A Graph-Based Approach to Designing Parallel Multipliers over Galois Fields Based on Normal Basis Representations

2013 ◽  
Author(s):  
K. Okamoto ◽  
N. Homma ◽  
T. Aoki
Keyword(s):  
Normal Basis ◽  
Galois Fields ◽  
Basis Representations

scholarly journals Normal Bases on Galois Ring Extensions

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 702
Author(s):  
Aixian Zhang ◽  
Keqin Feng
Keyword(s):  
Signal Processing ◽  
Finite Field ◽  
Block Cipher ◽  
Field Extension ◽  
Galois Ring ◽  
Normal Basis ◽  
Galois Rings ◽  
Ring Extension ◽  
Galois Fields ◽  
Normal Bases

Normal bases are widely used in applications of Galois fields and Galois rings in areas such as coding, encryption symmetric algorithms (block cipher), signal processing, and so on. In this paper, we study the normal bases for Galois ring extension R / Z p r , where R = GR ( p r , n ) . We present a criterion on the normal basis for R / Z p r and reduce this problem to one of finite field extension R ¯ / Z ¯ p r = F q / F p ( q = p n ) by Theorem 1. We determine all optimal normal bases for Galois ring extension.

Efficient parallel exponentiation in using normal basis representations

2005 ◽  
Vol 54 (2) ◽  
pp. 205-221 ◽  
Author(s):  
Mun-Kyu Lee ◽  
Yoonjeong Kim ◽  
Kunsoo Park ◽  
Yookun Cho
Keyword(s):  
Normal Basis ◽  
Basis Representations

scholarly journals Quantum binary field inversion: improved circuit depth via choice of basis representation

2013 ◽  
Vol 13 (1&2) ◽  
pp. 116-134
Author(s):  
Brittanney Amento ◽  
Martin Rotteler ◽  
Rainer Steinwandt
Keyword(s):  
Coding Theory ◽  
Fault Tolerant ◽  
Quantum Circuit ◽  
Normal Basis ◽  
Binary Fields ◽  
Resource Requirements ◽  
Gaussian Normal Basis ◽  
Circuit Depth ◽  
Quantum Arithmetic ◽  
Basis Representations

Finite fields of the form ${\mathbb F}_{2^m}$ play an important role in coding theory and cryptography. We show that the choice of how to represent the elements of these fields can have a significant impact on the resource requirements for quantum arithmetic. In particular, we show how the use of Gaussian normal basis representations and of `ghost-bit basis' representations can be used to implement inverters with a quantum circuit of depth $\bigO(m\log(m))$. To the best of our knowledge, this is the first construction with subquadratic depth reported in the literature. Our quantum circuit for the computation of multiplicative inverses is based on the Itoh-Tsujii algorithm which exploits that in normal basis representation squaring corresponds to a permutation of the coefficients. We give resource estimates for the resulting quantum circuit for inversion over binary fields ${\mathbb F}_{2^m}$ based on an elementary gate set that is useful for fault-tolerant implementation.

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