Goppa Codes Research Articles

AG Codes Achieve List-Decoding Capacity Over Constant-Sized Fields

The emergence of quantum computing poses a serious threat to conventional cryptographic schemes, necessitating the development of post-quantum cryptosystems. Simultaneously, covert communication demands innovative solutions to ensure secure and unobtrusive data transmission. This paper introduces a hybrid framework integrating the Goppa code-based Niederreiter cryptosystem with the Least Significant Bit (LSB) image steganography. The proposed scheme offers robust post-quantum confidentiality while effectively concealing ciphertext within images to achieve stealth communication. Detailed algorithms and system architecture are presented, underscoring the feasibility and security of the proposed model. Keywords: Niederreiter cryptosystem, Goppa codes, LSB Image- steganography, Hybrid post-quantum cryptography-steganography.

The article explores the challenges of ensuring data integrity and error resistance in Industrial Internet of Things (IIoT) systems, which are critically important for the functioning of automated manufacturing processes. A comprehensive analysis of key IIoT security issues is presented, along with an overview of current data protection approaches. The focus is placed on the use of error-correcting codes, particularly Goppa codes, which demonstrate high efficiency in detecting and correcting errors and show strong potential for integration into cryptographic systems. It is shown that Goppa codes can ensure not only error resistance but also data integrity due to the vast number of encoding rules, making them suitable for use in post-quantum cryptography scenarios. The results demonstrate that using Goppa codes allows for the preservation of data integrity and a significant reduction in the likelihood of introducing false data while maintaining the required level of error resistance. This confirms the feasibility of integrating such codes into the IIoT environment.

Faster List Decoding of AG Codes

The advancement of modern cryptography presents new challenges posed by quantum computers, necessitating the development of stronger encryption processes. One of the post-quantum cryptographic methods capable of providing protection against such threats is the Niederreiter cryptosystem based on binary Goppa codes. In this study, binary Goppa codes are utilized in the formation of public and private keys, as well as in the decoding process. The implementation employs a specific polynomial over a finite field of order sixteen, resulting in code parameters with a length of 12, a dimension of 4, and the capability to correct up to two errors. Goppa codes are applied in the error correction process through syndrome calculation, enabling the detection and correction of erroneous bits and accurate recovery of the original message. The results demonstrate that binary Goppa codes are effective in detecting and correcting errors, thereby ensuring message integrity. This research is expected to contribute to the development of more robust cryptosystems for maintaining information confidentiality in the rapidly evolving digital era.

The importance of data security in the digital era is growing, particularly in the face of quantum computing threats against classical cryptographic algorithms. One of the main candidates for post-quantum cryptography is the McEliece cryptosystem, which employs error-correcting codes to enhance encryption strength. This study implements Goppa codes within the McEliece cryptosystem to increase resistance against quantum attacks. A degree-two polynomial over a finite field with sixteen elements was used, resulting in code parameters with a length of twelve, a dimension of four, and the ability to correct two errors. Encryption is carried out by multiplying the binary message with the public key and adding a random error vector, while decryption utilizes the private key to correct errors through syndrome calculation. The results demonstrate that employing Goppa codes enhances system security by complicating the ciphertext structure, thereby strengthening resilience against quantum-based attacks. This implementation confirms that classical coding techniques remain relevant and effective in supporting modern cryptography.

Linear Complementary Dual Codes and Linear Complementary Pairs of AG Codes in Function Fields

Introducing locality in some generalized AG codes

Decoding Algorithms of Twisted GRS Codes and Twisted Goppa Codes

List-Decoding of AG Codes Without Genus Penalty

Abstract Let A be a local Artinian ring with residue field k(A). Let X be a curve over A and let be X′ = X ×spec A spec k(A) the fiber of X over k(A). Consider ℒ an invertible sheaf on X and ℒ ′ = ϕ *ℒ ∈ Pic(X′), where ϕ : X′ → X is the natural map. Let C and C′ be the algebraic geometric codes constructed using the groups of cohomology Γ(X, ℒ) and Γ(X′, ℒ ′) respectively. In this note, we first give the complete relation between Γ(X, ℒ) and Γ(X′, ℒ ′) without any condition and finally, we provide relations between C ⊗ A k(A) and C′ using exact sequences and dimensional theory. Therefore we extend, some results of Walker [18, 20] giving the characterization of WAG codes over rings.

