Abstract This paper considers coding for so-called partially stuck (defect) memory cells. Such memory cells can only store partial information as some of their levels cannot be used fully due to, e.g., wearout. First, we present new constructions that are able to mask u partially stuck cells while correcting at the same time t random errors. The process of “masking” determines a word whose entries coincide with writable levels at the (partially) stuck cells. For $$u>1$$ u > 1 and alphabet size $$q>2$$ q > 2 , our new constructions improve upon the required redundancy of known constructions for $$t=0$$ t = 0 , and require less redundancy for masking partially stuck cells than former works required for masking fully stuck cells (which cannot store any information). Second, we show that treating some of the partially stuck cells as erroneous cells can decrease the required redundancy for some parameters. Lastly, we derive Singleton-like, sphere-packing-like, and Gilbert–Varshamov-like bounds. Numerical comparisons state that our constructions match the Gilbert–Varshamov-like bounds for several code parameters, e.g., BCH codes that contain all-one word by our first construction.

We present a method to increase the dynamical range of a Residue Number System (RNS) by adding virtual RNS layers on top of the original RNS, where the required modular arithmetic for a modulus on any non-bottom layer is implemented by means of an RNS Montgomery multiplication algorithm that uses the RNS on the layer below. As a result, the actual arithmetic is deferred to the bottom layer. The multiplication algorithm that we use is based on an algorithm by Bajard and Imbert, extended to work with pseudo-residues (remainders with a larger range than the modulus). The resulting Recursive Residue Number System (RRNS) can be used to implement modular addition, multiplication, and multiply-and-accumulate for very large (2000+ bits) moduli, using only modular operations for small (for example 8-bits) moduli. A hardware implementation of this method allows for massive parallelization. Our method can be applied in cryptographic algorithms such as RSA to realize modular exponentiation with a large (2048-bit, or even 4096-bit) modulus. Due to the use of full RNS Montgomery algorithms, the system does not involve any carries, therefore cryptographic attacks that exploit carries cannot be applied. A full version of this paper is accessible at: http://www.arxiv.org/abs/1801.07561

Billions of devices are being connected to the Internet creating the Internet of Things (IoT). The IoT not only requires strong security, like current Internet applications, but also efficient operation. The recently introduced HIMMO scheme enables lightweight and collusion-resistant identity-based key sharing in a non-interactive way, so that any pair of Internet-connected devices can securely communicate. This paper firstly reviews the HIMMO scheme and introduces two extensions that e.g. enable implicit credential verification without the need of traditional digital certificates. Then, we show how HIMMO can be efficiently implemented even in resource-constrained devices, enabling combined key agreement and credential verification more efficiently than using ECDH-ECDSA. We further explain how HIMMO helps to secure the Internet and IoT by introducing the DTLS-HIMMO operation mode. DTLS, the datagram version of TLS, is becoming the standard security protocol in the IoT, although it is very frequently discussed that it does not offer the right performance for IoT scenarios. Our design, implementation, and evaluation show that DTLS-HIMMO operation mode achieves the security properties of the DTLS-Certificate security suite while exhibiting the overhead of symmetric-key primitives without requiring changes in the DTLS standard.

We consider a two polynomials analogue of the polynomial interpolation problem. Namely, we consider the Mixing Modular Operations (MMO) problem of recovering two polynomials f ∈ Zp[x] and g ∈ Zq[x] of known degree, where p and q are two (un)known positive integers, from the values of f(t) mod p+g(t) mod q at polynomially many points t ∈ Z. We show that if p and q are known, the MMO problem can be reduced to computing a close vector in a lattice with respect to the infinity norm. Using the Gaussian heuristic we also implemented in the SAGE system a polynomial-time algorithm. If p and q are kept secret, we do not know how to solve this problem. This problem is motivated by several potential cryptographic applications.

We consider the problem of determining the minimum number Nd of unit disks that is required to block all rays emanating from a point P in the two-dimensional space, where each disk has at least a distance d to point P and to any other disk. We study the asymptotic behavior of Nd, as d tends to infinity. By deriving upper bounds and lower bounds, we prove that pi2/16 ≤ lim_{d -> infinity} N_d/d2 ≤ 18/pi2, where the upper bound is based on establishing an interesting link between unit disks positioned on a regular triangular grid and Farey sequences from number theory. By positioning point P as well as the centers of the disks on the grid points of such a triangular grid, we create hexagonal rings of disks around P. We prove that we need exactly d-1 of these hexagons to block all rays emanating from P.

