Development of a Data Transmission System Based on Convolutional Codes with Error Llocalization

Abstract- The paper describes and investigates a new version of the McEliece scheme. The object of development is a public key cryptosystem with generalized error localization based on extended generalized Reed-Solomon codes. As a secret code, the code over the expanded field is taken and hidden by shortening its parity check matrix over the base field. Particular attention in this work is given to the representation of the extended field by means of the elements of the base field, since this procedure increases the efficiency of the proposed model from the point of view of cryptographic reliability. Considering the punctured parity check matrix over the extended field, where each element is represented as elements over the base field, protects the cryptosystem from hacking methods using the Schur algorithm. Also, by hiding the explicit cyclical form of the generalized Reed-Solomon code, by expanding and shortening the parity check matrix, we can reduce the key size of the cryptosystem approximately by 40% relative to the key sizes in the original version invented by mathematician Robert McEliece. The aim of the work is a mathematical and programmatic description of the field expansion and representing the elements of the extension field as elements above the base field algorithm, an algorithm for constructing a code over a field of this type, in particular, its parity check matrix, the cryptosystem key generation algorithm, the encryption and decryption algorithm. An analytical assessment of the effectiveness of the proposed cryptosystem in comparison with the classical version of the cryptosystem based on Goppa codes is also given. The result of the work is an algorithm implemented using the mathematical package MATLAB, which allows to encrypt and decrypt the code above the field, with the specified parameters. The software implementation of this algorithm is a confirmation of the working capacity of the described mathematical model. A comparative characteristic of the obtained algorithm with existing encryption schemes is also given. Keywords: code cryptography, McEliece cryptosystem, Reed-Solomon codes, extended codes, short codes.