# Finding differentials and space complexity

Most of article about differential cryptanalysis present the generic method, the way to find sub keys. But I can't find a clear explanation about how to find the differentials used in examples. In Differential Cryptanalysis of DES-like Cryptosystems by Biham and Shamir, the provide examples using differentials inputs such as $\Omega_p = 00\ 80\ 82\ 00\ 60\ 00\ 00\ 00_x$. I do get the idea about feeding S-box in order to retrieve the differentials but these S-box are only 6 bit input 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 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 thing can't be stored on current sized computer.

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