Nonlinear feedback shift register

Abstract: It is shown in this paper that the iterative algorithm introduced by Berlekamp for decoding BCH codes actually provides a general solution to the problem of synthesizing the shortest linear feedback shift register capable of generating a prescribed finite sequence of digits. The shift-register approach leads to a simple proof of the validity of the algorithm as well as providing additional insight into its properties. The equivalence of the decoding problem for BCH codes to a shift-register synthesis problem is demonstrated, and other applications for the algorithm are suggested.

Abstract: 1. Introduction.- 2. Stream Ciphers.- 2.1. Theoretical versus Practical Security.- 2.2. The Key Stream Generator.- 2.3. The Synchronization (Problem) of Stream Ciphers.- 3. Algebraic Tools.- 3.1. Finite Fields and Polynomials.- 3.2. Linear Feedback Shift Registers (LFSRs) and Sequences.- 3.3. Minimal Polynomial and Traces.- 4. Random Sequences and Linear Complexity.- 5. Nonlinear Theory of Periodic Sequences.- 5.1. Nonlinear Operations on Phases of a Sequence with Irreducible Minimal Polynomial.- 5.2. Nonlinear Operations on Sequences with Distinct Minimal Polynomials.- 5.3. Correlation-Immunity of Memoryless Combining Functions.- 5.4. Summary and Conclusions.- 6. Multiple Speed: An Additional Parameter in Secure Sequence Generation.- 6.1. The Simulated Linear Feedback Shift Register.- 6.2. A Random Number Generator Suggested by a Linear Cipher Problem.- 6.2.1. The Random Sequence Generator.- 6.2.2. Analysis of the Random Sequence Generator.- 6.2.3. Extensions and Comments.- 7. The Knapsack as a Nonlinear Function.- 7.1. The Significance of the Knapsack for Secrecy Systems.- 7.2. Addition is a Cryptographically Useful Function.- 7.3. The Knapsack in GF(2)-Arithmetic.- 8. The Hard Knapsack Stream Cipher.- 8.1. System Description.- 8.2. Analysis of the Knapsack Stream Cipher.- 8.3. Conclusions and Design Considerations.- 8.4. Simulation Results of Small Scale Knapsack Stream Ciphers.- 9. Nonlinear Combining Functions with Memory.- 9.1. Correlation Immunity.- 9.2. The Summation Principle.- 9.3. Summary and Conclusions.- Literature References.

Abstract: A new stream cipher, Grain, is proposed. The design targets hardware environments where gate count, power consumption and memory is very limited. It is based on two shift registers and a non-linear output function. The cipher has the additional feature that the speed can be increased at the expense of extra hardware. The key size is 80 bits and no attack faster than exhaustive key search has been identified. The hardware complexity and throughput compares favourably to other hardware oriented stream ciphers like E0 and A5/1.

Abstract: Shift registers have been used to generate sequences of 0’s and 1’s for over thirty years. A wide variety of applications has been made of these sequences. Principally, communications have made use of the sequences generated.One particular class of shift register sequences for which applications exist is the full length nonlinear shift register sequences. These sequences are periodic and of length $2^n $ and all $2^n $ different binary n-tuples appear exactly one time in a periodic portion of the sequence. In this paper we discuss various algorithms which have been suggested for generating these sequences.