry correspond to binary simplex codes of dimension k. For (kge 4) the associated collinearity graph contains maximal cliques different from maximal singular subspaces. We investigate maximal cliques corresponding to symmetric (n, 2m, m)-designs. The main results concern the case (k=4) and give a geometric interpretation of the five well-known symmetric (15, 8, 4)-designs.

For a set ({mathcal {L}}) of lines of (text{ PG }(n,q)), a set X of points of (text{ PG }(n,q)) is called an ({mathcal {L}})-blocking set if each line of ({mathcal {L}}) contains at least one point of X. Consider a possibly singular quadric Q of (text{ PG }(n,q)) and denote by ({mathcal {S}}) (respectively, ({mathcal {T}})) the set of all lines of (text{ PG }(n,q)) meeting Q in 2 (respectively, 1 or (q+1)) points. For ({mathcal {L}}in {{mathcal {S}},{mathcal {T}}cup {mathcal {S}}}), we find the minimal cardinality of an ({mathcal {L}})-blocking set of (text{ PG }(n,q)) and determine all ({mathcal {L}})-blocking sets of that minimal cardinality.

An n-server information-theoretic Distributed Point Function (DPF) allows a client to secret-share a point function (f_{alpha ,beta }(x)) with domain [N] and output group (mathbb {G}) among n servers such that each server learns no information about the function from its share (called a key) but can compute an additive share of (f_{alpha ,beta }(x)) for any x. DPFs with small key sizes and general output groups are preferred. In this paper, we propose a new transformation from share conversions to information-theoretic DPFs. By applying it to the share conversions from Efremenko’s PIR and Dvir–Gopi PIR, we obtain both an 8-server DPF with key size ( O(2^{10sqrt{log Nlog log N}}+log p)) and output group (mathbb {Z}_p) and a 4-server DPF with key size (O(tau cdot 2^{6sqrt{log Nlog log N}})) and output group (mathbb {Z}_{2^tau }). The former allows us to partially answer an open question by Boyle, Gilboa, Ishai, and Kolobov (ITC 2022) and the latter allows us to build the first DPFs that may take any finite Abelian groups as output groups. We also discuss how to further reduce the key sizes by using different PIRs, how to reduce the number of servers by resorting to statistical security or using nice integers, and how to obtain DPFs with t-security. We show the applications of the new DPFs by constructing new efficient PIR protocols with result verification.

Permutation and multi-permutation codes have been widely studied due to their potential applications in communications and storage systems, especially in flash memory. In this paper, we consider balanced multi-permutation codes correcting a single burst of stable deletions of length t and length at most t, respectively. Based on the properties of burst stable deletions and stabilizer permutation subgroups, we propose two constructions of multi-permutation codes correcting a single burst of stable deletions of length up to some parameter. The multi-permutation codes can achieve larger rates than available codes while maintaining simple interleaving structures. Moreover, the decoding methods are given in proofs and verified by examples.

We provide a new construction of quantum codes that enables integration of a broader class of classical codes into the mathematical framework of quantum stabilizer codes. Next, we present new connections between twisted codes and linear cyclic codes and provide novel bounds for the minimum distance of twisted codes. We show that classical tools such as the Hartmann–Tzeng minimum distance bound are applicable to twisted codes. This enabled us to discover five new infinite families and many other examples of record-breaking, and sometimes optimal, binary quantum codes. We also discuss the role of the (gamma ) value on the parameters of twisted codes and present new results regarding the construction of twisted codes with different (gamma ) values but identical parameters. Finally, we list many new record-breaking binary quantum codes that we obtained from additive twisted, linear cyclic, and constacyclic codes.

In this paper we study skew group codes as left ideals in some skew group rings. We have constructed a large class of LCD codes and a class of an LCD MDS codes. An important interest is given to the construction of idempotents generators of these codes.

We offer this tribute to our friend and colleague, Jenny Key. After describing her education and career, we comment on her areas of research. The paper concludes with a complete list of her publications.

We present a secret-key encryption scheme based on random rank metric ideal linear codes with a simple decryption circuit. It supports unlimited homomorphic additions and plaintext multiplications (i.e. the homomorphic multiplication of a clear plaintext with a ciphertext) as well as a fixed arbitrary number of homomorphic multiplications. We study a candidate bootstrapping algorithm that requires no multiplication but additions and plaintext multiplications only. This latter operation is therefore very efficient in our scheme, whereas bootstrapping is usually the main reason which penalizes the performance of other fully homomorphic encryption schemes. However, the security reduction of our scheme restricts the number of independent ciphertexts that can be published. In particular, this prevents to securely evaluate the bootstrapping algorithm as the number of ciphertexts in the key switching material is too large. Our scheme is nonetheless the first somewhat homomorphic encryption scheme based on random ideal codes and a first step towards full homomorphism. Random ideal codes give stronger security guarantees as opposed to existing constructions based on highly structured codes. We give concrete parameters for our scheme that shows that it achieves competitive sizes and performance, with a key size of 3.7 kB and a ciphertext size of 0.9 kB when a single multiplication is allowed.