arXiv subjects
Publications and source records attributed to San Ling.
New lower bounds on the minimum distance of quasi-twisted codes over finite fields are proposed. They are based on spectral analysis and eigenvalues of polynomial matrices. They generalize the Semenov-Trifonov and Zeh-Ling bounds in a manner similar to how the Roos and shift bounds extend the BCH and HT bounds for cyclic codes.
The focus of this work is to show how to combine Zech's logarithms and each of the cycle joining and cross-join pairing methods to construct binary de Bruijn sequences of any order. A basic implementation is supplied as a proof-of-concept. The cycles, in the cycle joining method, are typically generated by a linear feedback shift register. We prove a crucial characterization that determining Zech's logarithms is equivalent to identifying conjugate pairs shared by any two distinct cycles. This speeds up the task of building a connected adjacency subgraph that contains all vertices of the complete adjacency graph. Distinct spanning trees in either graph correspond to cyclically inequivalent de Bruijn sequences. As the cycles are being joined, guided by the conjugate pairs, we track the changes in the feedback function. Certificates of star or almost-star spanning trees conveniently handle large order cases. The characterization of conjugate pairs via Zech's logarithms, as positional markings, is then adapted to identify cross-join pairs. A modified $m$-sequence is initially used, for ease of generation. The process can be repeated on each of the resulting de Bruijn sequences. We show how to integrate an analytic tool, attributed to Fryers, in the process. Most prior constructions in the literature measure the complexity of the corresponding bit-by-bit algorithms. Our approach is different. We aim first to build a connected adjacency subgraph that is certified to contain all of the cycles as vertices. The ingredients are computed just once and concisely stored. Simple strategies are offered to keep the complexities low as the order grows.
Group signature is a fundamental cryptographic primitive, aiming to protect anonymity and ensure accountability of users. It allows group members to anonymously sign messages on behalf of the whole group, while incorporating a tracing mechanism to identify the signer of any suspected signature. Most of the existing group signature schemes, however, do not guarantee security once secret keys are exposed. To reduce potential damages caused by key exposure attacks, Song (ACMCCS 2001) put forward the concept of forward-secure group signature (FSGS), which prevents attackers from forging group signatures pertaining to past time periods even if a secret group signing key is revealed at the current time period. For the time being, however, all known secure FSGS schemes are based on number-theoretic assumptions, and are vulnerable against quantum computers. In this work, we construct the first lattice-based FSGS scheme. Our scheme is proven secure under the Short Integer Solution and Learning With Errors assumptions. At the heart of our construction is a scalable lattice-based key evolving mechanism, allowing users to periodically update their secret keys and to efficiently prove in zero-knowledge that key evolution process is done correctly. To realize this essential building block, we first employ the Bonsai tree structure by Cash et al. (EUROCRYPT 2010) to handle the key evolution process, and then develop Langlois et al.'s construction (PKC 2014) to design its supporting zero-knowledge protocol.
Self-dual and complementary dual cyclic/abelian codes over finite fields form important classes of linear codes that have been extensively studied due to their rich algebraic structures and wide applications. In this paper, abelian codes over Galois rings are studied in terms of the ideals in the group ring ${\rm GR}(p^r,s)[G]$, where $G$ is a finite abelian group and ${\rm GR}(p^r,s)$ is a Galois ring. Characterizations of self-dual abelian codes have been given together with necessary and sufficient conditions for the existence of a self-dual abelian code in ${\rm GR}(p^r,s)[G]$. A general formula for the number of such self-dual codes is established. In the case where $\gcd(|G|,p)=1$, the number of self-dual abelian codes in ${\rm GR}(p^r,s)[G]$ is completely and explicitly determined. Applying known results on cyclic codes of length $p^a$ over ${\rm GR}(p^2,s)$, an explicit formula for the number of self-dual abelian codes in ${\rm GR}(p^2,s)[G]$ are given, where the Sylow $p$-subgroup of $G$ is cyclic. Subsequently, the characterization and enumeration of complementary dual abelian codes in ${\rm GR}(p^r,s)[G]$ are established. The analogous results for self-dual and complementary dual cyclic codes over Galois rings are therefore obtained as corollaries.
