In this paper authors gave the following equations for $M_3$ and $M_4$ in section $3.2$ of Share distribution phase:

Shares Distribution

To distribute $n$ shares of a secret $K$ among the set of participants $P = \{ p_i : 1 \le i \le n\}$, the dealer $D$ does the following:

A set of integers $\{p, m_1 < \cdots < m _{n_1} , m_{n_1 +1} < \cdots < m_{ n_1 +n_2} ,\cdots, m_{n-n_r +1} < \cdots < m_n\}$,

where $0 \le K < p$, is chosen subject to the following

$\gcd(m_i , m_j)=1$ where for $i\neq j$

$\gcd(p , m_i)=1$ ,for all $i$

$$M_3 = min \Bigg{(}\prod \limits_{j=1}^{r} \prod \limits_{i=1}^{u_j} m_i \text{ for all } (u_1, u_2, \cdots ,u_r) \in \Omega(\tau_0) \Bigg{)}$$

$$M_4= min \Bigg{(} \prod_{j=1}^{r}\prod \limits_{i=1}^{v_j} m_{n_{j}+i-1}, \text{ for all } (v_1, v_2, \cdots ,v_r) \in\Omega(\Delta_1) \Bigg{)}$$

I am thinking that the given equation for $M_3$ is incorrect because vector $(u_1, u_2....u_r)$ uses some among all $m_i$'s instead of all

$u_1$ uses product of $m_1 \cdots m _{n_1}$ ,

$u_2$ uses product of $m_{n_1 +1} \cdots m_{ n_1 +n_2} ,$



$u_r$ uses product of $m_{n-n_r +1} \cdots m_n$

$u_1,u_2, \cdots , u_r$ use product of $m_1 \cdots m _{n_1}$ only, not product of $m_{n_1 +1} \cdots m_{ n_1 +n_2} \cdots m_{n-n_r +1} \cdots m_n$

My modified $M_3$ and $M_4$ are

$$ M_3=min \Bigg{(} \prod \limits_{j=1}^{r}\prod \limits_{i=1}^{u_j} m_{s_{j-1}+i}, \text{ for all } (u_1, u_2 ....u_r) \in\Omega(\tau_0) \Bigg{)} $$

where $0=s_0<s_1<s_2<s_3<s_4 \cdots <s_r=n, $ $s_i= \sum\limits_{j=1}^{i}n_j$ $$ M_4= max \Bigg{(}\prod \limits_{j=1}^{r}\prod \limits_{i=1}^{v_j} m_{s_{j-1}+i-1}, \text{ for all } (v_1, v_2 \cdots v_r) \in \Omega(\Delta_1) \Bigg{)}$$

$$M_3 > p M_4$$

Could you please check whether analysis on $M_3$ and $M_4$ given by the original paper is correct/ not


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