show/hide this revision's text 3 clarified further and made link better; deleted 2 characters in body; deleted 2 characters in body

$H$ is the binary (Shannon) entropyfunction.

In terms of the positive real number $t$, what distribution(s) $\hspace{.01 in}X$ on $\:[0\hspace{.005 in},\hspace{-0.03 in}\scriptsize+\normalsize\infty\hspace{-0.02 in})$
with mean $1$ will minimize the value of the following (one-player) game?


Player chooses a real number $s$ such that $\: 0 < s \leq t \:$ and a
finite partition $\:\left\langle B_0,...,B_n \right\rangle\:$ of $\:[0\hspace{.005 in},\hspace{-0.03 in}\scriptsize+\normalsize\infty\hspace{-0.02 in})^2$ into universally measurable sets
and a real number $s$ such that $\: 0 < s \leq t \;\;$..

Have $\;\;\; f \: : \: [0\hspace{.005 in},\hspace{-0.03 in}\scriptsize+\normalsize\infty\hspace{-0.02 in})^2 \: \to \: \{0\hspace{.005 in},...,\hspace{-0.01 in}n\} \;\;\;$ be given by $\;\; \langle x,\hspace{-0.03 in}y\rangle \: \in \: B_{f(\langle x,y\rangle)} \;\;\;\;$.;\;\;$.

Challanger samples $x$ and $y$ independently from $X$, samples $b$ uniformly from $\{0\hspace{.005 in},\hspace{-0.04 in}1\}$,$\:$ then gives Player $\langle w,\hspace{-0.01 in}z\hspace{.01 in}\rangle \;\; := \;\; \begin{cases} \langle x,y+s\rangle & \text{if } \: b=0 \\ \langle x+s,y\rangle & \text{if } \: b=1 \end{cases}$

Player's score is $\:-\big(H\big(b \: | \; f(\langle w,\hspace{-0.01 in}z\hspace{.01 in}\rangle\big)\big) \;\;$.

show/hide this revision's text 2 made definition more accurate

$H$ is the binary entropy function.

In terms of the positive real number $t$, what distribution(s) $\hspace{.01 in}X$ on $\:[0\hspace{.005 in},\hspace{-0.03 in}\scriptsize+\normalsize\infty\hspace{-0.02 in})$
with mean $1$ will minimize the value of the following (one-player) game?


Player chooses a universally measurable subset finite partition $B$ \:\left\langle B_0,...,B_n \right\rangle\:$ of $\:[0\hspace{.005 in},\hspace{-0.03 in}\scriptsize+\normalsize\infty\hspace{-0.02 in})^2$ into universally measurable sets
and a real number $s$ such that $\: 0 < s \leq t \;\;$.

Have $\;\;\; f \: : \: [0\hspace{.005 in},\hspace{-0.03 in}\scriptsize+\normalsize\infty\hspace{-0.02 in})^2 \: \to \: \{0\hspace{.005 in},\hspace{-0.04 in}1\in},...,\hspace{-0.01 in}n\} \;\;\;$ is defined to be the characteristic function of given by $B$.\;\; \langle x,\hspace{-0.03 in}y\rangle \: \in \: B_{f(\langle x,y\rangle)} \;\;\;\;$.

Challanger samples $x$ and $y$ independently from $X$, samples $b$ uniformly from $\{0\hspace{.005 in},\hspace{-0.04 in}1\}$,$\:$ then gives Player $\langle w,\hspace{-0.01 in}z\hspace{.01 in}\rangle \;\; := \;\; \begin{cases} \langle x,y+s\rangle & \text{if } \: b=0 \\ \langle x+s,y\rangle & \text{if } \: b=1 \end{cases}$

Player's score is $\:-\big(H\big(b \: | \; f(\langle w,\hspace{-0.01 in}z\hspace{.01 in}\rangle\big)\big) \;\;$.

show/hide this revision's text 1

What distribution(s) of delays make(s) timing attacks hardest?

$H$ is the binary entropy function.

In terms of the positive real number $t$, what distribution(s) $\hspace{.01 in}X$ on $\:[0\hspace{.005 in},\hspace{-0.03 in}\scriptsize+\normalsize\infty\hspace{-0.02 in})$
with mean $1$ will minimize the value of the following (one-player) game?


Player chooses a universally measurable subset $B$ of $\:[0\hspace{.005 in},\hspace{-0.03 in}\scriptsize+\normalsize\infty\hspace{-0.02 in})^2$
and a real number $s$ such that $\: 0 < s \leq t \;\;$.

$f \: : \: [0\hspace{.005 in},\hspace{-0.03 in}\scriptsize+\normalsize\infty\hspace{-0.02 in})^2 \: \to \: \{0\hspace{.005 in},\hspace{-0.04 in}1\} \;\;\;$ is defined to be the characteristic function of $B$.

Challanger samples $x$ and $y$ independently from $X$, samples $b$ uniformly from $\{0\hspace{.005 in},\hspace{-0.04 in}1\}$,$\:$ then gives Player $\langle w,\hspace{-0.01 in}z\hspace{.01 in}\rangle \;\; := \;\; \begin{cases} \langle x,y+s\rangle & \text{if } \: b=0 \\ \langle x+s,y\rangle & \text{if } \: b=1 \end{cases}$

Player's score is $\:-\big(H\big(b \: | \; f(\langle w,\hspace{-0.01 in}z\hspace{.01 in}\rangle\big)\big) \;\;$.