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Normally, coin tosses are random variables $r$ drawn (usually uniformly unless otherwise specified) from a finite set $R.$ In this case however $R=g^{k^{-1}}$ where $r=H’(R)$ and $H’$ is a hash function from a cyclic group of prime order into $Z_q$ and $k$ is a uniformly distributed random variable from $Z_q.$ So $R$ is a random variable which is a function ...


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There are $p^k$ possible polynomials and $p^k$ possible tuples of shares, each of which uniquely determines a polynomial, so the distribution of share values is a permutation of the distribution of polynomials. If you know nothing about $s$, i.e. you model it as uniformly distributed like the $t_i$, then all polynomials are equiprobable and therefore so are ...


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