Feedback with carry shift registers: 2-adic models and summation combiners

Pseudorandom sequences with a variety of statistical properties are important in many-areas of communications and computing. Sequences such as m-sequences, nonlinear feedback shift register sequences, and summation combiner sequences, have been successfully studied largely because there is an algebraic framework based on finite fields for analysing them. We describe a new type of feedback register, feedback with carry shift registers (FCSRs). These easily implemented devices relate summation combiner sequences, arithmetic codes and 1/q sequences. We describe an algebraic framework, based on algebra over the 2-adic numbers, in which FCSR sequences can be analyzed. This gives a method for cracking the summation combiner which has been suggested for generating cryptographically secure binary sequences