We give recursive constructions, valid for any field, of [n,k] codes capable of correcting a (wrap-around) burst of n - k erasures.

Recently there has been interest in the construction of small parity-check sets for iterative decoding of the Hamming code with the property that each uncorrectable (or stopping) set of size three is the support of a codeword and hence uncorrectable anyway. Here we reformulate and generalize the problem and improve on this construction. We show that a parity-check collection that corrects all correctable erasure patterns of size m for the Hamming code with codimension r provides, in fact, for all codes of codimension r a corresponding "generic" parity-check collection with this property. This leads in a natural way to a necessary and sufficient condition for such generic parity-check collections. We use this condition to construct a generic parity-check collection for codes of codimension r correcting all correctable erasure patterns of size at most m, for all r and mlesr, thus generalizing the known construction for m=3. Then we discuss optimality of our construction and show that it can be improved for mges3 and r large enough. Finally, we discuss some directions for further research

A generic (r,m)-erasure correcting set is a collection of vectors in F2rwhich can be used to generate, for each binary linear code of codimension r, a collection of parity check equations that enables iterative decoding of all correctable erasure patterns of size at most m. That is to say, the only stopping sets of size at most m for the generated parity check equations are the erasure patterns for which there is more than one manner to fill in the erasures to obtain a codeword. We give an explicit construction of generic (r,m)-erasure correcting sets of cardinality Sigmai=0m-1(ir-1). Using a random-coding-like argument, we show that for fixed m, the minimum size of a generic (r,m)-erasure correcting set is linear in r

The famous max-flow min-cut theorem states that a source node s can send information through a network (V, E) to a sink node t at a rate determined by the min-cut separating s and t. Recently, it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to re-encode the information they receive. We demonstrate examples of networks where the achievable rates obtained by coding at intermediate nodes are arbitrarily larger than if coding is not allowed. We give deterministic polynomial time algorithms and even faster randomized algorithms for designing linear codes for directed acyclic graphs with edges of unit capacity. We extend these algorithms to integer capacities and to codes that are tolerant to edge failures.

In this article, the computation of the entropy rate H(y) of a binary-valued stochastic process (Y/sub 1/, Y/sub 2/,...) which is a function of a stationary, time-invariant and irreducible Markov chain (X/sub 1/, X/sub 2/,..) is considered. The central idea of this article is to replace the summation over all words of length n by a summation over a complete set of prefixes (or prefixset for brevity). A prefixset W is a finite set of words (not necessarily of equal length) containing a unique prefix for each word of sufficient length. The method of prefixsets is of interest beyond computing the entropy rate. For the problem of estimating the next state of a Markov chain from observed output sequences, we can precompute a prefixset W of these sequences and associate a unique estimate of the state with each of the elements of W. The method also has a strong relation with variable-to-fixed length (Tunstall) codes. It replaces the set of all words of a given length by a prefixset of "more typical" words, effectively balancing the contributions of all words in the bounds.

The famous max-flow min-cut theorem states that a source node s can send information through a network (V,E) to a sink node t at a data rate determined by the min-cut separating s and t. Recently it has been shown that this rate can also be achieved for multicasting to several sinks provided that the intermediate nodes are allowed to reencode the information they receive. In contrast, we present graphs where without coding the rate must be a factor Ω(log|V|) smaller. However, so far no fast algorithms for constructing appropriate coding schemes were known. Our main result are polynomial time algorithms for constructing coding schemes for multicasting at the maximal data rate.

We consider lists of distinct q-ary addresses of length n. We wish that any b consecutive addresses in such a list agree in many positions. We give upper bounds on what can be achieved. Moreover, for each q and n, we give explicit constructions of address lists, among which is the conventional q-ary reflected Gray code, that attain these bounds for all b simultaneously. This work has applications in address retrieval on optical disc.