Group signatures allow users of a group to sign messages anonymously in the name of the group, while incorporating a tracing mechanism to revoke anonymity and identify the signer of any message. Since its introduction by Chaum and van Heyst (EUROCRYPT 1991), numerous proposals have been put forward, yielding various improvements on security, efficiency and functionality. However, a drawback of traditional group signatures is that the opening authority is given too much power, i.e., he can indiscriminately revoke anonymity and there is no mechanism to keep him accountable. To overcome this problem, Kohlweiss and Miers (PoPET 2015) introduced the notion of accountable tracing signatures (ATS) - an enhanced group signature variant in which the opening authority is kept accountable for his actions. Kohlweiss and Miers demonstrated a generic construction of ATS and put forward a concrete instantiation based on number-theoretic assumptions. To the best of our knowledge, no other ATS scheme has been known, and the problem of instantiating ATS under post-quantum assumptions, e.g., lattices, remains open to date. In this work, we provide the first lattice-based accountable tracing signature scheme. The scheme satisfies the security requirements suggested by Kohlweiss and Miers, assuming the hardness of the Ring Short Integer Solution (RSIS) and the Ring Learning With Errors (RLWE) problems. At the heart of our construction are a lattice-based key-oblivious encryption scheme and a zero-knowledge argument system allowing to prove that a given ciphertext is a valid RLWE encryption under some hidden yet certified key. These technical building blocks may be of independent interest, e.g., they can be useful for the design of other lattice-based privacy-preserving protocols.
Holographic representations of data encode information in packets of equal importance that enable progressive recovery. The quality of recovered data improves as more and more packets become available. This progressive recovery of the information is independent of the order in which packets become available. Such representations are ideally suited for distributed storage and for the transmission of data packets over networks with unpredictable delays and or erasures. Several methods for holographic representations of signals and images have been proposed over the years and multiple description information theory also deals with such representations. Surprisingly, however, these methods had not been considered in the classical framework of optimal least-squares estimation theory, until very recently. We develop a least-squares approach to the design of holographic representation for stochastic data vectors, relying on the framework widely used in modeling signals and images.
We propose a construction of de Bruijn sequences by the cycle joining method from linear feedback shift registers (LFSRs) with arbitrary characteristic polynomial $f(x)$. We study in detail the cycle structure of the set $Ω(f(x))$ that contains all sequences produced by a specific LFSR on distinct inputs and provide a fast way to find a state of each cycle. This leads to an efficient algorithm to find all conjugate pairs between any two cycles, yielding the adjacency graph. The approach is practical to generate a large class of de Bruijn sequences up to order $n \approx 20$. Many previously proposed constructions of de Bruijn sequences are shown to be special cases of our construction.
In this work, we provide the first lattice-based group signature that offers full dynamicity (i.e., users have the flexibility in joining and leaving the group), and thus, resolve a prominent open problem posed by previous works. Moreover, we achieve this non-trivial feat in a relatively simple manner. Starting with Libert et al.'s fully static construction (Eurocrypt 2016) - which is arguably the most efficient lattice-based group signature to date, we introduce simple-but-insightful tweaks that allow to upgrade it directly into the fully dynamic setting. More startlingly, our scheme even produces slightly shorter signatures than the former, thanks to an adaptation of a technique proposed by Ling et al. (PKC 2013), allowing to prove inequalities in zero-knowledge. Our design approach consists of upgrading Libert et al.'s static construction (EUROCRYPT 2016) - which is arguably the most efficient lattice-based group signature to date - into the fully dynamic setting. Somewhat surprisingly, our scheme produces slightly shorter signatures than the former, thanks to a new technique for proving inequality in zero-knowledge without relying on any inequality check. The scheme satisfies the strong security requirements of Bootle et al.'s model (ACNS 2016), under the Short Integer Solution (SIS) and the Learning With Errors (LWE) assumptions. Furthermore, we demonstrate how to equip the obtained group signature scheme with the deniability functionality in a simple way. This attractive functionality, put forward by Ishida et al. (CANS 2016), enables the tracing authority to provide an evidence that a given user is not the owner of a signature in question. In the process, we design a zero-knowledge protocol for proving that a given LWE ciphertext does not decrypt to a particular message.
