Brutomolecule: hacking biological defences (Part III)

As per Part I the necessary and sufficient condition for $h(x;p)$ to exist under all non-trivial $p$ was that $A_{\alpha}:=\{i\in\omega\mid x(i)=\alpha\}$ be a set of asymptotic density for each $\alpha\in\Sigma$ . Under this condition, one has

$h(x;~p)=-\sum_{\alpha\in\Sigma}d(A_{\alpha})\log(p(\alpha)).$

Working with the inequality (basic result in Analysis) $\log(x)\leq x-1$ for all $x\in(0,\infty)$ , one has $-x\log(y/x)\geq -x(y/x-1)=x-y$ . From this one obtains the following identity

$\begin{array}{c}[mc]{rcl} -\sum_{\alpha\in\Sigma}d(A_{\alpha})\log(p(\alpha)) &= &-\sum_{\alpha\in\Sigma}d(A_{\alpha})\log(d(A_{\alpha})) +\sum_{\alpha\in\Sigma}-d(A_{\alpha})\log(p(\alpha)/d(A_{\alpha}))\\ &\geq &-\sum_{\alpha\in\Sigma}d(A_{\alpha})\log(d(A_{\alpha})) +\sum_{\alpha\in\Sigma}d(A_{\alpha})-p(\alpha)\\ &= &-\sum_{\alpha\in\Sigma}d(A_{\alpha})\log(d(A_{\alpha})) +\sum_{\alpha\in\Sigma}d(A_{\alpha})-\sum_{\alpha\in\Sigma}p(\alpha)\\ &= &-\sum_{\alpha\in\Sigma}d(A_{\alpha})\log(d(A_{\alpha}))+1-1\\ \end{array}$

Hence $h(x;~p)$ is bounded below by $h(x;~p_{x})$ , where $p_{x}(\alpha)=d(A_{\alpha})$ . Thus it is necessary and sufficient, in order to minimise $h(x;\cdot)$ , that the distribution of letters in $\Sigma$ coincides with the asymptotic distribution of letters in $x$ . Under this choice one has

$h_{\min}(x) = h(x;~p_{x}) = -\sum_{\alpha\in\Sigma}d(A_{\alpha})\log(d(A_{\alpha})) = \mathcal{H}(x)$

Thus the minimised exponential growth-exponent for waiting times coincides with the asymptotic entropy of the sequence $x$ . This completes Task 1.

Towards Task 2 we have that the letters in $x$ are asymptotically distributed with $d(A_{G})=\frac{1}{2}$ , $d(A_{T})=\frac{1}{4}$ and $d(A_{A})=d(A_{C})=\frac{1}{8}$ . The minimised value of $h_{\min}(x)$ is as above given by the asymptotic entropy

$h_{\min}(x)=\mathcal{H}(x)=-\frac{1}{2}\log\frac{1}{2}+-\frac{1}{4}\log\frac{1}{4}+-\frac{1}{8}\log\frac{1}{8}+-\frac{1}{8}\log\frac{1}{8}\approx 1.213008$

In terms of Part II , the sequence $x$ represents the signature of all life forms in a solar system. The Brutomolecule can only steer how it randomly transmits its sequence of molecules. According to this, if the Brutomolecule can work out the asymptotic distribution of the letters in the growing key $x\vert n$ , even without knowledge of the exact sequence of those letters, then it can adjust its strategy (choice of $p$ ) accordingly and minimise the exponential growth rate of its randomise brute force attack. The asymptotic entropy of the sequence $x$ thus represents a physical boundary on efficiency of these randomise brute force attacks. Accordingly. since entropy is maximised under uniformly distributed letters, the best protection against such randomised brute force attacks is for solar systems to choose keys with asymptotically uniformly distribute letters.

{G C A T G A} = 6.21 & (CATAGACA) == 29.22 29.22 (+) 6.21 = 79.31622 sqrt(79.31622) is not equal to 1.213 Thus the answer is 1.213