13 questions linked to/from Galois fields in cryptography
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### Multiplication/Division in Galois Field (2^8)

I'm attempting to implement multiplication and division in $GF(2^8)$ using log and exponential tables. I'm using the exponent of 3 as my generator, using instructions from here. However I'm having ...
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### Shamir's Secret Sharing Scheme - prime security?

Does the prime (which needs to be larger than the message) have to be chosen randomly for security, or can it just be a huge constant? Also, the Wikipedia article for SSSS states The size of each ...
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### How are these AES MixColumn multiplication tables calculated?

I'm using the mul2, mul3, mul9, mul11, mul13 and mul14 tables for the MixColumn and InvMixColumn steps in AES-128. However, I got these off some Github repository, and now I'm looking for an actual ...
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### How to generate a random polynomial of degree $m$?

I am trying to do a homework in which implement a variation of this paper and I don't know how to generate a polynomial of degree $m-1$. This polynomial is used to generate the $y_0$ and $y_1$ values. ...
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### Multiplicative Inverse in AES [duplicate]

I want to calculate the S-box for AES. But I have a problem with the multiplicative inverse step. For example, the inverse of 95 is ...
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### How to do Hexadecimal multiplication in GF(2^8)

I'm trying to do a hexadecimal multiplication in GF(2^8). Now, I know the technique of shifting left the binary numbers in some cases, for example: 02 * 9E 0000 0010 * 1001 1110 you shift one time ...
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### Does GHASH use XOR or addition?

I've been looking into GHASH and some sites describe it as a polynomial: A1*(H* *M-1)+A2*(H* *M-2)...C1*H(H* *M-N)+C2*H(H* *(M-N-1))...+len(A||C)*H+E(0)=t while ...
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### Galois field step in XEX mode

I'm reading through the XEX mode spec, and I want to make sure I've understood the $\alpha^j$ step correctly: $X = E_K(I) \oplus \alpha^j$ $C = E_K(M \oplus X) \oplus X$ where: $M$ is the ...
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### What is the inverse function of the finite field GF($2^3$) and GF($2^4$)?

I am currently reading a paper Cryptanalysis of a Theorem Decomposing the Only Known Solution to the Big APN Problem. In this paper, they mention that they used $I$ which is the inverse of the finite ...
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### How are Rijndael's S-Boxes generated?

Wikipedia: Rijndael's Forward S-box I'm writing C Code to generate S-boxes but I'm stuck. Would you mind explaining one entry of the S-box? (say for x = 0x2). ...
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### XEX mode - how to perturb the tweak

The first part of the classical description of XEX is $X = E_{k}(I) \otimes \alpha^j$ What I see a lot in sample code on the Internet is that you start with $T_ = E_{k}(I)$ and use that for the ...
I was watching this class about AES in this LINK and I was trying to grasp the concept of prime fields, which is a finite field with prime order $p$. The non-prime field part (order is $p^n$) is ...