Summary of Random Numbers Generated by Linear Recurrence
Modulo Two
Yu Zhang
yz08@fsu.edu
There is a kind of numbers which appear to be random. But they show regularities
under closer and longer observation. In this paper, the author presented a random
number generator of this type. Here, the numbers are generated by modulo two
linear recurrence techniques.
First, the author introduced linear recurrence relations over GF(2). A sequence
consists of 0’s and 1’s is generated by the linear recursion relation
ak=c1ak-1+c2ak-2+…+cnak-n (mod 2) for any given set of integers ci (0 or 1). Obviously,
the period of the sequence cannot be longer than 2n-1. If the polynomial
f(x)=1+c1x+c2x2+…+xn be primitive over GF(2), the sequence can reach the maximum
p
period. There are three interest properties of the sequence: (1) ∑k=1 ak =
p+1
2
=
2n−1 . (2) “cycle-and-add” property: for every different two nonzero set, there is a
unique integer v which is no less than p-1 and no larger than 0 such that for every k,
s1ak-1+s2ak-2+…+snak-n=ak+v (mod 2). (3) Every nonzero vector (e1, e2, … en) only shows
one time per period as n consecutive binary digits in a. Moreover, if we change the
definition slightly, we will get a different sequence version: α k=(-1) α k=1-2α k.
Obviously, ak consists of 0 and 1 whileαk takes the values 1 and -1. And the properties
p
become to (1) ∑k=1 αk = −1; (2) For every different two nonzero set, there is a
unique integer v which is no less than p-1 and no larger than 0 such that
s1
s2
sn
αk−1 αk−2 … αk−n = αk+v . (3) Except the all 1s vector, every n-vector shows noly
one time per period.
Second, the author shows how to generate the random number. Given a sequence a
generated
above,
we
can
define
a
sequence:
yk = 0 ∙ aqk+r−1 aqk+r−2 ∙∙∙
aqk+r−L (basw 2), where r is an integer which is choosen randomly and 0<=r<=2 n-1.
L<=n. q>=L and (q, 2n-1)=1. Another form of yk is yk = ∑Lt=1 2−t aqk+r−t . From the
previous session, we can define a more convenient way of generating a sequence.
Using the sequence α which is corresponding to a, we can define wk =
∑Lt=1 2−t αqk+r−t . The relation between yk and wk is wk=1-2-L-2yk.
Finally, the author pointed that there are many properties of this kind of random
numbers. For example, the mean value μ of the sequence wk is μ =
1−2−L
−2−n (1−2−n )
1−2−L
2
≈ 0, and the variance σ
1
2−n (1−2−n )2 )] ≈ 3.
2
1
is σ = 3 +
1
1−2−2L
2−n [3 (( 1−2−n )
−
(1−2−L )
1−2−n
2
−