scholarly journals Formal Design of Arithmetic Circuits over Galois Fields Based on Normal Basis Representations

2014 ◽  
Vol E97.D (9) ◽  
pp. 2270-2277 ◽  
Author(s):  
Kotaro OKAMOTO ◽  
Naofumi HOMMA ◽  
Takafumi AOKI
Keyword(s):  
Normal Basis ◽  
Arithmetic Circuits ◽  
Galois Fields ◽  
Formal Design ◽  
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.

scholarly journals Parallel Construction of Irreducible Polynomials

1991 ◽  
Vol 20 (358) ◽  
Author(s):  
Gudmund Skovbjerg Frandsen
Keyword(s):  
Irreducible Polynomial ◽  
Field Extension ◽  
Normal Basis ◽  
Arithmetic Circuits ◽  
Irreducible Polynomials ◽  
Prime Field ◽  
Parallel Construction ◽  
Finite Prime Field

Let arithmetic pseudo-<strong>NC</strong>^k denote the problems that can be solved by log space uniform arithmetic circuits over the finite prime field GF(p) of depth O(log^k (n + p)) and size polynomial in (n + p). We show that the problem of constructing an irreducible polynomial of specified degree over GF(p) belongs to pseudo-<strong>NC</strong>^2.5. We prove that the problem of constructing an irreducible polynomial of specified degree over GF(p) whose roots are guaranteed to form a normal basis for the corresponding field extension pseudo-<strong>NC</strong>^2 -reduces to the problem of factor refinement. We show that factor refinement of polynomials is in arithmetic <strong>NC</strong>^3. Our algorithm works over any field and compared to other known algorithms it does not assume the ability to take <em>p</em>'th roots when the field has characteristic <em>p</em>.

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