Maximal and Near-Maximal Shift Register Seqyences: Efficient Event Counters and Easy Discrete Logarithms

A Linear Feedback Shift Register, or LFSR, can implement an event counter by shifting whenever an event occurs. A single two-input exclusive-OR gate is often the only additional hardware necessary to allow a shift register to generate, by successive shifts, all of its possible nonzero values. The counting application requires that the number of shifts be recoverable from the LFSR contents so that further processing and analysis may be done. Recovering this number from the shift register value corresponds to a problem from number theory and cryptography known as the discrete logarithm. For some sizes of shift register, the maximal-length LFSR implementation requires more than a single gate, and for some the discrete logarithm calculation is hard. This paper proposes for such sizes the use of certain one-gate LFSRs whose sequence lengths are nearly maximal, and which support easy discrete logarithms. These LFSRs have a concise mathematical characterization , and are quite common. The paper concludes by describing an application of these ideas in a computer hardware monitor, and by presenting a table that describes eecient LFSRs of size up to 64 bits.