DES S-Box properties Two of the properties that lead to the design of DES S-Boxes are:


For any non-zero 6-bit difference between inputs, no more than 8 of the 32 pairs of 
inputs exhibiting that difference may result in the same output difference.
This is a criterion similar to the previous one, but for the case of three S-boxes.


The book i am reading (Understanding Cryptography from Christof Paar and Jan Pelzl) translates the second one to: A collision (zero output difference) at the 32-bit output of the eight S-boxes is only possible for three adjacent S-boxes.

Can anyone explain me the first property? I have searched for that property on google but the statement is always the same and i can't understand it. At first i thought that the non-zero 6-bi difference was between two different inputs to an s-box. But then it talks about 8 pairs of the 32 pairs (the only 32-bit input is the one before the expansion) that exhibit the 6-bit difference. Am i the only one finding this explanation confuse as hell?

As for the second criterion and assuming the book explanation i assume that you can't have something like (O = output of s box):

O(s1) = O(s2) = O(s3) = O(s4) = O(s5) = O(s6) = O(s7) = O(s8) 

but you can have:

O(s1) = O(s2) = O(s3) &amp;&amp; O(s4) = O(s5) = O(s6) or some combination of three adjacent s-boxes.

I hope you understand my questions and you can help because i would like to clarify this.
First property says- 
If you consider a 6-bit Inpput Difference(Difference means XOR in case of DES).i.e  6-bit binary number which is the xor of two inputs(obviously both 6 bits).When you enter both of these inputs to Sbox and get two outputs each of 4 bits and xor the outputs what you get is called Output Difference.

Total 64 inputs are possible hence 32 pairs having a same difference.

So this property says for any  non-zero 6-bit input difference there can be maximum 8 possible different 4 bit output difference and not more than that.

Hope i am clear