Error reconciliation is an important technique for learning with error (LWE) and ring-LWE (RLWE)-based constructions. In this paper, we present a comparison analysis on two error reconciliation-based RLWE key exchange protocols: Ding et al. in 2012 (DING12) and Bos et al. in 2015 (BCNS15). We take them as examples to explain the core idea of error reconciliation, building key exchange over RLWE problem, implementation and real-world performance, and compare them comprehensively. We also analyse a LWE key exchange 'Frodo' that uses an improved error reconciliation mechanism in BCNS15. To the best of our knowledge, our work is the first to present at least 128-bit classic (80-bit quantum) and 256-bit classic (> 200-bit quantum) secure parameter choices for DING12 with efficient portable C/C++ implementations. Benchmark shows that our efficient implementation is 11× faster than BCNS15 and one key exchange execution only costs 0.07 ms on a four-year-old middle range CPU. Error reconciliation is 1.57× faster than BCNS15.

In this thesis, we consider some aspects of$noncommutative classical invariant theory$, i.e., noncommutative invariants of$the classical group SL(2, k)$. We develop a$symbolic method$for invariants and covariants, and we use the method to compute some invariant algebras. The subspace$Ĩ$_{d}^{m}of the noncommutative invariant algebra$Ĩ$_{$d$}consisting of homogeneous elements of degree$m$has the structure of a module over the$symmetric group S$_{$m$}. We find the explicit decomposition into irreducible modules. As a consequence, we obtain the$Hilbert series$of the commutative classical invariant algebras. The$Cayley—Sylvester theorem$and the$Hermite reciprocity law$are studied in some detail. We consider a new power series H($Ĩ$_{d},$t$) whose coefficients are the number of irreducible$S$_{$m$}-modules in the decomposition of$Ĩ$_{d}^{m}, and show that it is rational. Finally, we develop some analogues of all this for covariants.

Graph theory is important in information theory. We introduce a quantization process on graphs and apply the quantized graphs in quantum information. The quon language provides a mathematical theory to study such quantized graphs in a general framework. We give a new method to construct graphical quantum error correcting codes on quantized graphs and characterize all optimal ones. We establish a further connection to geometric group theory and construct quantum low-density parity-check stabilizer codes on the Cayley graphs of groups. Their logical qubits can be encoded by the ground states of newly constructed exactly solvable models with translation-invariant local Hamiltonians. Moreover, the Hamiltonian is gapped in the large limit when the underlying group is infinite.

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