Normal Bases on Galois Ring Extensions

• Main Authors: Aixian Zhang, Keqin Feng
• Format: Article
• Language: English
• Published: MDPI AG 2018-12-01
• Series: Symmetry
• Subjects: Galois ring; optimal normal basis; multiplicative complexity; finite field
• Online Access: https://www.mdpi.com/2073-8994/10/12/702

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 <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="bold">R</mi> <mo>/</mo> <msub> <mi mathvariant="normal">Z</mi> <msup> <mi>p</mi> <mi>r</mi> </msup> </msub> </mrow> </semantics> </math> </inline-formula>, where <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="bold">R</mi> <mo>=</mo> <mi>GR</mi> <mo stretchy="false">(</mo> <msup> <mi>p</mi> <mi>r</mi> </msup> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> We present a criterion on the normal basis for <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="bold">R</mi> <mo>/</mo> <msub> <mi mathvariant="normal">Z</mi> <msup> <mi>p</mi> <mi>r</mi> </msup> </msub> </mrow> </semantics> </math> </inline-formula> and reduce this problem to one of finite field extension <inline-formula> <math display="inline"> <semantics> <mrow> <mover> <mi mathvariant="bold">R</mi> <mo>&#175;</mo> </mover> <mo>/</mo> <msub> <mover> <mi mathvariant="normal">Z</mi> <mo>&#175;</mo> </mover> <msup> <mi>p</mi> <mi>r</mi> </msup> </msub> <mo>=</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo>/</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>p</mi> </msub> <mspace width="4pt"></mspace> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>=</mo> <msup> <mi>p</mi> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> by Theorem 1. We determine all optimal normal bases for Galois ring extension.