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21

There is a good article from Coppersmith which explains it. Basically, the designers of DES had envisioned differential cryptanalysis (a good 15 years before differential cryptanalysis was rediscovered by Biham and Shamir, and published); they could measure how well DES could resist differential cryptanalysis for a given set of S-boxes. So they generated a ...


16

They are both linear, but in different algebraic Groups. Which is to say, xor is linear in any finite field of characteristic 2, while 'ordinary' addition is linear in the infinite field of the Real numbers. Addition modulo $n$ (which is more cryptologically significant than addition over the Reals) is also a linear operation, but in the ring of integers ...


16

Before it was the standard, the NSA proposed some changes to the S-boxes and didn't explain them. The explanation (which turned out to be correct when differential cryptanalysis was "rediscovered" by the non-spy community) was that if you changed a single bit of the input, every bit of output should have a 50% chance of changing (this is called the "strict ...


12

Let a "block cipher" be defined with a fixed S-box $S$ (i.e. a permutation of some space) and a key $K$ (same size than a block), such that the encryption of a block $M$ is $C = S[P\oplus K]$. Everybody knows $S$ and can apply and invert it (that's a "S-box", not a "key" -- if the S-box is "key dependent" then the S-box is itself a block cipher in its own ...


10

If a block cipher is linear with respect to some field, then, given a few known plaintext-ciphertext pairs, it is possible to recover the key using a simple Gaussian elimination. This clearly contradicts the security properties one expects from a secure block cipher.


8

The security of a block cipher is, based on what we know, invariant to the permutation cycles of the S-box. This is because the values are always transformed (by a fixed function or a keyed function) before going through the S-box again in the next round. Furthermore, many S-boxes are functions not permutations (i.e., output size is different from input ...


8

The following information about the DES S-Box might be useful (taken from here): DES Design Criteria there were 12 criterion used, resulting in about 1000 possible S-Boxes, of which the implementers chose 8 these criteria are CLASSIFIED SECRET however, some of them have become known The following are design criterion: R1: Each ...


8

What is the definition of linearity? Linearity is defined for maps between vector spaces. If you have a field $F$ and two vector spaces $U$ and $V$ over the field $F$, a map $$T:U\rightarrow V$$ is said to be linear if $$T(\gamma_1\odot u_1\oplus\gamma_2\odot u_2)=\gamma_1 \odot T(u_1)\oplus\gamma_2\odot T(u_2)$$ whenever $\gamma_1,\gamma_2\in F$ and ...


6

You should think of Rijndael's S-box as a function that maps bytes to bytes, where a byte (octet) is considered to be a member of a finite field of size $2^8$ (with xor as addition). It's not seen as a 16x16 octet array, really. The substitution is then just done byte-wise: every octet in the 4x4 block is replaced by its function value under the S-box ...


6

The affine transformation works similar to MixColumns, but operates on an array of 8 bits instead of 4 bytes. Confusion in various descriptions of the affine transform in AES comes from where the LSB of the input byte is located. Some show it at the top of the column, others show it at the bottom. I will be using the version shown in the Rijndael paper, with ...


6

Here's the cryptography theory perspective. We want block ciphers to resemble pseudo-random permutations (PRPs). PRPs are a desirable modeling goal because a block cipher under a given key is a permutation on the input, and a PRP is simply a random collection of permutations. The block cipher's key can never be better at creating permutations than an actual ...


6

I'm going to ignore the "CBC" part of the question and focus on "What are the strengths and weaknesses of a s-box and xor cipher. I'm going to assume that the s-box size is smaller than the message block size since any cipher that has a block size that is equal to it's s-box size is going to have a block size small enough to be brute forced. Using xor and ...


5

There are 256! possible 8x8 S-boxes (i.e., bijective functions from $\{0,1\}^8$ to $\{0,1\}^8$. This is an absolutely enormous number. You couldn't possibly enumerate all of them within the lifetime of the universe. So, yes, this is one reason why it is not straightforward to determine whether there exists such a S-box with differential uniformity 2. ...


5

This is only a partial answer to the question, but still: The S-boxes where chosen to maximize confusion and to create an avalanche of change. For example, there were specific properties chosen to make the S-boxes resistant against differential cryptanalysis, by making sure that small differences between different inputs lead to larger differences in the ...


5

The S-Box was generated when Rijndael was designed, not in any step. It's used in every round in the SubBytes step. The S-box is constant. You could see it as a function taking a byte and returning a byte. It is used to reduce algebraic properties of Rijndael. In fact, this is it: | 0 1 2 3 4 5 6 7 8 9 a b c d e f ...


