Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let $G$ be a finite group of order $g$. A probability distribution $Z$ on $G\ $is called $\varepsilon$-uniform if $\left\vert Z(x)-1/g\right\vert \leq\varepsilon/g$ for each $x\in G$. If $x_{1},x_{2},\dots,x_{m}$ is a list of elements of $G$, then the  random cube  $Z_{m}:=Cube(x_{1},\dots,x_{m}) $ is the probability distribution where $Z_{m}(y)$ is proportional to the number of ways in which $y$ can be written as a product $x_{1}^{\varepsilon_{1}}x_{2}^{\varepsilon_{2}}\cdots x_{m}^{\varepsilon_{m}}$ with each $\varepsilon _{i}=0$ or $1$. Let $x_{1},\dots,x_{d} $ be a list of generators for $G$ and consider a sequence of cubes $W_{k}:=Cube(x_{k}^{-1},\dots,x_{1}^{-1},x_{1},\dots,x_{k})$ where, for $k>d$, $x_{k}$ is chosen at random from $W_{k-1}$. Then  prove that for each $\delta>0$ there is a constant $K_{\delta}>0$ independent of $G$ such that, with probability at least $1-\delta$, the distribution $W_{m}$ is $1/4$-uniform when $m\geq d+K_{\delta }\lg\left\vert G\right\vert $. This justifies a proposed algorithm of Gene Cooperman for constructing random generators for groups.  also consider modifications of this algorithm which may be more suitable in practice.
['John D. Dixon']
I am writing a research paper on a idea Generating Random Elements in Finite Groups
[{"from": "human", "value": "I am writing a research paper on a idea Generating Random Elements in Finite Groups"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Let $G$ be a finite group of order $g$. A probability distribution $Z$ on $G\\ $is called $\\varepsilon$-uniform if $\\left\\vert Z(x)-1/g\\right\\vert \\leq\\varepsilon/g$ for each $x\\in G$. If $x_{1},x_{2},\\dots,x_{m}$ is a list of elements of $G$, then the  random cube  $Z_{m}:=Cube(x_{1},\\dots,x_{m}) $ is the probability distribution where $Z_{m}(y)$ is proportional to the number of ways in which $y$ can be written as a product $x_{1}^{\\varepsilon_{1}}x_{2}^{\\varepsilon_{2}}\\cdots x_{m}^{\\varepsilon_{m}}$ with each $\\varepsilon _{i}=0$ or $1$. Let $x_{1},\\dots,x_{d} $ be a list of generators for $G$ and consider a sequence of cubes $W_{k}:=Cube(x_{k}^{-1},\\dots,x_{1}^{-1},x_{1},\\dots,x_{k})$ where, for $k>d$, $x_{k}$ is chosen at random from $W_{k-1}$. Then  prove that for each $\\delta>0$ there is a constant $K_{\\delta}>0$ independent of $G$ such that, with probability at least $1-\\delta$, the distribution $W_{m}$ is $1/4$-uniform when $m\\geq d+K_{\\delta }\\lg\\left\\vert G\\right\\vert $. This justifies a proposed algorithm of Gene Cooperman for constructing random generators for groups.  also consider modifications of this algorithm which may be more suitable in practice."}]