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. N e t w o r k s a n d C o m m u n i c a t i o n s ; V o l u m e 7 , N o . 2 , A p r i l 2 0 1 9 C o p y r i g h t © S o c i e t y f o r S c i e n c e a n d E d u c a t i o n , U n i t e d K i n g d o m 21 Section 3 reviews the existing studies related to NLFSR functions with optimal periods. Section 4 describes the proposed construction method. Section 5 presents the new NLFSR feedback functions. Finally, Section 6 concludes the present study and offers recommendations for future work. T r a n s a c t i o n s o n