TITLE: Is there any theory why (for Bitcoin) the discrete logarithm problem is so hard to solve?
QUESTION [31 upvotes]: Note I am an active member and contributor at the sister site https://bitcoin.stackexchange.com while studying Bitcoin and as a person who studied mathematics 10 years ago there is one thing I kept asking people and I am getting unsatisfying answers. 
Why is the discrete logarithm problem (in particular in the case of bitcoin) so hard?
In Bitcoin we take a an elliptic curve as the zeros of $E: y^2 = x^3 + 7$ over $F_p$ with:
 p = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F

which has a nice representation as 
$p = 2^{256} - 2^{32} - 2^9 - 2^8 - 2^7 - 2^6 - 2^4 - 1$
The we chose a generator point (with the standard encoding in hex):
g = 04 79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB 2DCE28D9 59F2815B 16F81798 483ADA77 26A3C465 5DA4FBFC 0E1108A8 FD17B448 A6855419 9C47D08F FB10D4B8

This is used to select a cyclic subgroup $C$ of cofactor $h=1$ and order $n$ with
n=115792089237316195423570985008687907852837564279074904382605163141518161494337

We already know there is a group homomorphism 
$f: F_n \rightarrow C$ with $x\mapsto g^x$ 
In particular by our construction we know that $f$ has to be bijective but apparently it is very hard to give an analytical construction of the inverse map.  
In Bitcoin / Computer science we chose $p$ and $g$ in such a way to make $n$ large enough that with pure brute force calculation we won't be able to solve the discrete logarithm. But why can't we construct an inverse map explicitly? 
Would anyone here please share some insights why the construction of an analytic closed form for this is so hard?

REPLY [5 votes]: As commented by the two answers, Pollard Rho is the best known algorithm for discrete logarithms in a generic cyclic group (where no other special structure is used, and no such special structure, e.g., amenability to index calculus). 
The so called baby step giant step algorithm can also be used with essentially the same time complexity $O(\sqrt{n})$ where $|G|=n,$ as Pollard Rho. Unfortunately the bsgs needs memory of the same order as well, while Pollard Rho requires negligible memory. 
So, if $p$ has size $b$ bits, the time complexity for both Pollard Rho and bsgs is $O(2^{b/2}),$ and thus still exponential in input size $b.$ The bsgs is based on a very neat idea, see below:
Input: $x=g^k,$ where $g$ is a generator of a multiplicative cyclic group of size $n$, say $\mathbb{Z}_p^\ast$ for simplicity. The goal is to recover $k,$ and $g$ is public as well as $p$ and the group operation. Let $m=\lceil \sqrt{n}~\rceil.$
Step 1. Precomputation: Form the list $$L=\{(j,g^{jm}):j=0,1,\ldots,m-1\}$$ and store it sorted on the second component (or you could use a hash table and a lookup to find an entry in step 2 below). Complexity: $O(\sqrt{n}\log n)$ time (with hash sorting time complexity would be $O(\sqrt{n})$ but generally additional memory is needed to control collisions in that case) and $O(\sqrt{n})$ memory.
Think of the elements of $G$ in a $\lceil m\rceil \times \lceil m \rceil$ array (with some repeats at the end):
$$
\begin{array}{cccccc}
1 & g & g^2 & g^3 &\cdots & g^{m-1} \\
g^m & g^{m+1} & g^{m+2} & g^{m+3} & \cdots & g^{2m-1} \\
\vdots & &  & & \vdots\\
g^{m(m-1)} & g^{m(m-1)+1} & \cdots & g^{n-1} & 1 & \cdots\\
\end{array}
$$
Step 2. Online Phase Note that the list $L$ has the entries in the first column sorted as integers.
Now, form the elements $x,xg,\ldots, xg^i,\ldots,$ sequentially and lookup in $L$ until the element is found in $L$ (clearly this is guaranteed, as long as we continue until $xg^{m-1}$, since this operation spans two consecutive rows of the array starting at $x$ and ending at $x g^m$ which is below $x$ and one position to the left).
When we find an element in $L,$ at the $i_0$th iteration of Step 2, we then know
$$
xg^{i_0-1}=g^{j_0m}
$$ 
where $i_0,j_0$ are known. Solving we get $x=g^{j_0 m-i_0+1}$ so $k=j_0 m-i_0+1\pmod n.$
Note that if you have a new $x,$ you can just repeat Step 2.
One final comment which may be of interest. About 6 years ago there was quite a lot of progress in the DLP algorithms for composite order fields with special exponents (I am quoting a cryptography stack exchange question here  below):

A recent paper by Göloğlu, Granger, McGuire, and Zumbrägel: Solving a 6120-bit DLP on a Desktop Computer seems to "demonstrate a practical DLP break in the finite field of $2^{6120}$ elements, using just a single core-month". They credit a 2012 paper by Antoine Joux: Faster index calculus for the medium prime case. Application to 1175-bit and 1425-bit finite field for paving the way they explore. In 2013 Joux published A new index calculus algorithm with complexity $L(1/4+o(1))$ in very small characteristic, and very recently announced he "is able to compute discrete logarithms in $GF(2^{6168})=GF({(2^{257})}^{24})$ using less than 550 CPU.hours". 

This puts some pairing-based cryptographic schemes relying on the hardness of DLP in fields of characteristic 2 at risk, but not prime field based schemes, be it classical integer residue field DLP or elliptic curve DLP.