arXiv subjects

Ludo M. G. M. Tolhuizen

Publications and source records attributed to Ludo M. G. M. Tolhuizen.

4 recordsLinked to original sources

arXiv2018

A Multi-layer Recursive Residue Number System

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.

cs.CR↗

arXiv2007

Optimal codes for correcting a single (wrap-around) burst of errors

In 2007, Martinian and Trott presented codes for correcting a burst of erasures with a minimum decoding delay. Their construction employs [n,k] codes that can correct any burst of erasures (including wrap-around bursts) of length n-k. The raised the question if such [n,k] codes exist for all integers k and n with 1<= k <= n and all fields (in particular, for the binary field). In this note, we answer this question affirmatively by giving two recursive constructions and a direct one.

cs.IT↗

arXiv2006

Generating parity check equations for bounded-distance iterative erasure decoding

A generic $(r,m)$-erasure correcting set is a collection of vectors in $\bF_2^r$ which 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 theerasures to obtain a codeword. We give an explicit construction of generic $(r,m)$-erasure correcting sets of cardinality $\sum_{i=0}^{m-1} {r-1\choose i}$. 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 $r$. Keywords: iterative decoding, binary erasure channel, stopping set

cs.IT↗

arXiv2005

On parity check collections for iterative erasure decoding that correct all correctable erasure patterns of a given size

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 generalise the problem, and improve on this construction. First we show that a parity check collection that corrects all correctable erasure patterns of size m for the r-th order Hamming code (i.e, the Hamming code with codimension r) provides for all codes of codimension $r$ a corresponding ``generic'' parity check collection with this property. This leads naturally to a necessary and sufficient condition on 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 m <= r, thus generalising the known construction for m=3. Then we discussoptimality of our construction and show that it can be improved for m>=3 and r large enough. Finally we discuss some directions for further research.

cs.IT↗