An Authentication Code over Galois Rings with Optimal Impersonation and Substitution Probabilities

Juan Ku-Cauich,Guillermo Morales-Luna,Horacio Tapia-Recillas

doi:10.3390/mca23030046

Abstract

Two new systematic authentication codes based on the Gray map over a Galois ring are introduced. The first introduced code attains optimal impersonation and substitution probabilities. The second code improves space sizes, but it does not attain optimal probabilities. Additionally, it is conditioned to the existence of a special class of bent maps on Galois rings.

Highlights

Resilient maps were introduced in 1985 by Chor et al [1] and independently by Bennett et al [2], in the context of key distribution and quantum cryptography protocols
Within the systematic authentication codes, two main problems arise: the first problem consists of getting optimal minimal attack probabilities, the second problem consists of keeping the size of the key spaces as low as possible in comparison with the size of the message space—namely, the product of the sizes of the source state space and the tag space
In [10] a family of bent maps is introduced over Galois rings of characteristic p2, with p a prime number. The class of these maps is closed under multiplication by units in the Galois ring, under the assumption that there exists a similar class of bent functions in Galois rings of characteristic pr, with r > 2

Summary

Introduction

Resilient maps were introduced in 1985 by Chor et al [1] and independently by Bennett et al [2], in the context of key distribution and quantum cryptography protocols. Within the systematic authentication codes, two main problems arise: the first problem consists of getting optimal minimal attack probabilities, the second problem consists of keeping the size of the key spaces as low as possible in comparison with the size of the message space—namely, the product of the sizes of the source state space and the tag space. Two new systematic authentication codes based on the Gray map on a Galois ring are introduced with the purpose of optimally reducing the impersonation and substitution probabilities. The class of these maps is closed under multiplication by units in the Galois ring, under the assumption that there exists a similar class of bent functions in Galois rings of characteristic pr, with r > 2 For this hypothetical code we obtain spaces of acceptable size, similar to sizes in former constructions but with improved impersonation and substitution probabilities. The reader can find it in [11]

The Gray Map over Galois Rings

General Systematic Authentication Codes

A New Systematic Authentication Code

Main Characteristics of the New Code

Parameter Comparison with Other Codes

Conclusions