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In this case, the base does not matter as the $\log$ terms are wrapped in an $O$ expression. The $O$ expression lets you throw away constant factors and to convert something base $X$ to base $Y$ is simply $\log_X Z = \frac{\log_Y Z}{\log_Y X}$, well, $\log_Y X$ is a constant, so it can be thrown out in the $O$ expression. The reason the second reference ...


Yes, this logic works. You have shown that given the discrete log $x$ lets you easily find collisions. Now you need to show that any collision lets you extract the discrete log $x$ and then you are done (then you have shown the equivalence).


I will assume for simplicity that you're talking about the full multiplicative group of $F_p$ instead of a proper subgroup, thus there are no problems with $g^a+1$ (except when $g^a=p-1$ which can be trivially ruled out by comparing to $p$). The quantity $\log_g(g^a+1)$ is sometimes referred to as the Zech logarithm (strictly speaking, it is defined for ...


There is 3 kind of discrete log problem as you explained : Diffie-Hellman problem (Dlog): Pick $a \in \{1,\ldots,q\}$. Compute $A = g^a (mod\ p)$ Given $(p,q,g,A)$ find $a$. Assumed hard. Computational Diffie-Hellman problem (CDH) : Pick $a,b \in \{1,\ldots,q\}$. Compute $A = g^a (mod\ p)$ and $B = g^b (mod\ p)$ Given $(p,q,g,A,B)$ find $g^{ab}$. Note ...

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