On the group of multiplication modulo large prime $p$ of $m$ bits
Given $$ b_1^k=g_1 \;\text {mod}\;p$$ $$ b_2^k=g_2 \;\text{mod}\;p$$ $$ b_3^k=g_3 \;\text{mod}\;p$$ $$ \vdots $$ $$ b_n^k=g_n \;\text{mod}\;p$$
How many pairs are enough to computationally solve $k$ of this discrete logarithm problem? I have this question because if we erroneously use the random number generator in Elgamal encryption The masking key may have relation, e.g. the masking key is the square of previous one modulo $p$. We can decrypt the ciphertext easily, but we may still not know the private key of Bob (the receiver.), but we can know many pairs of relation like equations above. Are the pairs let we solve the private key of Bob(receiver)?