248 research outputs found
A simple combinatorial treatment of constructions and threshold gaps of ramp schemes
We give easy proofs of some recent results concerning threshold gaps in ramp schemes. We then generalise a construction method for ramp schemes employing error-correcting codes so that it can be applied using nonlinear (as well as linear) codes. Finally, as an immediate consequence of these results, we provide a new explicit bound on the minimum length of a code having a specified distance and dual distance
Multiplicative codes of Reed-Muller type
This is a comprehensive study of multiplicative codes of Reed-Muller type and their applications.
Our codes apply to the elds of cryptography and coding theory, especially to multiparty computa-
tion and secret sharing schemes. We also study the AB method to analyze the minimum distance
of linear codes. The multiplicative codes of Reed-Muller type and the AB method are connected
when we study the distance and dual distance of a code and its square. Generator matrices for our
codes use a combination of blocks, where a block consists of all columns of a given weight. Several
interesting linear codes, which are best known linear codes for a given length and dimension, can
be constructed in this way.
On squares of cyclic codes
The square of a linear error correcting code is the linear code
spanned by the component-wise products of every pair of (non-necessarily
distinct) words in . Squares of codes have gained attention for several
applications mainly in the area of cryptography, and typically in those
applications one is concerned about some of the parameters (dimension, minimum
distance) of both and . In this paper, motivated mostly by the
study of this problem in the case of linear codes defined over the binary
field, squares of cyclic codes are considered. General results on the minimum
distance of the squares of cyclic codes are obtained and constructions of
cyclic codes with relatively large dimension of and minimum distance of
the square are discussed. In some cases, the constructions lead to
codes such that both and simultaneously have the largest
possible minimum distances for their length and dimensions.Comment: Accepted at IEEE Transactions on Information Theory. IEEE early
access version available at https://ieeexplore.ieee.org/document/8451926
On Computational Shortcuts for Information-Theoretic PIR
Information-theoretic private information retrieval (PIR) schemes have attractive concrete efficiency features. However, in the standard PIR model, the computational complexity of the servers must scale linearly with the database size.
We study the possibility of bypassing this limitation in the case where the database is a truth table of a simple function, such as a union of (multi-dimensional) intervals or convex shapes, a decision tree, or a DNF formula. This question is motivated by the goal of obtaining lightweight homomorphic secret sharing (HSS) schemes and secure multiparty computation (MPC) protocols for the corresponding families.
We obtain both positive and negative results. For first-generation PIR schemes based on Reed-Muller codes, we obtain computational shortcuts for the above function families, with the exception of DNF formulas for which we show a (conditional) hardness result. For third-generation PIR schemes based on matching vectors, we obtain stronger hardness results that apply to all of the above families.
Our positive results yield new information-theoretic HSS schemes and MPC protocols with attractive efficiency features for simple but useful function families. Our negative results establish new connections between information-theoretic cryptography and fine-grained complexity
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