Henk D. L. Hollmann

Kristiina Oksner, Henk D. L. Hollmann, Ago-Erik Riet et al.

Batch codes are of potential use for load balancing and private information retrieval in distributed data storage systems. Recently, a special case of batch codes, termed functional batch codes, was proposed in the literature. In functional batch codes, users can query linear combinations of the information symbols, and not only the information symbols themselves, as is the case for standard batch codes. In this work, we consider linear functional batch codes with the additional property that every query is answered by using only a small number of coded symbols. We derive bounds on the minimum length of such codes, and evaluate the results by numerical computations.

Baran Düzgün, Henk D. L. Hollmann, Ago-Erik Riet et al.

Various types of recovery algorithms for batch codes have been investigated, such as asynchronous recovery or recovery as afforded by batch codes obtained from Almost Affinely Disjoint (AAD) families. In this paper, we offer the first systematic investigation of linear batch codes equipped with particular recovery algorithms. We introduce and investigate various known and new types of algorithms, and we investigate the order hierarchy of these types of batch codes. The simplest known recovery algorithms are those associated with graph-based batch codes. We investigate the resulting batch codes for arbitrary bipartite graphs, thereby generalizing some known results.

Irina Bocharova, Boris Kudryashov, Henk D. L. Hollmann et al.

In modern data storage systems, non-binary LDPC codes for recovering from disk failures are increasingly considered strong competitors to MDS codes such as Reed-Solomon codes. Since disk failures can be modeled as erasures, we analyze non-binary LDPC codes over a $q$-ary field in the $q$-ary erasure channel, relative to MDS codes. Our focus is on non-binary LDPC codes whose parity-check matrix is obtained by replacing the non-zero entries of a binary base matrix by elements of a $q$-ary finite field. For such LDPC codes, we introduce the notion of ultimate distance, which upper-bounds their minimum distance. We derive a random-coding bound on the number of non-correctable erasure patterns for the Gallager ensemble of regular non-binary LDPC codes under maximum-likelihood decoding. An algorithm for finding the ultimate distance is presented. A low-complexity algorithm for searching for the minimum distance of the non-binary LDPC code is proposed. Finally, we construct examples of non-binary LDPC codes achieving the ultimate distance.

Henk D. L. Hollmann, Ronald Rietman, Sebastiaan de Hoogh et al.

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.