# Finding differentials and space complexity

In Differential Cryptanalysis of DES-like Cryptosystems by Biham and Shamir, they provide examples using differentials inputs such as $$\Omega_p = 00\ 80\ 82\ 00\ 60\ 00\ 00\ 00_x$$. Most of article is about differential cryptanalysis and presents the generic method, the way to find subkeys. But I can't find a clear explanation about how to find the differentials used in examples.

I do get the idea about feeding an S-box in order to retrieve the differentials, but these S-boxes only have 6-bit inputs for 4-bit outputs (in DES). So the memory space required in order to do such analysis is $$2^6 \times 2^4 = 1024$$ integers $$\approx 4\ Ko$$. This can be easily placed in the RAM. The method is presented in Modern Cryptanalysis (2008, Wiley Publishing; Chapter 4, pp. 122-145) by Christopher Swenson.

However, given an S-box with a bigger input such as the one used in FEAL-X (while remaining broken), we need a table of the following given size : $$2^{32} \times 2^{32} = 2^{64}$$ integers $$\approx 2^{32}\times 16\ Go$$. Such a thing is too big to store on one of today's computers.

Hence my question: how is it possible to find the right differential for bigger input round functions without the size constraint of this analysis.