In Handbook of applied cryptography, Golomb's randomness postulates are given: Let $s$ be a periodic sequence of period $N$. Golomb’s randomness postulates are the following.

R1: In the cycle $s_N$ of $s$, the number of $1$s differs from the number of $0$s by at most $1$.

R2: In the cycle $s_N$, at least half the runs have length $1$, at least one-fourth have length $2$, at least one-eighth have length $3$, etc., as long as the number of runs so indicated exceeds $1$. Moreover, for each of these lengths, there are (almost) equally many gaps and blocks.6

R3: The autocorrelation function $C(t)$ is two-valued. That is for some integer $K$, $C(t)=K$ for $t\neq 0 \pmod N$ while $C(0)=N.$

But why do those postulates make sense?

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    $\begingroup$ I fixed your autocorrelation definition that seemed to have dropped off. $\endgroup$ – kodlu Feb 12 at 4:48

They make sense as a starting point, for pseudorandomness.

R1. This is a strict balancedness condition, the difference is 1, in case $N$ is odd and zero is impossible.

R2. If you have an i.i.d. and unbiased binary source, the run length distribution is geometric, which in a finite length segment should follow the profile given in R2, due to the law of large numbers.

R3. Having an autocorrelation $C(t)$ of the form $$ C(t)=N I\{t=0\}+ f(t) I\{t \neq 0\} $$ where $I$ is the indicator function, with $f(t)$ small in absolute value is a feature of an i.i.d. independent sequence. By parseval $|f(t)|\leq 1$ is possible for odd lengths. Golomb chooses this off peak correlation function $f(t)$ as constant so phase information is not leaked.

Of course he also knew that algbraic structures, in particular cyclic difference sets, allowed one to mechanise this two level correlation.


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