A Deterministic Construction of Normal Bases With Complexity O(n3 + n log n log(log n) log q)

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.

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Optimal Normal Bases

Applications of Finite Fields ◽

10.1007/978-1-4757-2226-0_5 ◽

1993 ◽

pp. 93-114

Author(s):

Ian F. Blake ◽

XuHong Gao ◽

Ronald C. Mullin ◽

Scott A. Vanstone ◽

Tomik Yaghoobian

Keyword(s):

Normal Bases

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Normal Bases

Applications of Finite Fields ◽

10.1007/978-1-4757-2226-0_4 ◽

1993 ◽

pp. 69-92

Author(s):

Ian F. Blake ◽

XuHong Gao ◽

Ronald C. Mullin ◽

Scott A. Vanstone ◽

Tomik Yaghoobian

Keyword(s):

Normal Bases

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Optimal normal bases

Designs Codes and Cryptography ◽

10.1007/bf00125200 ◽

1992 ◽

Vol 2 (4) ◽

pp. 315-323 ◽

Cited By ~ 59

Author(s):

Shuhong Gao ◽

Hendrik W. Lenstra

Keyword(s):

Normal Bases

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Semi-Deterministic Construction of Scale-Free Networks with Designated Parameters

Journal of Interconnection Networks ◽

10.1142/s0219265918500019 ◽

2018 ◽

Vol 18 (01) ◽

pp. 1850001

Author(s):

NAOKI TAKEUCHI ◽

SATOSHI FUJITA

Keyword(s):

Degree Distribution ◽

Interconnection Networks ◽

Maximum Degree ◽

Minimum Degree ◽

Scale Free ◽

Scale Free Networks ◽

Ideal Number ◽

Message Propagation ◽

Deterministic Construction ◽

The Ideal

Scale-free networks have several favorable properties as the topology of interconnection networks such as the short diameter and the quick message propagation. In this paper, we propose a method to construct scale-free networks in a semi-deterministic manner. The proposed algorithm extends the Bulut's algorithm for constructing scale-free networks with designated minimum degree k and maximum degree m, in such a way that: (1) it determines the ideal number of edges derived from the ideal degree distribution; and (2) after connecting each new node to k existing nodes as in the Bulut’s algorithm, it adjusts the number of edges to the ideal value by conducting add/removal of edges. We prove that such an adjustment is always possible if the number of nodes in the network exceeds [Formula: see text]. The performance of the algorithm is experimentally evaluated.

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