Time/memory/data tradeoff attack
A time/memory/data tradeoff attack is a type of cryptographic attack where an attacker tries to achieve a situation similar to the space–time tradeoff but with the additional parameter of data, representing the amount of data available to the attacker. An attacker balances or reduces one or two of those parameters in favor of the other one or two. This type of attack is very difficult, so most of the ciphers and encryption schemes in use were not designed to resist it.

History 

Tradeoff attacks on symmetric cryptosystems date back to 1980, when Martin Hellman suggested a time/memory tradeoff method to break block ciphers with  possible keys in time  and memory  related by the tradeoff curve  where . Later, in 1995, Babbage and Golic devised a different tradeoff attack for stream ciphers with a new bound such that  for  where  is the output data available to the cryptanalyst at real time.

Attack mechanics 

This attack is a special version of the general cryptanalytic time/memory tradeoff attack, which has two main phases: 
 Preprocessing:  During this phase, the attacker explores the structure of the cryptosystem and is allowed to record their findings in large tables. This can take a long time.
 Realtime:  In this phase, the cryptanalyst is granted real data obtained from a specific unknown key. They then try to use this data with the precomputed table from the preprocessing phase to find the particular key in as little time as possible.

Any time/memory/data tradeoff attack has the following parameters: 
 search space size
 time required for the preprocessing phase
 time required for the realtime phase
 amount of memory available to the attacker
 amount of realtime data available to the attacker

Hellman's attack on block ciphers 

For block ciphers, let  be the total number of possible keys and also assume the number of possible plaintexts and ciphertexts to be . Also let the given data be a single ciphertext block of a specific plaintext counterpart. If we consider the mapping from the key  to the ciphertext  as a random permutation function  over an  point space, and if this function  is invertible; we need to find the inverse of this function . 
Hellman's technique to invert this function:
During the preprocessing stage
Try to cover the  point space by an  rectangular matrix that is constructed by iterating the function  on  random starting points in  for  times. The start points are the leftmost column in the matrix and the end points are the rightmost column. Then store the pairs of start and end points in increasing order of end points values.

Now, only one matrix will not be able to cover the whole  space. But if we add more rows to the matrix, we will end up with a huge matrix that includes recovered points more than once. So, we find the critical value of  at which the matrix contains exactly  different points. Consider the first  paths from start points to end points are all disjoint with  points, such that the next path which has at least one common point with one of those previous paths and includes exactly  points. Those two sets of  and  points are disjoint by the birthday paradox if we make sure that . We achieve this by enforcing the matrix stopping rule: .
Nevertheless, an  matrix with  covers a portion  of the whole space. To generate  to cover the whole space, we use a variant of  defined:  and  is simple out manipulation such as reordering of bits of   (refer to the original paper for more details). And one can see that the total preprocessing time is . Also  since we only need to store the pairs of start and end points and we have  matrices each of  pairs.

During the real time phase
The total computation required to find  is  because we need to do  inversion attempts as it is likely to be covered by one matrix and each of the attempts takes  evaluations of some . The optimum tradeoff curve is obtained by using the matrix stopping rule  and we get  and choice of  and  depends on the cost of each resource.

According to Hellman, if the block cipher at hand has the property that the mapping from its key to cipher text is a random permutation function  over an  point space, and if this  is invertible, the tradeoff relationship becomes much better: .

Babbage-and-Golic attack on stream ciphers 

For stream ciphers,  is specified by the number of internal states of the bit generatorprobably different from the number of keys.  is the count of the first pseudorandom bits produced from the generator. Finally, the attacker's goal is to find one of the actual internal states of the bit generator to be able to run the generator from this point on to generate the rest of the key. Associate each of the possible  internal states of the bit generator with the corresponding string that consists of the first  bits obtained by running the generator from that state by the mapping  from states  to output prefixes . This previous mapping is considered a random function over the  points common space. To invert this function, an attacker establishes the following. 
During the preprocessing phase, pick  random  states and compute their corresponding  output prefixes. 
Store the pairs  in increasing order of  in a large table.
During the realtime phase, you have  generated bits. Calculate from them all  possible combinations of  of consecutive bits with length .
Search for each  in the generated table which takes log time. 
If you have a hit, this  corresponds to an internal state  of the bit generator from which you can forward run the generator to obtain the rest of the key.
By the Birthday Paradox, you are guaranteed that two subsets of a space with  points have an intersection if the product of their sizes is greater than .
This result from the Birthday attack gives the condition  with attack time  and preprocessing time  which is just a particular point on the tradeoff curve . We can generalize this relation if we ignore some of the available data at real time and we are able to reduce  from  to  and the general tradeoff curve eventually becomes  with  and .

Shamir and Biryukov's attack on stream ciphers 

This novel idea introduced in 2000 combines the Hellman and Babbage-and-Golic tradeoff attacks to achieve a new tradeoff curve with better bounds for stream cipher cryptoanalysis. Hellman's block cipher technique can be applied to a stream cipher by using the same idea of covering the  points space through matrices obtained from multiple variants  of the function  which is the mapping of internal states to output prefixes. Recall that this tradeoff attack on stream cipher is successful if any of the given  output prefixes is found in any of the matrices covering . This cuts the number of covered points by the matrices from  to  points. This is done by reducing the number of matrices from  to  while keeping  as large as possible (but this requires  to have at least one table). 
For this new attack, we have  because we reduced the number of matrices to  and the same for the preprocessing time . The realtime required for the attack is  which is the product of the number of matrices, length of each iteration and number of available data points at attack time.

Eventually, we again use the matrix stopping rule to obtain the tradeoff curve  for  (because ).

Attacks on stream ciphers with low sampling resistance 

This attack, invented by Biryukov, Shamir, and Wagner, relies on a specific feature of some stream ciphers: that the bit generator undergoes only few changes in its internal state before producing the next output bit.
Therefore, we can enumerate those special states that generate  zero bits for small values of  at low cost. But when forcing large number of output bits to take specific values, this enumeration process become very expensive and difficult.
Now, we can define the sampling resistance of a stream cipher to be  with  the maximum value which makes such enumeration feasible.

Let the stream cipher be of  states each has a full name of  bits and a corresponding output name which is the first  bits in the output sequence of bits. If this stream cipher has sampling resistance , then an efficient enumeration can use a short name of  bits to define the special states of the generator. Each special state with  short name has a corresponding short output name of  bits which is the output sequence of the special state after removing the first  leading bits. Now, we are able to define a new mapping over a reduced space of  points and this mapping is equivalent to the original mapping. If we let , the realtime data available to the attacker is guaranteed to have at least one output of those special states. Otherwise, we relax the definition of special states to include more points. If we substitute for  by  and  by  in the new time/memory/data tradeoff attack by Shamir and Biryukov, we obtain the same tradeoff curve  but with . This is actually an improvement since we could relax the lower bound on  since  can be small up to  which means that our attack can be made faster. This technique reduces the number of expensive disk access operations from  to  since we will be accessing only the special  points, and makes the attack faster because of the reduced number of expensive disk operations.

References 

Cryptographic attacks