International Core Journal of Engineering 2020-26 | Page 186
the stochastic characteristics of pseudo-random sequences
are: periodicity, balance, run-length characteristics,
autocorrelation properties, cross-correlation properties,
linear complexity, and k-error linear complexity [5]. shift register. N is both the number of memory cells and the
number of stages of the feedback shift register. The Boolean
function f( , , ⋯, ) is a function about the initial state and
is called a feedback function of the feedback shift register.
Period: Set on the finite field GF(q), there is a sequence
=
… . If a positive integer n satisfies
= (i ≥ 0),
is said to be a sequence of period n.
the sequence The feedback shift register state is , and the feedback
function f( ) of the LFSR is a linear function with respect to
the state of the register. The n-level structure is
⊕
⊕⋯⊕
, and the
f( , , ⋯ , )=
is 0 or 1. In the n-level linear feedback shift
constant
is always 1 by default, otherwise if
register, always
=0, the nth-level memory cell is equivalent to non-
existence, and the n-th order is degraded to n-1.
Autocorrelation function: Set SF(q) to have sequences
s, period is n, and record q times until original root ε =
/
, then the periodic autocorrelation function of s
defined as
(ω)=∑
,0≤ ω ≤ n − 1.
Figure 2 is a schematic diagram of the n-stage LFSR. In
represents each feedback switch, 0 is open and
the figure,
1 is closed.
(1)
K different values can also be expressed as
autocorrelation functions of periodic sequences, but this is
called a k-value autocorrelation sequence.
If A meets
|
(ω)|=
1, ω = 0
, other
(2)
It means that s has ideal autocorrelation properties.
Linear complexity: Set on GF(q) with sequence s, period
n, the linear complexity LC(s) of s is defined as the minimum
positive integer l that is satisfied::
+
+…+
=0, l≤