5

What are the disadvantages of using random s-boxes? This relates to the "why" behind some of the rules for s-boxes. AES, for example, requires an invertible s-box. A random s-box will not necessarily be invertible. In an s-box we also want non-linearity to thwart linear and differential cryptanalysis. This might not be the case with a random s-box. ...


5

Is it valid to call this table an S-Box? Not really; at least, not with the meaning we usually give to "S-box". The "S" in "S-box" stands for Substitution; we take the data, and replace it with a value from the S-box (using the data as an index into the S-box). The classical (if not the original) example is the S-boxes within AES; at certain points ...


4

I understand the question as you have a single 4-bit S-box, which you first apply rowwise, and then columnwise. As already mentioned, this is equivalent to a large S-box $\mathcal{S}$ $$ c = \mathcal{S}(m\oplus k_1)\oplus k_2. $$ This is a well-known Even-Mansour cipher, and it can be broken with complexity $2^{n/2}$, which is $2^8$ for your $n=16$. The ...


4

Your code is an attempt to implement the function $f$ which is a polynomial representation of the $\text{GF}(2)$ affine part of the S-box of the AES (usually referred to as $A$). Function $f$ is described on page 7 of the paper and your coefficients seems to be OK. Your code is mapping $\text a$ to $\text q$ such that $\text q=f(\text a)$. You're however ...


3

As I already said in my comment, what you have here is not what one usually calls a block cipher. A (standard) block cipher is a pair of functions $$E : K \times M \to M \text{ and } D : K \times M \to M$$ This means $E$ takes an element of $K$ and an element of $M$ as input, and gives an element of $M$ as output – same for $D$. (where $K$ is called the key ...


3

The answer is: it depends. It depends on how you plan to use your S-box. Presumably you are going to use your S-box in some block cipher. In that case, you have to look at what properties you need from the S-box, and then generate the S-box accordingly. You can't separate the design of the S-box from the design of the rest of the cipher. There is no ...


3

The answer is: it depends upon the cipher. You cannot design a S-box in isolation; the S-box properties need to be crafted together with the design of the block cipher, to make sure they work well together. See also Desirable S-box properties. Don't forget to use "search" before asking questions. The 3-Way cipher also uses 3x3 S-boxes.


3

If you are looking specifically for $3 \times 3$ S-boxes, you might want to take a look at the printCipher specification since it uses one. You can trust the people who chose it to have picked the best one. If you still want to go for the generation, keep in mind that many S-boxes have identical cryptographic properties (differential uniformity/spectrum, ...


3

Ciphers that use S-boxes are typically in the form of a Feistel network. This has the property that inverting the cipher does not involve inverting the round function, but simply applying the rounds in the opposite order. Therefore one doesn't need to invert the S-box to decrypt.


3

Item 2 has been answered satisfactorily, so this will focus on point 1: the s-box. The size of the s-box is not a 16x16 array unless it is viewed as such. The s-box is actually an 8-bit non linear transformation of the input, and is only viewed as a 16x16 array if you arrange it as a table of such dimensions. This array would then be a 1 to 1 representation ...


3

standard AES disclaimer: Given the questions you've asked, you should not implement AES yourself in a real-world system because there are lots of security considerations when implementing ciphers. Think of the S-box as a function from byte $ \to $ byte. So, to look up the image of $x$ under the s-box transformation, you simply use $S_\text{box}(x)$, which ...


3

To answer your questions in order: You won't find test vectors for the s-boxes in the submission - the s-box functions are implementation specific optimisations, especially the bit-sliced s-box functions like the Osvik and Gladman/Simpson, which actually compute multiple s-box lookups in parallel. If you need to test your s-box implementations, I would ...


3

The reason it is taking 4 32-bit integers into the round function is because it IS a bitsliced implementation. It bitsclices 32 4-bit sboxes into 4 32-bit inputs and uses standard logical operations on the words to get the job done. The sbox you posted was not generated by Osvik, but he generated a set of optimized blitsliced sboxes for 32-bit ...


3

I'm not seeing any particularity between the two field that they build, but exists? First off, one basic truth about finite fields is that there is at most one finite field of a given size. Given that both $x^7 + x + 1$ and $x^7 + x^4 + x^3 + x^2 + 1$ are both irreducible, they both generate the same finite field. Where they differ is the ...


3

They are referring to the formula for Bézout's identity which calculates the greatest common divisor, extended to a finite field, where 1 is the multiplicative neutral element. From the relation $b(x)a(x)+m(x)c(x)=1$, $$b^{-1}(x)=a(x)\mod m(x)$$ I cant figure out what $a(x)$ and $c(x)$ are? In this case, $b(x)$ is the element for which the inverse is ...



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