Access Restriction

Author Fitzpatrick, G. B.
Source ACM Digital Library
Content type Text
Publisher Association for Computing Machinery (ACM)
File Format PDF
Copyright Year ©1960
Language English
Subject Domain (in DDC) Computer science, information & general works ♦ Data processing & computer science
Abstract A number of papers have been written from time to time about logical counters of a certain type which have quite simple logic and have been variously referred to as Binary Ring Counters, Shift Register Counters, Johnson Counters, etc. To my knowledge, most of these papers confine themselves to certain special cases and usually leave the subject with some speculation as to the possibility of generating periods of any desired length by the use of these special types. The point of view of this paper is to consider all possible counters of this general type to see how one would obtain a particular period. Special emphasis is placed on determining the least number of bits, $\textit{n},$ required to produce a given period, $\textit{K}.The$ rules for counting are as follows. If an $\textit{n}-bit$ counter is in state $(\textit{a}\textit{n}-1,$ $\textit{a}\textit{n}-2$ ···, $\textit{a}2,$ $\textit{a}1,$ $\textit{a}0)$ at a given time, $\textit{T},$ then at $\textit{T}$ + 1 its state is $(\textit{b}\textit{n}-1,$ $\textit{b}\textit{n}-2,$ ···, $\textit{b}1,$ $\textit{b}0)$ where $\textit{b}0$ = $\textit{a}\textit{n}-1,$ $\textit{bi}$ = $\textit{a}\textit{i}-1$ + $\textit{ci}\textit{a}\textit{n}-1$ for $\textit{i}$ = 1, 2, ···, $\textit{n}$ - 1. The $\textit{a}'s,$ $\textit{b}'s,$ and $\textit{c}'s$ are all 0's or 1's, the $\textit{c}'s$ being constants, and the indicated operations are carried out using modulo 2 arithmetic. This is equivalent to considering the state of the counter as an $(\textit{n}$ - 1)th degree polynomial in $\textit{X},$ multiplying said polynomial by $\textit{X}$ and reducing it modulo $\textit{m}(\textit{X}),$ where $\textit{m}(\textit{X})$ is a polynomial of degree $\textit{n}$ which is relatively prime to $\textit{X}.$ At time $\textit{T}$ the state of the counter corresponds to: $\textit{A}(\textit{X})$ = $\textit{a}\textit{n}-1\textit{X}\textit{n}-1$ + $\textit{a}\textit{n}-2\textit{X}\textit{n}-2$ + ··· + $\textit{a}1\textit{X}$ + $\textit{a}0.$ The polynomial which corresponds to the state of the counter at time $\textit{T}$ + 1 is obtained by forming $\textit{X}·\textit{A}$ $(\textit{X})$ and reducing, if necessary, modulo $\textit{m}$ $(\textit{X})$ = $\textit{Xn}$ + $\textit{c}\textit{n}-1\textit{X}\textit{n}-1$ + $\textit{c}\textit{n}-2\textit{X}\textit{n}-2$ + ··· + $\textit{c}1\textit{X}$ + 1. Since $\textit{a}\textit{n}-1·\textit{m}(\textit{X})$ = 0 mod $\textit{m}(\textit{X}),$ $\textit{X}·\textit{A}(\textit{X})$ = $\textit{X}·\textit{A}(\textit{X})+$ $\textit{a}\textit{n}-1\textit{m}(\textit{X})$ mod $\textit{m}(\textit{X}),$ so $\textit{X}·\textit{A}(\textit{X})$ = $(\textit{a}\textit{n}-2$ + $\textit{c}\textit{n}-1·\textit{a}\textit{n}-1)\textit{X}\textit{n}-1$ + $(\textit{a}\textit{n}-3$ + $\textit{c}\textit{n}-2\textit{a}\textit{n}-1)\textit{X}\textit{n}-2$ + ··· + $(\textit{a}0$ + $\textit{c}1\textit{a}\textit{n}-1)\textit{X}$ + $\textit{a}\textit{n}-1$ = $\textit{b}\textit{n}-1\textit{X}\textit{n}-1$ + $\textit{b}\textit{n}-2\textit{X}\textit{n}-2$ + ··· + $\textit{b}1\textit{X}$ + $\textit{b}0.$ It is well known that more than one possible period may be obtained depending upon the initial state of the counter. Several examples are given by Young [4]. However, starting with $\textit{X}$ itself will always yield the longest possible period for any given $\textit{m}(\textit{X})$ and, furthermore, any other periods possible will always be divisors of the major period (Theorem I below). Since these minor periods can always be obtained with moduli of lower degree they are of no real interest here, and throughout the remainder of this paper the expression “period of the counter” will be assumed to refer to the major period.The set of all polynomials whose coefficients are the integers modulo 2 is the polynomial domain $\textit{GF}(2,$ $\textit{X}),$ which has among other things unique factorization into primes (irreducibles). If $\textit{m}(\textit{X})$ is in $\textit{GF}(2,$ $\textit{X}),$ then $\textit{GF}(2,$ $\textit{X})$ modulo $\textit{m}(\textit{X})$ is a commutative ring. Thus it is closed under multiplication, but it may have proper divisors of zero. However, any element which is relatively prime to $\textit{m}(\textit{X})$ in $\textit{GF}(2,$ $\textit{X})$ has an inverse in $\textit{GF}(2,$ $\textit{X})/\textit{m}(\textit{X})$ [1].
ISSN 00045411
Age Range 18 to 22 years ♦ above 22 year
Educational Use Research
Education Level UG and PG
Learning Resource Type Article
Publisher Date 1960-07-01
Publisher Place New York
e-ISSN 1557735X
Journal Journal of the ACM (JACM)
Volume Number 7
Issue Number 3
Page Count 11
Starting Page 287
Ending Page 297

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Source: ACM Digital Library