DOI : 10.14738/tnc.72.6473 Publication Date: 30 th April 2019 URL: http://dx.doi.org/10.14738/tnc.72.6473 V OLUME 7, N O . 2 ISSN: 2054 -7420 SOCIETY FOR SCIENCE AND EDUCATION UNITED KINGDOM T RANSACTIONS ON N ETWORKS AND C OMMUNICATIONS TNC A New Type of NLFSR Functions with Maximum Periods Ibraheem Al-Hejri, Talal Al-Kharobi College of Computer Sciences and Engineering, King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia alhejri87@gmail.com; talalkh@kfupm.edu.sa ABSTRACT Nonlinear feedback shift registers (NLFSRs) have received much attention in designing various cryptographic algorithms such as stream ciphers and light weight block ciphers in the provision of high-level security in communication systems. The main purpose of NLFSRs is to generate pseudorandom sequences of bits. NLFSRs are known to be more secure than their linear counterparts. However, there is no mathematical foundation on how to construct an NLFSR with optimal period. In this paper, we propose a new type of NLFSR function of degree 2 with optimal periods. Using our construction method, we propose 639 new functions of this type with optimal periods. Keywords: NLFSR; Stream Ciphers; Pseudorandom; Feedback Functions; Optimal Period. 1 Introduction Feedback shift register (FSR) is a kind of implementation possessing the property of randomness. FSR is one of the most efficient ways to generate pseudorandom sequences. FSR has several applications such as authentication [1], cryptography [2], testing [3], and data compression [4]. FSR has mainly two types: Linear Feedback Shift Register (LFSR) and Non- Linear Feedback Shift Register (NLFSR). Notwithstanding the ease of implementation and speed of LFSR, it can be cryptanalyzed easily due to its linearity and the simplicity of its structure. LFSRs are well studied in literature, with a well-established theory. It is easy to mathematically obtain an LSFR feedback function with an optimal period by using primitive generator polynomials. An NLFSR is a generalization of LFSR in which the current state is a nonlinear transformation of the previous state. This feature gives NLFSRs the ability to generate a very secure pseudorandom sequence which is hard to break compared to the LFSRs. Thus, to determine the structure of NLFSR, we need at least Θ (2 n ) bits, while we need just 2n bits in order to break an LFSR sequence. While the theory behind LFSR is well understood, NLFSRs still have many open fundamental problems. Probably one of the most important issues is to devise a systematic way to construct NLFSRs with optimal periods. Existing algorithms are either only applicable to small NLFSRs or consider certain special cases. In this paper, we propose a new type of feedback functions of degree 2 that provides optimal periods. Our construction method constructs 639 functions of this type for NLFSRs of sizes 8 ≤ n ≤ 23. The importance of this work lies in generating NLFSR functions that can significantly improve the cryptographic strength of the generators of stream cipher cryptosystems and pseudorandom number generators of many cryptographic algorithms. This paper is organized as follows. Section 2 presents the necessary preliminary information and related definitions.