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@@ -68,6 +68,44 @@ the probability that a party holding all the stake will be selected to
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be a leader. Stakeholder is selected as leader for slot j with
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probability $\phi_f(\alpha_i)$, $\alpha_i$ is $U_i$ stake.
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+
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+### linear aggregation dependent leader selection
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+in the previous leader selection function, it has the unique property of
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+independent aggregation of the stakes, meaning the property of a leader winning leadership with stakes $\sigma$ is independent of whether the stakeholder would act as a pool of stakes, or distributed stakes on competing coins.
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+"one minus the probability" of winning leadership with aggregated stakes is
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+$1-\phi(\sum_{i}\sigma_i)=1-(1+(1-f)^{\sigma_i})=-(1-f)^{\sum_{i}\sigma_i}$,
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+the joint "one minus probability" of all the stakes (each with probability $\phi(\sigma_i))$
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+winning aggregated winning the leadership
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+$\prod_{i}^{n}(1-\phi(\sigma_i))=-(1-f)^{\sum_{\sigma_i}}$
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+thus:
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+$$ 1-\phi(\sum_{i}\sigma_i) =\prod_{i}^{n}(1-\phi(\sigma_i)) $$
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+
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+#### linear leader selection
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+$$y < T $$
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+$$y = 2^lk \mid 0 \le k \le 1$$
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+$$T = 2^l\phi(v)$$
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+$$ \phi(v)=\frac{1}{v_{max}}v $$
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+
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+#### dependent aggregation
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+linear leader selection has the dependent aggregation property, meaning it's favorable to compete in pools with sum of the stakes over aggregated stakes of distributed stakes:
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+
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+$$\phi(\sum_{i}{\sigma_i})>\prod_{i}^{n}{\sigma_i}$$
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+$$\sum_{i}{\sigma_i}>(\frac{1}{v_{max}})^{n-1}v_1v_2 \dots v_n$$
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+let's assume the stakes are divided to stakes of value $\sigma_i=1$ for $\Sigma>1 \in \mathbb{Z}$, $\sum_{i}{\sigma_i}=V$
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+$$V>(\frac{1}{v_{max}})^{n-1}$$
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+note that $(\frac{1}{v_{max}})^{n-1} < 1, V>1$, thus competing with single coin of the sum of stakes held by the stakeholder is favourable.
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+
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+### scalar linear aggregation dependent leader selection
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+ a target function T with scalar coefficients can be formalized as
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+ $$T=2^lk\phi(\Sigma)=2^l(\frac{1}{v_{max}})\Sigma$$
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+ let's assume $v_{max}=2^v$, then:
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+ $$T=2^lk\phi(\Sigma)=2^{\frac{l}{v}}\Sigma$$
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+ then the lead statement is
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+ $$y<2^{\frac{l}{v}}\Sigma$$ for example for a group order or l=24 bits, and maximum value of $v_{max}=2^{10}$, then lead statement:
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+ $$y<2^{14}\Sigma$$
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+
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+
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+
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## Leaky non-resettable beacon
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Built on top of globally synchronized clock, that leaks the nonce
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mohab