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

2018 ◽  
Vol 23 (3) ◽  
pp. 46
Author(s):  
Juan Ku-Cauich ◽  
Guillermo Morales-Luna ◽  
Horacio Tapia-Recillas
Keyword(s):  
Special Class ◽  
Galois Ring ◽  
Gray Map ◽  
Galois Rings ◽  
Authentication Codes ◽  
Authentication Code

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.

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

2018 ◽  
Author(s):  
Juan Carlos Ku-Cauich ◽  
Guillermo Morales-Luna ◽  
Horacio Tapia-Recillas
Keyword(s):  
Special Class ◽  
Galois Ring ◽  
Gray Map ◽  
Galois Rings ◽  
Authentication Codes ◽  
Authentication Code

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. Besides it is conditioned to the existence of a special class of bent maps on Galois rings.

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 Efficient Secret Image Sharing Scheme with Authentication and Cheating Prevention

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Lina Zhang ◽  
Xuan Dang ◽  
Li Feng ◽  
Junhan Yang
Keyword(s):  
Distributed Storage ◽  
Limited Resources ◽  
Secret Image Sharing ◽  
Authentication Codes ◽  
Authentication Code ◽  
Secret Image ◽  
Image Sharing ◽  
Sharing Scheme ◽  
Hash Algorithm ◽  
Security Levels

Due to the widespread adoption and popularity of digital images in distributed storage, Secret Image Sharing (SIS) has attracted much attention. However, preventing the cheating of shares is an important problem that needs to be solved in the traditional SIS scheme. An adversary without image shares may participate in the restoration phase as a share owner. In this phase, the adversary can obtain real shares or prevent recovering real images by submitting fake shadows. Our schemes are based on the original Thien-Lin’s scheme. In the scheme I, we use some XOR operations to get two authentication codes through all secret pixel values to achieve a lightweight and fast-calculated authentication scheme for cheating prevention. This scheme is suitable for small devices with limited resources. In scheme II, we use a hash algorithm to generate the authentication code. This scheme is suitable for environments with larger storage space and higher security levels. Since all pixel values are involved in the authentication in our proposed schemes, it can prevent fake shadow images from cheating. Meanwhile, the shadow size is almost the same as the original Thien-Lin’s scheme. Experimental results and theoretical analysis show that the proposed schemes are feasible and effective.

Symbol-Pair Distances of Repeated-Root Negacyclic Codes of Length 2s over Galois Rings

2021 ◽  
Vol 28 (04) ◽  
pp. 581-600
Author(s):  
Hai Q. Dinh ◽  
Hualu Liu ◽  
Roengchai Tansuchat ◽  
Thang M. Vo
Keyword(s):  
Finite Field ◽  
Galois Ring ◽  
Maximum Distance ◽  
Galois Rings ◽  
Constacyclic Codes ◽  
Chain Ring ◽  
Negacyclic Codes ◽  
Singleton Bound ◽  
Maximum Distance Separable ◽  
Special Case

Negacyclic codes of length [Formula: see text] over the Galois ring [Formula: see text] are linearly ordered under set-theoretic inclusion, i.e., they are the ideals [Formula: see text], [Formula: see text], of the chain ring [Formula: see text]. This structure is used to obtain the symbol-pair distances of all such negacyclic codes. Among others, for the special case when the alphabet is the finite field [Formula: see text] (i.e., [Formula: see text]), the symbol-pair distance distribution of constacyclic codes over [Formula: see text] verifies the Singleton bound for such symbol-pair codes, and provides all maximum distance separable symbol-pair constacyclic codes of length [Formula: see text] over [Formula: see text].

scholarly journals Cogredient Standard Forms of Symmetric Matrices over Galois Rings of Odd Characteristic

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Yonglin Cao
Keyword(s):  
Prime Number ◽  
Explicit Formula ◽  
Galois Ring ◽  
Symmetric Matrices ◽  
Galois Rings ◽  
Positive Integers

Let R=GR(ps,psm) be a Galois ring of characteristic ps and cardinality psm, where s and m are positive integers and p is an odd prime number. Two kinds of cogredient standard forms of symmetric matrices over R are given, and an explicit formula to count the number of all distinct cogredient classes of symmetric matrices over R is obtained.

scholarly journals Three authentication schemes without secrecy over finite fields and Galois rings

2021 ◽  
Author(s):  
Juan Carlos Ku-Cauich ◽  
Miguel Angel Márquez-Hidalgo
Keyword(s):  
Finite Fields ◽  
Gray Map ◽  
Galois Rings ◽  
Main Achievement ◽  
The Third ◽  
Key Space ◽  
Authentication Schemes ◽  
The Given ◽  
Message Space

We give three new authentication schemes without secrecy. The first two on finite fields and Galois rings, using Gray map for this link. The third construction is given on Galois rings. The main achievement in this work is to obtain optimal impersonation and substitution probabilities in the schemes. Additionally, in the first and second scheme, we simplify the source space and bring a better relationship between the size of the message space and the key space than the given in [8]. Finally, we provide a third scheme on Galois rings, which generalizes the scheme over finite fields constructed in [9].

scholarly journals A New Construction of Multisender Authentication Codes from Polynomials over Finite Fields

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Xiuli Wang
Keyword(s):  
Finite Fields ◽  
Authentication Codes ◽  
Authentication Code ◽  
Polynomials Over Finite Fields ◽  
New Construction

Multisender authentication codes allow a group of senders to construct an authenticated message for a receiver such that the receiver can verify the authenticity of the received message. In this paper, we construct one multisender authentication code from polynomials over finite fields. Some parameters and the probabilities of deceptions of this code are also computed.

EISENSTEIN LATTICES, GALOIS RINGS AND QUATERNARY CODES

2006 ◽  
Vol 02 (02) ◽  
pp. 289-303 ◽  
Author(s):  
PHILIPPE GABORIT ◽  
ANN MARIE NATIVIDAD ◽  
PATRICK SOLÉ
Keyword(s):  
Modular Lattice ◽  
Orthogonal Basis ◽  
Galois Ring ◽  
Galois Rings ◽  
Type Ii ◽  
Dual Codes ◽  
Type Ii Codes ◽  
Quaternary Codes ◽  
Double Circulant Codes ◽  
Circulant Codes

Self-dual codes over the Galois ring GR(4,2) are investigated. Of special interest are quadratic double circulant codes. Euclidean self-dual (Type II) codes yield self-dual (Type II) ℤ4-codes by projection on a trace orthogonal basis. Hermitian self-dual codes also give self-dual ℤ4-codes by the cubic construction, as well as Eisenstein lattices by Construction A. Applying a suitable Gray map to self-dual codes over the ring gives formally self-dual 𝔽4-codes, most notably in length 12 and 24. Extremal unimodular lattices in dimension 38, 42 and the first extremal 3-modular lattice in dimension 44 are constructed.

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