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In this paper we study the distribution properties of d-FCSR sequences. These sequences have ecient generators and have several good statistical properties. We show that for d = 2 the number of occurrences of an xed size subsequence diers from the average number of occurrences by at most a small constant times the square root of the average.

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Abstract. The Feedback with Carry Shift Registers (FCSRs) have been proposed as an alternative to Linear Feedback Shift Registers (LFSRs) for the design of stream ciphers. FCSRs have good statistical properties and they provide a built-in non-linearity. However, two attacks have shown that the current representations of FCSRs can introduce weaknesses in the cipher. We propose a new “ring ” representation of FCSRs based upon matrix definition which generalizes the Galois and Fibonacci representations. Our approach preserves the statistical properties and circumvents the weaknesses of the Fibonacci and Galois representations. Moreover, the ring representation leads to automata with a quicker diffusion characteristic and better implementation results. As an application, we describe a new version of F-FCSR stream ciphers. Keywords: Stream cipher, FCSRs, ℓ-sequence, ring FCSRs. 1

Abstract. Maximal-length binary sequences have been known for a long time. They have many interesting properties, one of them is that when taken in blocks of n consecutive positions they form 2 n − 1 different codes in a closed circular sequence. This property can be used for measuring absolute angular positions as the circle can be divided in as many parts as different codes can be retrieved. This paper describes how can a closed binary sequence with arbitrary length be effectively designed with the minimal possible block-length, using linear feedback shift registers (LFSR). Such sequences can be used for measuring a specified exact number of angular positions, using the minimal possible number of sensors that linear methods allow. 1.

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The feedback-with-carry shift register (FCSR) is an important primitive in the design of stream ciphers. In the first part of this thesis, we propose efficient methods to search for FCSR architectures of guaranteed period and 2-adic complexity. We devise extended versions of these methods that yield architectures of guaranteed period and 2-adic complexity, given additional design constraints such as a fixed number of feedback tap connections. We also propose a search algorithm for a generalisation of the basic FCSR architecture called the d-FCSR, and discuss the difficulty of finding valid architectures for values of the parameter d other than d = 2.

Stream ciphers are often used when it is necessary to encrypt large amounts of data very quickly; they are private-key encryption algorithms that operate on the plaintext one bit or one machine word at a time. Stream ciphers are considered secure if knowledge of a small number of bits of keystream cannot be used to recover the entire keystream. A good stream cipher should have good randomness, high period, linear span, and security against any known attack. Hence, the development of a good random number generator has been a hot topic in cryptology. Cryptographers have liked stream ciphers made up of shift registers: They are easily implemented in digital hardware. The Feedback with carry shift register (FCSR) is introduced as a source for long pseudo random binary sequences. Therefore, this paper provides the design of a new stream cipher model, it is an FCSR based stream cipher model, main building blocks and modules of the cipher system model are presented in details. The design considerations are provided. The initialization of the new cipher model and the operational scenario of the new model as a whole are introduced. Finally, the new cipher model is being implemented using VHDL. Keywords: Stream ciphers, feedback shift registers, FCSR. 1

the security of FCSR-based pseudorandom generators