Efficient user revocation is a necessary but challenging problem in many multi-user cryptosystems. Among known approaches, server-aided revocation yields a promising solution, because it allows to outsource the major workloads of system users to a computationally powerful third party, called the server, whose only requirement is to carry out the computations correctly. Such a revocation mechanism was considered in the settings of identity-based encryption and attribute-based encryption by Qin et al. (ESORICS 2015) and Cui et al. (ESORICS 2016), respectively. In this work, we consider the server-aided revocation mechanism in the more elaborate setting of predicate encryption (PE). The latter, introduced by Katz, Sahai, and Waters (EUROCRYPT 2008), provides fine-grained and role-based access to encrypted data and can be viewed as a generalization of identity-based and attribute-based encryption. Our contribution is two-fold. First, we formalize the model of server-aided revocable predicate encryption (SR-PE), with rigorous definitions and security notions. Our model can be seen as a non-trivial adaptation of Cui et al.'s work into the PE context. Second, we put forward a lattice-based instantiation of SR-PE. The scheme employs the PE scheme of Agrawal, Freeman and Vaikuntanathan (ASIACRYPT 2011) and the complete subtree method of Naor, Naor, and Lotspiech (CRYPTO 2001) as the two main ingredients, which work smoothly together thanks to a few additional techniques. Our scheme is proven secure in the standard model (in a selective manner), based on the hardness of the Learning With Errors (LWE) problem.
In this paper, we propose new classes of trapdoor functions to solve the closest vector problem in lattices. Specifically, we construct lattices based on properties of polynomials for which the closest vector problem is hard to solve unless some trapdoor information is revealed. We thoroughly analyze the security of our proposed functions using state-of-the-art attacks and results on lattice reductions. Finally, we describe how our functions can be used to design quantum-safe encryption schemes with reasonable public key sizes. In particular, our scheme can offer around $106$ bits of security with a public key size of around $6.4$ $\texttt{KB}$. Our encryption schemes are efficient with respect to key generation, encryption and decryption.
We determine the cycle structure of linear feedback shift register with arbitrary monic characteristic polynomial over any finite field. For each cycle, a method to find a state and a new way to represent the state are proposed.
We study a class of Linear Feedback Shift Registers (LFSRs) with characteristic polynomial $f(x)=p(x)q(x)$ where $p(x)$ and $q(x)$ are distinct irreducible polynomials in $\F_2[x]$. Important properties of the LFSRs, such as the cycle structure and the adjacency graph, are derived. A method to determine a state belonging to each cycle and a generic algorithm to find all conjugate pairs shared by any pair of cycles are given. The process explicitly determines the edges and their labels in the adjacency graph. The results are then combined with the cycle joining method to efficiently construct a new class of de Bruijn sequences. An estimate of the number of resulting sequences is given. In some cases, using cyclotomic numbers, we can determine the number exactly.
