Construction of Multiblock Space–Time Codes From Division Algebras With Roots of Unity as Nonnorm Elements

Lahtonen, Jyrki; Markin, Nadya; McGuire, Gary

doi:10.1109/tit.2008.929963

Cited by 7 publications

(17 citation statements)

References 14 publications

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“…Proof: Note that the ideal 7Z remains inert in Z[i], in other words, its residue field Z[i]/7Z[i] is the finite field F 49 of 49 elements. We further note that the ideal 7Z[i] is completely ramified in Q(ζ 7 , i), hence its residue field remains to be of size 49 in K. By local class field theory, the reciprocity map induces an epimorphism F × 49 → C 3 ∼ = Gal(L/K), whose kernel is a subgroup of F × 49 of order 16, which contains the image of norms of L/K (see [20] for more detailed treatment of this type of argument). Hence if an element is a norm of Q(ζ 7 , i)/Q(i,…”

Section: Claimmentioning

confidence: 99%

“…Hence v 2 = σ(v 2 ) and we have therefore just shown that θ = zσ(z). Note that since our construction allows the ground field F to be of arbitrary degree over Q, it can moreover be used for the multiblock construction in [20].…”

Section: This Givesmentioning

confidence: 99%

“…whose kernel is a subgroup of F 49 of order 16, which contains the image of norms of L/K (see [20] for more detailed treatment of this type of argument). Hence if an element is a norm of…”

Section: B Examples Examplementioning

confidence: 99%

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Iterated Space-Time Code Constructions From Cyclic Algebras

Markin

Oggier

2013

IEEE Trans. Inform. Theory

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We propose a full-rate iterated space-time code construction, to design codes of Q-rank 2n from cyclic algebra based codes of Q-rank n. We give a condition for determining whether the resulting codes satisfy the full diversity property, and study their maximum likelihood decoding complexity with respect to sphere decoding. Particular emphasis is given to the asymmetric MIDO (multiple input double output) codes. In the process, we derive an interesting way of obtaining division algebras, and study their center and maximal subfield.

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Section: Claimmentioning

confidence: 99%

Section: This Givesmentioning

confidence: 99%

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Iterated Space-Time Code Constructions From Cyclic Algebras

Markin

Oggier

2013

IEEE Trans. Inform. Theory

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“…This paper is inspired by the abundant literature of space-time coding based on cyclic division algebras (see the monographs [9,32] and references therein). On a high level, our construction is reminiscent of multiblock space-time codes [21,23], with the caveat of scaling up the number of blocks to make the codes practically undecodable. In the context of space-time coding, our construction generalizes [21] and offers greater flexibility in the code parameters (the number of blocks vs. the number of antennas).…”

Section: Related Work and Organizationmentioning

confidence: 99%

Non-commutative Ring Learning with Errors from Cyclic Algebras

Grover

Ling²,

Vehkalahti

2022

J Cryptol

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The Learning with Errors (LWE) problem is the fundamental backbone of modern lattice-based cryptography, allowing one to establish cryptography on the hardness of well-studied computational problems. However, schemes based on LWE are often impractical, so Ring LWE was introduced as a form of ‘structured’ LWE, trading off a hard to quantify loss of security for an increase in efficiency by working over a well-chosen ring. Another popular variant, Module LWE, generalizes this exchange by implementing a module structure over a ring. In this work, we introduce a novel variant of LWE over cyclic algebras (CLWE) to replicate the addition of the ring structure taking LWE to Ring LWE by adding cyclic structure to Module LWE. We show that the security reductions expected for an LWE problem hold, namely a reduction from certain structured lattice problems to the hardness of the decision variant of the CLWE problem (under the condition of constant rank d). As a contribution of theoretic interest, we view CLWE as the first variant of Ring LWE which supports non-commutative multiplication operations. This ring structure compares favorably with Module LWE, and naturally allows a larger message space for error correction coding.

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“…In the STBC literature two large classes of algebras have been studied: cyclic algebras (e.g. in [2], [3], [5]) and crossed-product algebras (e.g. in [6] and [7]).…”

mentioning

confidence: 99%

Quadratic Forms and Space-Time Block Codes From Generalized Quaternion and Biquaternion Algebras

Unger

Markin

2011

IEEE Trans. Inform. Theory

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In the context of space-time block codes (STBCs), the theory of generalized quaternion and biquaternion algebras (i.e., tensor products of two quaternion algebras) over arbitrary base fields is presented, as well as quadratic form theoretic criteria to check if such algebras are division algebras. For base fields relevant to STBCs, these criteria are exploited, via Springer's theorem, to construct several explicit infinite families of (bi-)quaternion division algebras. These are used to obtain new 2 × 2 and 4 × 4 STBCs.

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Construction of Multiblock Space–Time Codes From Division Algebras With Roots of Unity as Nonnorm Elements

Cited by 7 publications

References 14 publications

Iterated Space-Time Code Constructions From Cyclic Algebras

Iterated Space-Time Code Constructions From Cyclic Algebras

Non-commutative Ring Learning with Errors from Cyclic Algebras

Quadratic Forms and Space-Time Block Codes From Generalized Quaternion and Biquaternion Algebras

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