In this study, the authors study a turbo‐coding (TC) scheme, whose constituent codes are designed using convolutional encoders. These encoders are finite‐state sequence machines (FSSMs) operating over the Galois Field, GF(4). The scheme includes encryption polynomials whose coefficients are selected every L steps, from the set of optimal polynomials of GF(4). Two cases are considered for the polynomial selection: periodical and random. This kind of encoder was studied in a previous study and a correspondence between the randomness of the encoded sequence and performance of the TC was conjectured. The main contribution of this study is to systematically confirm this correspondence, by analysing the randomness of the output and performance of the TC using several randomness quantifiers. Three of the quantifiers are defined on the basis of recurrence plots. Other two quantifiers are defined on the basis of the information theory. All the quantifiers allow one to justify why the proposed TC works better with random selection of the optimal polynomials and with small values of L . In summary, it is shown that a random selection of polynomials and a small L produce FSSMs with enhanced randomness properties and it is also shown that they produce the best quality of the TC, measured by means of the corresponding bit error rate.

The behaviour of nonlinear turbo codes that use nonlinear finite state sequential machines (FSSMs), designed over the finite field GF(4), as constituent encoders is studied. These FSSMs show a nonzero output for the all-zero input case. The state-transitions structure and the randomness of the output for the all-zero input response of these FSSMs are studied for different initial state conditions. A relationship is found between these two characteristics and the BER performance or, equivalently, the form of the extrinsic information transfer chart of the corresponding turbo code. It is found that FSSMs with responses of the form of long closed cycles, and good randomness properties of the corresponding output are the best options as constituent encoders of nonlinear turbo codes designed over GF(4). A coefficient called the random behaviour coefficient is defined, to provide a quantitative measure of the behaviour of these FSSMs as constituent encoders of a turbo code.

In this paper, we present quasi-chaotic (QC) schemes for secure digital communication systems, designed over Galois fields (GFs) with optimal randomness properties. Schemes with the maximum-output sequence length (all-zero input response) are presented for different GFs. Two coefficients are used to quantitatively measure the proximity of the behavior of these schemes with respect to the ideal white noise behavior. The proposed schemes outperform those presented by Frey in his paper, and achieve the optimal QC properties available for a given chaotic digital scheme. The proposed schemes have also a very little loss in bit-error rate performance so that they are a good alternative to the design of systems for which encryption and error correction are important joint goals. They can be implemented with time-variant coefficients in order to highly improve the cryptographic properties of the transmission.