The study of frequency-hopping sequences (FHSs) has been focused on the establishment of theoretical bounds for the parameters of FHSs as well as on the construction of optimal FHSs with respect to the bounds. Peng and Fan (2004) derived two lower bounds on the maximum nontrivial Hamming correlation of an FHS set, which is an important indicator in measuring the performance of an FHS set employed in practice. In this paper, we obtain two main results. We study the construction of new optimal frequency-hopping sequence sets by using cyclic codes over finite fields. Let $\mathcal{C}$ be a cyclic code of length $n$ over a finite field $\mathbb{F}_q$ such that $\mathcal{C}$ contains the one-dimensional subcode $ \mathcal{C}_0=\{(α,α,\cdots,α)\in \mathbb{F}_q^n\,|\,α\in \mathbb{F}_q\}. $ Two codewords of $\mathcal{C}$ are said to be equivalent if one can be obtained from the other through applying the cyclic shift a certain number of times. We present a necessary and sufficient condition under which the equivalence class of any codeword in $\mathcal{C}\setminus\mathcal{C}_0$ has size $n$. This result addresses an open question raised by Ding {\it et al.} in \cite{Ding09}. As a consequence, three new classes of optimal FHS sets with respect to the Singleton bound are obtained, some of which are also optimal with respect to the Peng-Fan bound at the same time. We also show that the two Peng-Fan bounds are, in fact, identical.
This paper considers a linear quasi-cyclic product code of two given quasi-cyclic codes of relatively prime lengths over finite fields. We give the spectral analysis of a quasi-cyclic product code in terms of the spectral analysis of the row- and the column-code. Moreover, we provide a new lower bound on the minimum Hamming distance of a given quasi-cyclic code and present a new algebraic decoding algorithm.More specifically, we prove an explicit (unreduced) basis of an l\_a l\_b-quasi-cyclic product code in terms of the generator matrix in reduced Gr{ö}bner basis with respect to the position-over-term order (RGB/POT) form of the l\_a-quasi-cyclic row- and the l\_b-quasi-cyclic column-code, respectively. This generalizes the work of Burton and Weldon for the generator polynomial of a cyclic product code (where l\_a =l\_b=1). Furthermore, we derive the generator matrix in Pre-RGB/POT form of an l\_a l\_b-quasi-cyclic product code for two special cases: (i) for l\_a=2 and l\_b=1, and (ii) if the row-code is a 1-level l\_a-quasi-cyclic code (for arbitrary l\_a) and l\_b=1.For arbitrary l\_a and l\_b, the Pre-RGB/POT form of the generator matrix of an l\_a l\_b-quasi-cyclic product code is conjectured.The spectral analysis is applied to the generator matrix of the product of an l-quasi-cyclic and a cyclic code, and we propose a new lower bound on the minimum Hamming distance of a given l-quasi-cyclic code. In addition, we develop an efficient syndrome-based decoding algorithm for l-phased burst errors with guaranteed decoding radius.
Linear quasi-cyclic product codes over finite fields are investigated. Given the generating set in the form of a reduced Gr{ö}bner basis of a quasi-cyclic component code and the generator polynomial of a second cyclic component code, an explicit expression of the basis of the generating set of the quasi-cyclic product code is given. Furthermore, the reduced Gr{ö}bner basis of a one-level quasi-cyclic product code is derived.
For any given positive integer $m$, a necessary and sufficient condition for the existence of Type I $m$-adic constacyclic codes is given. Further, for any given integer $s$, a necessary and sufficient condition for $s$ to be a multiplier of a Type I polyadic constacyclic code is given. As an application, some optimal codes from Type I polyadic constacyclic codes, including generalized Reed-Solomon codes and alternant MDS codes, are constructed.
Let $R$ be a finite commutative chain ring with unique maximal ideal $\langle γ\rangle$, and let $n$ be a positive integer coprime with the characteristic of $R/\langle γ\rangle$. In this paper, the algebraic structure of cyclic codes of length $n$ over $R$ is investigated. Some new necessary and sufficient conditions for the existence of nontrivial self-dual cyclic codes are provided. An enumeration formula for the self-dual cyclic codes is also studied.
A new lower bound on the minimum Hamming distance of linear quasi-cyclic codes over finite fields is proposed. It is based on spectral analysis and generalizes the Semenov- Trifonov bound in a similar way as the Hartmann-Tzeng bound extends the BCH approach for cyclic codes. Furthermore, a syndrome-based algebraic decoding algorithm is given.