# Is my analysis regarding $M_3$ and $M_4$ correct?

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 $$m_{n_1 +1} \cdots m_{ n_1 +n_2} ,$$

$$\cdots$$

$$\cdots$$

$$u_r$$ 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_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