31,054 research outputs found
GCSA Codes with Noise Alignment for Secure Coded Multi-Party Batch Matrix Multiplication
A secure multi-party batch matrix multiplication problem (SMBMM) is
considered, where the goal is to allow a master to efficiently compute the
pairwise products of two batches of massive matrices, by distributing the
computation across S servers. Any X colluding servers gain no information about
the input, and the master gains no additional information about the input
beyond the product. A solution called Generalized Cross Subspace Alignment
codes with Noise Alignment (GCSA-NA) is proposed in this work, based on
cross-subspace alignment codes. The state of art solution to SMBMM is a coding
scheme called polynomial sharing (PS) that was proposed by Nodehi and
Maddah-Ali. GCSA-NA outperforms PS codes in several key aspects - more
efficient and secure inter-server communication, lower latency, flexible
inter-server network topology, efficient batch processing, and tolerance to
stragglers. The idea of noise alignment can also be combined with N-source
Cross Subspace Alignment (N-CSA) codes and fast matrix multiplication
algorithms like Strassen's construction. Moreover, noise alignment can be
applied to symmetric secure private information retrieval to achieve the
asymptotic capacity
Searching for Minimum Storage Regenerating Codes
Regenerating codes allow distributed storage systems to recover from the loss
of a storage node while transmitting the minimum possible amount of data across
the network. We present a systematic computer search for optimal systematic
regenerating codes. To search the space of potential codes, we reduce the
potential search space in several ways. We impose an additional symmetry
condition on codes that we consider. We specify codes in a simple alternative
way, using additional recovered coefficients rather than transmission
coefficients and place codes into equivalence classes to avoid redundant
checking. Our main finding is a few optimal systematic minimum storage
regenerating codes for and , over several finite fields. No such
codes were previously known and the matching of the information theoretic
cut-set bound was an open problem
On Linear Operator Channels over Finite Fields
Motivated by linear network coding, communication channels perform linear
operation over finite fields, namely linear operator channels (LOCs), are
studied in this paper. For such a channel, its output vector is a linear
transform of its input vector, and the transformation matrix is randomly and
independently generated. The transformation matrix is assumed to remain
constant for every T input vectors and to be unknown to both the transmitter
and the receiver. There are NO constraints on the distribution of the
transformation matrix and the field size.
Specifically, the optimality of subspace coding over LOCs is investigated. A
lower bound on the maximum achievable rate of subspace coding is obtained and
it is shown to be tight for some cases. The maximum achievable rate of
constant-dimensional subspace coding is characterized and the loss of rate
incurred by using constant-dimensional subspace coding is insignificant.
The maximum achievable rate of channel training is close to the lower bound
on the maximum achievable rate of subspace coding. Two coding approaches based
on channel training are proposed and their performances are evaluated. Our
first approach makes use of rank-metric codes and its optimality depends on the
existence of maximum rank distance codes. Our second approach applies linear
coding and it can achieve the maximum achievable rate of channel training. Our
code designs require only the knowledge of the expectation of the rank of the
transformation matrix. The second scheme can also be realized ratelessly
without a priori knowledge of the channel statistics.Comment: 53 pages, 3 figures, submitted to IEEE Transaction on Information
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