diff options
author | Jeff Burdges <burdges@gnunet.org> | 2018-04-25 08:15:56 +0200 |
---|---|---|
committer | Jeff Burdges <burdges@gnunet.org> | 2018-04-25 08:15:56 +0200 |
commit | bad189ee87a5724a79622b0fe0e07ed3620fe068 (patch) | |
tree | bb3f73ef0e7ac0d0b7f650b1808df33fb2d964c8 /games | |
parent | cf4086163362d68011d093bc683732bacac5aba1 (diff) | |
download | papers-bad189ee87a5724a79622b0fe0e07ed3620fe068.tar.gz papers-bad189ee87a5724a79622b0fe0e07ed3620fe068.tar.bz2 papers-bad189ee87a5724a79622b0fe0e07ed3620fe068.zip |
Fix income transperency
Diffstat (limited to 'games')
-rw-r--r-- | games/games.tex | 47 |
1 files changed, 28 insertions, 19 deletions
diff --git a/games/games.tex b/games/games.tex index ebe551a..a609624 100644 --- a/games/games.tex +++ b/games/games.tex @@ -904,36 +904,45 @@ choice of $\gamma$, as otherwise the refresh always fails. As our $\gamma$ are chosen randomly, any given refresh with one false planchet has only a $1\over\kappa$ chance of succeeding. +% We recall ... MAYBE COPY FROM REMOVED TEXT IN cf4086163362d68011d093bc683732bacac5aba1 If such a malicious refresh is detected, the coin becomes unspendable, and contributes to the lost coins $b := w - s$ instead of the winnings $p := L - w'$ aka $|X|$. We observe $b + p$ gives the value of refreshes attempted with false planchets. Of course $p = 0$ and $b = 0$ if the adversary never -uses false planchets, so no $R_C$ occur, leaving $p \over b + p$ undefined. +uses false planchets, i.e. no $R_C$ occur, leaving $p \over b + p$ undefined. -We shall prove $E({p \over b + p}) = {1\over\kappa}$ (\dag) -with the expectation taken over games with false planchets. -We obtain a valid game by terminating a game early, so - we work by induction on the length of the game. - -% https://math.stackexchange.com/questions/852890/expectation-of-random-variables-ratio -If we group together the games identical up until some $R_C$ with value $v$, -then we find $R_C$ -... -contributes $v\over\kappa$ to $p$ and $\kappa v$ to $b + p$. - - - by grouping games identical up to an $R_C$ - according to the simulator's random choice of $\gamma$. -Of course We defined $R_C$ to be an anti-chain in the refresh forest, but emphasize that an adversary who wins an $R_C$ should not gamble again, -while an adversary who looses an $R_C$ looses $C$. -It follows that each $R_C$ contributes ${1\over\kappa}$ towards the total expectation. -{RETHINK WHAT HAPPENS HERE AND ABOVE} +while an adversary who looses an $R_C$ looses $C$ completely. +There is no way to influence $p$ or $b$ through withdrawals or spends +by corrupted users of course. In principle, one could decrease $b$ by + sharing from a corrupted user to a non-corrupted users, +but we may assume this does not occur either. + +We obtain a valid game run by terminating a game early, or even by +blocking oracle calls on decedents of a particular coin $C$, + which we refer to as a strictly simpler game run. +We shall prove $E({p \over b + p}) = {1\over\kappa}$ (\dag) +with the expectation taken over games with false planchets + in which the adversary plays optimally + in that no strictly simpler game run increases $p \over b + p$. + +For any $R_C$, +there are $\kappa$ game runs identical up through the commitment phase +of $R_C$ that diverge in simulator's random choice of $\gamma$, + and blocks $C$ afterwards. +If $v$ is the value of $R_C$ then one run adds $v$ to $p$, while + the remaining $\kappa-1$ runs add $v$ to $b$. + +An adversary may respond adaptively to $\gamma$ of course, but +induction on the length of the game now still produces disjoint groupings +$\mathbb{G}$ of optimal games in which + $\sum_{\mathbb{G}} {p \over b + p} = {1\over\kappa} |\mathbb{G}|$. We conclude that $E({p \over b + p}) = {1\over\kappa}$, as desired. \end{proof} +% https://math.stackexchange.com/questions/852890/expectation-of-random-variables-ratio %%% $L - w' \over (L - w') + (w - s)$ % $E(b/p) + 1 = E(b/p + p/p) = E((b + p)/p) = \kappa$ % $E(b/p) = \kappa-1$ |