Abstract-Minimum-Storage Regenerating (MSR) codes have emerged as a viable alternative to Reed-Solomon (RS) codes as they minimize the repair bandwidth while they are still optimal in terms of reliability and storage overhead. Although several MSR constructions exist, so far they have not been practically implemented mainly due to the big number of I/O operations. In this paper, we analyze high-rate MDS codes that are simultaneously optimized in terms of storage, reliability, I/O operations, and repair-bandwidth for single and multiple failures of the systematic nodes. The codes were recently introduced in [1] without any specific name. Due to the resemblance between the hashtag sign # and the procedure of the code construction, we call them in this paper HashTag Erasure Codes (HTECs). HTECs provide the lowest data-read and data-transfer, and thus the lowest repair time for an arbitrary sub-packetization level α, where α ≤ r k/r , among all existing MDS codes for distributed storage including MSR codes. The repair process is linear and highly parallel. Additionally, we show that HTECs are the first high-rate MDS codes that reduce the repair bandwidth for more than one failure. Practical implementations of HTECs in Hadoop release 3.0.0-alpha2 demonstrate their great potentials.

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We present MQQ-SIG, a signature scheme based on "Multivariate Quadratic Quasigroups". The MQQ-SIG signature scheme has a public key consisting of n 2 quadratic polynomials in n variables where n = 160, 192, 224 or 256. Under the assumption that solving systems of n 2 MQQ's equations in n variables is as hard as solving systems of random quadratic equations, we prove that in the random oracle model our signature scheme is CMA (Chosen-Message Attack) resistant. From efficiency point of view, the signing and verification processes of MQQ-SIG are three orders of magnitude faster than RSA or ECDSA. Compared with other MQ signing schemes, MQQ-SIG has both advantages and disadvantages. Advantages are that it has more than three times smaller private keys (from 401 to 593 bytes), and the signing process is an order of magnitude faster than other MQ schemes. That makes it very suitable for implementation in smart cards and other embedded systems. However, MQQ-SIG has a big public key (from 125 to 512 Kb) and it is not suitable for systems where the size of the public key has to be small.

We construct an explicit family of locally repairable and locally regenerating codes whose existence was proven in a recent work by Kamath et al. about codes with local regeneration but no explicit construction was given. This explicit family of codes is based on HashTag codes. HashTag codes are recently defined vector codes with different vector length α (also called a sub-packetization level) that achieve the optimal repair bandwidth of MSR codes or near-optimal repair bandwidth depending on the sub-packetization level. We applied the technique of paritysplitting code construction. We show that the lower bound on the size of the finite field for the presented explicit code constructions can be lower than the one given in the work of Kamath et al. Finally, we discuss the importance of having two ways for node repair with locally regenerating HashTag codes: repair only with local parity nodes or repair with both local and global parity nodes. To the best of the authors' knowledge, this is the first work where this duality in repair process is discussed. We give a practical example and experimental results in Hadoop where we show the benefits of having this repair duality.

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