We consider the product code C/sub p/ of q-ary linear codes with minimum distances d/sub c/ and d/sub r/. The words in C/sub p/ of weight less than d/sub r/d/sub c/+max(d/sub r//spl lceil/(d/sub c//g)/spl rceil/,d/sub c//spl lceil/(d/sub r//q)/spl rceil/) are characterized, and their number is expressed in the number of low-weight words of the constituent codes. For binary product codes, we give an upper bound on the number of words in C/sub p/ of weightless than min(d/sub r/(d/sub c/+/spl lceil/(d/sub c//2)/spl rceil/+1)), d/sub c/(d/sub r/+/spl lceil/(d/sub r//2)/spl rceil/+1) that is met with equality if C/sub c/ and C/sub r/ are (extended) perfect codes.

We construct uniquely decodable (UD) code pairs for the binary multiplying channel without feedback, using pairs of binary codes. By taking appropriate cosets of linear codes with many information sets for these binary codes, we obtain new rate pairs in the zero-error capacity region Z of this channel. In particular, the rate pair (log(3/2), log(3/2)) is in Z and yields the largest known sum of the rates of pairs in Z. As this rate pair can be achieved with UD pairs with equal members, we have obtained an asymptotically optimal construction for the combinatorial concept of cancellative families of sets.

Let S consist of all words of a code C for which each symbol is in a stipulated subalphabet, possibly different for distinct positions. We consider the special case where C is a linear maximum-distance separable (MDS) code, and the subalphabets are linear subspaces over the ground field with equal dimensions. We give an explicit algorithm for selecting the subspaces in such a way that a straightforward systematic encoding algorithm, based on an encoder for C, can be applied. The number of information symbols that can be encoded with this algorithm equals a well-known lower bound on the dimension of S.

For original paper see ibid., vol.44, p.413-15 (1996 April). The present authors provide counterexamples to the expressions for the minimum Hamming distance of the projection codes in the original paper of Li et al.

A Diamond code is an error-correcting code obtained from two component codes. As in a product code, any symbol in a word of a Diamond code is checked by both component codes. However, the "code directions" for the component codes have been selected to minimize the memory that is required between successive decoding stages for the component codes. Diamond codes combine the error correcting power of a product code with the reduced memory requirements of the cross interleaved Reed-Solomon code (CIRC), applied in the compact disk system. We discuss encoding, decoding, and minimum distance properties of Diamond codes. Variations on the Diamond code construction are proposed that result in codes that are suited for use in rewritable block-oriented applications.

The generalization of the Gilbert-Varshamov bound due to Gu and Fuja (1993) is a direct consequence of Turan's theorem on the existence of a clique in a graph with many edges. Turan's theorem allows a slight improvement of Gu and Fuja's result. This improved generalized Gilbert-Varshamov bound is in fact equivalent to Turan's theorem.

We report on block-coding techniques for partial-response channels with transfer function (1/spl mnplus/D/sup m/), m=1, 2, ... . We consider various constructions of block codes with prescribed minimum Euclidean distance. Upper and lower bounds to the size of a code with minimum squared Euclidean distance greater than unity are furnished. A table is presented of cardinalities of codes of small length with prescribed minimum squared Euclidean distance.

J.W. Schwartz and J.K. Wolf (ibid., vol. 39, no. 11, pp. 1403-1404, Nov. 1990) gave a parity check matrix for a systematic (12,8) binary code that corrects all single errors and detects eight of the nine double adjacent errors within any of the three 4-bit nibbles. We present a parity check matrix for a systematic (12,8) binary code that corrects all single errors and detects any pair of errors within a nibble.>

The authors construct a (75,13,30) code and a (75,11,32) code. The minimum distances of these codes are one larger than the largest known (T. Verhoeff 1987) minimum distances of codes with the same length and dimension. As a (75,11,33) code does not exist, the (75,11,32) code is, in a sense, optimal. Both codes are obtained by judiciously shortening a (80,14,32) code that is obtained as a generalized concatenated code.>

A conjecture of V.C. Da Rocha (see Electron. Lett., vol.21, no.21, p.949-50, 1985) concerning the minimum distance of a class of combinatorial codes is proved.>

For a linear block code it is shown how the number of computations needed for the determination of an optimal encoder/decoder pair can be reduced by using the code's automorphism group. Furthermore, it is shown that the use of an unequal-error-protection-optical generator matrix and a minimum-weight coset leader decoder is suboptimal for a q-ary symmetry channel, but their determination needs less computational effort.>

Using some known techniques, several new binary linear codes are constructed, i.e., codes with a greater minimum distance than any previously known binary code of the same length and dimension [8]. A detailed description of one of the constructions is given.