Squares of matrix-product codes

Cascudo, Ignacio; Gundersen, Jaron Skovsted; Ruano, Diego

doi:10.1016/j.ffa.2019.101606

Cited by 10 publications

(2 citation statements)

References 28 publications

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“…The best binary construction available in the literature is the one in [2] obtained from cyclic codes, but their constellation is quite limited. A larger constellation can be found at [4].…”

Section: Introductionmentioning

confidence: 95%

“…Such a SSS is enough to construct an information theoretic secure secret sharing scheme if at most t players are corrupted [1,8,11]. This application shows the importance of finding linear codes where d(C ⊥ ) is also high relative to the length of the code, where C ⊥ is the dual code of C. Although in this work we have not focused in maximizing d(C ⊥ ) (this is also the case of other articles in the literature as [2,4]), we note that for the affine variety codes considered in this article, the dual of C is again an affine variety code that can be easily constructed by [17,Proposition 1]. Moreover, its minimum distance can also be estimated using the footprint bound.…”

Section: Introductionmentioning

confidence: 99%

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High dimensional affine codes whose square has a designed minimum distance

García-Marco¹,

Márquez-Corbella²,

Ruano³

2019

Preprint

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Given a linear code C, its square code C (2) is the span of all component-wise products of two elements of C. Motivated by applications in multi-party computation, our purpose with this work is to answer the following question: which families of affine variety codes have simultaneously high dimension k(C) and high minimum distance of C (2) , d(C (2) )? More precisely, given a designed minimum distance d we compute an affine variety code C such that d(C (2) ) ≥ d and that the dimension of C is high. The best construction that we propose comes from hyperbolic codes when d ≥ q and from weighted Reed-Muller codes otherwise.

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Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

High dimensional affine codes whose square has a designed minimum distance

García-Marco¹,

Márquez-Corbella²,

Ruano³

2019

Preprint

Self Cite

View full text Add to dashboard Cite

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A Secret-Sharing Based MPC Protocol for Boolean Circuits with Good Amortized Complexity

Cascudo

Gundersen

2020

Lecture Notes in Computer Science

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We present a new secure multiparty computation protocol in the preprocessing model that allows for the evaluation of a number of instances of a boolean circuit in parallel, with a small online communication complexity per instance of 10 bits per party and multiplication gate. Our protocol is secure against an active dishonest majority, and can also be transformed, via existing techniques, into a protocol for the evaluation of a single "well-formed" boolean circuit with the same complexity per multiplication gate at the cost of some overhead that depends on the topology of the circuit. Our protocol uses an approach introduced recently in the setting of honest majority and information-theoretical security which, using an algebraic notion called reverse multiplication friendly embeddings, essentially transforms a batch of evaluations of an arithmetic circuit over a small field into one evaluation of another arithmetic circuit over a larger field. To obtain security against a dishonest majority we combine this approach with the well-known SPDZ protocol that operates over a large field. Structurally our protocol is most similar to MiniMAC, a protocol which bases its security on the use of error-correcting codes, but our protocol has a communication complexity which is half of that of MiniMAC when the best available binary codes are used. With respect to certain variant of MiniMAC that utilizes codes over larger fields, our communication complexity is slightly worse; however, that variant of MiniMAC needs a much larger preprocessing than ours. We also show that our protocol also has smaller amortized communication complexity than Committed MPC, a protocol for general fields based on homomorphic commitments, if we use the best available constructions for those commitments. Finally, we construct a preprocessing phase from oblivious transfer based on ideas from MASCOT and Committed MPC.

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Properties of constacyclic codes under the Schur product

Falk

Heninger

Rudow

2020

Des. Codes Cryptogr.

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For a subspace W of a vector space V of dimension n, the Schur-product space W k for k ∈ N is defined to be the span of all vectors formed by the componentwise multiplication of k vectors in W . It is well known that repeated applications of the Schur product to the subspace W creates subspaces W,W 2 ,W 3 , . . . whose dimensions are monotonically non-decreasing. However, quantifying the structure and growth of such spaces remains an important open problem with applications to cryptography and coding theory. This paper characterizes how increasing powers of constacyclic codes grow under the Schur product and gives necessary and sufficient criteria for when powers of the code and or dimension of the code are invariant under the Schur product.

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Squares of matrix-product codes

Cited by 10 publications

References 28 publications

High dimensional affine codes whose square has a designed minimum distance

High dimensional affine codes whose square has a designed minimum distance

A Secret-Sharing Based MPC Protocol for Boolean Circuits with Good Amortized Complexity

Properties of constacyclic codes under the Schur product

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