The characteristics of a McEliece-type code cryptosystem on a special sum of tensor products of base codes, called D-code, are investigated. Binary Reed — Muller codes were chosen as the base codes. Previously, conditions were found for these D-codes, under which the corresponding cryptosystem is resistant to known structural attacks based on the Schur — Hadamard product. However, when using a decoder operating within half the code distance, a McEliece-type system on D-codes provides security comparable to the strength of the classical McEliece cryptosystem on Goppa codes, with a significantly larger key size. In this paper, two probabilistic decoders for D-codes are constructed. In the case of using these decoders, parameters of some D-codes have been found that provide comparable resistance to information set decoding type attacks, while having a smaller key size than in the classical system. However, the presence of a non-negligible decoding failure rate currently limits the scope of application of the D-code cryptosystem to ephemeral session key encapsulation mechanisms (IND-CPA KEM).

Galois Hulls of a kind of Goppa Codes with Applications to EAQECCs

Goppa Codes Arising from Quasihermitian Curves

On the matrix code of quadratic relationships for a Goppa code

Quantum Error Correction with Goppa Codes from Maximal Curves: Design, Simulation, and Performance

Stopping sets are useful for analyzing the performance of a linear code under an iterative decoding algorithm over an erasure channel. In this paper, we consider stopping sets of one-point algebraic geometry codes defined by a hyperelliptic curve of genus g=2 defined by the plane model y2=f(x), where the degree of f(x) was 5. We completely classify the stopping sets of the one-point algebraic geometric codes C=CΩ(D,mP∞) defined by a hyperelliptic curve of genus 2 with m≤4. For m=3, we proved in detail that all sets S⊆{1,2,⋯,n} of a size greater than 3 are stopping sets and we give an example of sets of size 2,3 that are not.

Cryptography is the science used to protect information from unauthorized access. One promising cryptographic algorithm is the McEliece algorithm, which uses code-based cryptography. This algorithm was introduced by Robert McEliece in 1978 and is known for its resistance to attacks from quantum computers, which are expected to be able to break most current cryptographic algorithms. The McEliece algorithm uses binary Goppa code for encryption and decryption, offering high execution speed and resistance to various types of attacks. Although one of its main drawbacks is the large public key size, recent developments in research have shown progress in reducing the key size without sacrificing security. This study aims to explore the working mechanism of the McEliece algorithm, analyze its advantages and disadvantages, and discuss its potential applications in modern technology. The results of this study indicate that the McEliece algorithm has great potential in the field of quantum-safe cryptography, with applications ranging from secret communication to secure data storage.

The McEliece algorithm is an asymmetric cryptosystem based on error-correcting codes, relying on the complexity of the syndrome decoding problem for its security. This study discusses the implementation of the McEliece algorithm using the Hamming(7,4) code in the encryption and decryption process of binary messages. Encryption is done by generating a public key consisting of a disguised generator matrix G′, a permutation matrix P, and a non-singular matrix SSS. The binary message is encrypted by adding controlled noise to increase security. In the decryption phase, the received message is processed using reverse permutation and error detection with a parity check matrix to recover the original message. Experiments are carried out by implementing the algorithm in Python, with results showing successful encryption and decryption of messages according to the McEliece theoretical framework. This study confirms that the Hamming code can be used as a simplified approach to the implementation of McEliece, although with security limitations compared to Goppa codes.

Classical Goppa codes are a well-known class of codes with applications in code-based cryptography, which are a special case of alternant codes. Many papers are devoted to the search for Goppa codes with a cyclic extension or with a cyclic parity-check subcode. Let F q be a finite field with q = 2 l elements, where l is a positive integer. In this paper, we determine all the generator polynomials of cyclic expurgated or extended Goppa codes under some prescribed permutations induced by the projective general linear automorphism A ∈ P GL 2 (F q ). Moreover, we provide some examples to support our findings. 2010 Mathematics Subject Classification. 94B05.