# What are the expected values of a particular rotational-XOR property of a sequence of random bitstrings?

Assuming that $$x$$ is a sequence of $$l$$ bits and $$0 \le n < l$$, let $$R(x, n)$$ denote the result of the left bitwise rotation of $$x$$ by $$n$$ bits. For example, if $$x = 0100110001110000$$, then $$\begin{array}{l} R(x,0) = {\rm{0100110001110000}},\\ R(x,1) = {\rm{1001100011100000}},\\ R(x,2) = {\rm{0011000111000001}},\\ \ldots \\ R(x,15) = {\rm{0010011000111000}}. \end{array}$$

Let $$A \oplus B$$ denote the result of the XOR operation for two sequences of $$l$$ bits. For example, $$0100110001110000 \oplus 1010010001000010 = 1110100000110010.$$

Let $$H(x)$$ denote the number of non-zero bits in $$x$$ (i.e. the Hamming weight of $$x$$).

Assuming that $$x$$ and $$y$$ are two bitstrings of the same length $$l$$, let $$f(x, y)$$ denote the minimal element (the smallest number) in the tuple $$\begin{array}{l} (H(x \oplus y),\\ H(x \oplus R(y,1)),\\ H(x \oplus R(y,2)),\\ \ldots \\ H(x \oplus R(y,l - 1))). \end{array}$$

Suppose that we have a TRNG which generates sequences of random bits. Generate a sequence of $$L = k \times l$$ bits. Split this sequence into $$k$$ words (so the length of each word is $$l$$): $$w_0, w_1, \ldots, w_{k-1}$$. Then compute the following tuple $$T$$ of numbers:

$$\begin{array}{l} (f({w_0},{w_1}),\\ f({w_0},{w_2}),\\ \ldots \\ f({w_0},{w_{k - 1}}),\\ f({w_1},{w_2}),\\ f({w_1},{w_3}),\\ \ldots \\ f({w_1},{w_{k - 1}}),\\ f({w_2},{w_3}),\\ \ldots \\ f({w_{k - 2}},{w_{k - 1}})). \end{array}$$

In other words, for any pair of words $$(w_i, w_j)$$ such that $$i \neq j$$, compute the corresponding $$f({w_i},{w_j})$$.

Question 1: given $$k$$ and $$l$$, how to compute the expected value of the minimal number $$M_T$$ in $$T$$?

Question 2: given $$k$$ and $$l$$, how to compute the expected value of the average number $$A_T$$ in $$T$$? Here the number $$A_T$$ is computed as follows: sum all elements of $$T$$, then divide the sum by the total number of elements in $$T$$.

The expected number here implies the number with the maximum probability. For example, the expected number of zero bits in a sequence of $$l$$ random bits is $$l/2$$.

• "The expected number here implies the number with the maximum probability. For example, the expected number of zero bits in a sequence of 𝑙 random bits is 𝑙/2." This statement need not be true. Nov 12, 2021 at 2:51
• The question as asked is very difficult. I will type up an answer to a related question when I have more time. Nov 12, 2021 at 2:52
• @kodlu: I am interested in an explanation of the first comment. If the probability of either 0 or 1 is 50%, what is the expected number of zero bits in a sequence of $l$ bits, assuming that $l$ is even? Nov 13, 2021 at 13:08
• I only meant that for more general distributions that can arise in the analysis, that property need not hold. you are taking weights and doing minimizations of Hamming weights. Nov 13, 2021 at 20:08

Edit: TL;DR

Editing this since a closed form answer is not tractable but a computation is.

All those pairwise hamming distances are samples from a binomial distribution, specifically $$\textrm{Bin}(\ell,1/2).$$ Then we need to consider the expectation, as well as the minimum of a set of binomial random variables, under the assumption that a TRNG is used.

Question 1: See page 68-69 in reference A, where the mean of the minimum $$M_T$$ is computed. Their $$N$$ is your $$\ell$$ and their $$n$$ is your total number of samples $$k\binom{k}{2}.$$ You can use the formula for $$M_T=\mu_{1:n}=\sum_{x=0}^{N-1}(1-F(x))^n,$$ where $$F(x)$$ is the cumulative binomial distribution given in (4.4.1).

A. Barry C. Arnold, N. Balakrishnan, H. N. Nagaraja, A First Course in Order Statistics, Classics in Applied Mathematics, SIAM, 2008.

Question 2:

By the law of large numbers, and assumption that the sequences are generated by a TRNG, the average $$A_T$$ will approach $$\ell/2$$ with very high probability very rapidly.

Since you say the input strings are outputs of a TRNG, I shall take each of the terms as uniformly distributed random vectors.

You define
$$f(x, y)=\min\{H(x \oplus y),H(x \oplus R(y,1)),\ldots,H(x \oplus R(y,\ell-1)\}$$ which I shall model as the minimum Hamming weight of a set of $$k$$ randomly uniformly independently chosen binary $$\ell-$$tuples.

Each Hamming weight in this set is distributed as $$\textsf{Binomial}(\ell,1/2).$$ So we have the minimum of $$k$$ unbiased binomial samples on $$\{0,1,\ldots,\ell\}$$. The minimum is also called the first Order Statistic and letting $$F(u)=\mathbb{Pr}[X\leq u]$$ where $$X$$ is binomially distributed as above (you are already using $$x$$ as a variable hence the $$u$$), we have $$\mathbb{Pr}[f(x,y)\leq x]=1-(1-F(u))^{k}.$$

If the randomness hypothesis holds, then your next step looks at all $$\binom{k}{2}$$ pairs of these minima, so in effect you are looking at the minimum of a collection of $$\binom{k}{2}k$$ binomial samples. This is $$O(k^3)$$ samples and the minimum will go to zero quite fast, if the randomness hypothesis above holds, so you'd want to consider $$1-\left[1-(1-F(u))^k\right]^{\binom{k}{2}}$$

Based on this my tentative answers are:

Question 1: given $$k$$ and $$l$$, how to compute the expected value of the minimal number $$M_T$$ in $$T$$?

This will be close to zero for any sizable $$k$$ compared to $$\ell.$$ You might want to use the Gaussian approximation to the binomial and use the Gaussian cumulative distribution function to evaluate an estimate.

Question 2: given $$k$$ and $$l$$, how to compute the expected value of the average number $$A_T$$ in $$T$$? Here the number $$A_T$$ is computed as follows: sum all elements of $$T$$, then divide the sum by the total number of elements in $$T$$.

This is now the average of the individual order statistics instead of the minimum of the minima. It will be very likely comparable to $$1-(1-F(u))^k.$$

Remark: If you want to directly use approximations to the binomial you can see my answer to the following mathoverflow question. Note that for the binomial $$\textsf{Binomial}(\ell,1/2),$$ and for any $$u \in \{0,1,\ldots,\ell\}$$ we have $$2^{-\ell}\sum_{j\leq u-1} \binom{\ell}{j}\leq F(u)\leq 2^{-\ell}\sum_{j\leq u} \binom{\ell}{j}$$

• Is it possible to have an explicit formula to find the expected value of $M_T$, assuming that the formula must only contain $k$ and $l$ as variables? Nov 13, 2021 at 12:56
• @lyricallywicked, I doubt a simple expression exists, given the time required to search for one, I really have no motivation to proceed further without a single upvote Sep 4 at 18:21