# Tagged Questions

A finite field is a mathematical construct based on a set of axioms which are held to be true. A number of interesting and useful properties arise from finite fields that makes them particularly suitable for use in cryptography, notably in block ciphers. Questions concerning finite fields should use ...

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### Why does Shamir's Secret Sharing Scheme need a finite field?

I read ampersand's question "Necessity for finite field arithmetic and the prime number p in Shamir's Secret Sharing Scheme", where he asked why Shamir's Secret Sharing Scheme uses arithmetic in a ...
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### Galois fields in cryptography

I don't really understand Galois fields, but I've noticed they're used a lot in crypto. I tried to read into them, but quickly got lost in the mess of heiroglyphs and alien terms. I understand they're ...
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### How to determine the order of an elliptic curve group from its parameters?

Let $\quad E:\; y^2 = x^3 + ax + b \quad$ be an elliptic curve defined over a finite field $\mathbb F_q$ where $q = p^n$, $a,b \in \mathbb F_q$ and $p \neq 2, 3$. By Hasse's theorem we know that the ...
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### Choice of multiplication polynomial in Rijndael s-box affine mapping

The Rijndael specification details the design choices for the s-box in section 7.2. They describe the choice of affine mapping as follows: We have chosen an affine mapping that has a very simple ...
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### Do recent announcements about solving the DLP in $GF(2^{6120})$ apply to schemes proposed for cryptographic use?

A recent paper by Göloğlu, Granger, McGuire, and Zumbrägel: Solving a 6120-bit DLP on a Desktop Computer seems to "demonstrate a practical DLP break in the finite field of $2^{6120}$ elements, using ...
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### Secure degree reduction for Shamir's secret sharing

I understand the basic Shamir Secret Sharing protocol, and when two shares are multiplied, the degree of the polynomial increases. I've seen in a number of papers a reference to a degree reduction ...
The Wikipedia article on Shamir's Secret Sharing says to that to have information theoretical security the splitting algorithm should be evaluated using finite field arithmetic on the field $\rm{GF}(p)... 1answer 1k views ### Necessity for finite field arithmetic and the prime number p in Shamir's Secret Sharing Scheme Shamir's original paper (PDF, 197kb) describing a threshold secret sharing scheme states: To make this claim more precise, we use modular arithmetic instead of real arithmetic. The set of ... 2answers 2k views ### Design properties of the Rijndael finite field So we've already had a question on replacing the Rijndael S-Box. My question is - can we use a different finite field other than the one given by$x^8 + x^4 + x^3 + x + 1$in$GF(2^8)$. In other words,... 2answers 149 views ### What is the difference between the standard representants of$\mathbb Z/q\mathbb Z$? The symbol$\mathbb Z/q\mathbb Z$(given that$q$is prime) represents the prime field$\mathbb Z_q$. Basically, the elements of this field are represented by$\{0, 1, \dots, q-1\}$, let's call this ... 1answer 443 views ### Security of pairing-based cryptography over binary fields regarding new attacks In the last week, the discrete logarithm problem was broken for the binary fields$\mathbb{F}_{2^{(14 \times 127)}}$and$\mathbb{F}_{2^{(27 \times 73)}}$. Pairing-based cryptography using binary ... 3answers 1k views ### How robust is discrete logarithm in$GF(2^n)$? "Normal" discrete logarithm based cryptosystems (DSA, Diffie-Hellman, ElGamal) work in the finite field of integers modulo a big prime$p$. However, there exist other finite fields out there, in ... 3answers 426 views ### Mapping between subgroups and the integers This question is a companion to the equivalent question on elliptic curves. Preliminaries Diffie-Hellman, Elgamal, DSA, etc. are examples of protocols that work in the integers modulus a large prime ... 1answer 1k views ### AES mixcolumn stage I'm studying AES, and am having problems with the "mixcolumn" stage. I read about finite fields, but I still don't know. How do I construct$GF(2^8)$? ... 3answers 1k views ### Complexity of arithmetic in a finite field? I am wondering what the complexities are of adding/subtracting and muliplying/dividing numbers in a finite field$\mathbb{F}_q$. I need it to understand an article I am reading. Thank you 1answer 112 views ### Does$i^n=j^n$for$i, j \in GF(2^q)$and$i \neq j$for some$n<2^q-1$Let$i, j \in GF(2^q)$and$i \neq j$and$i,j\neq0$. Is that possible that$i^n=j^n$for some$n$such that$0 < n < 2^q-1$? I am looking for a proof if the answer is no, or for a method to ... 1answer 563 views ### Solve a system of non linear equations over GF I have the following set of equations: $$M_{1}=\frac{y_1-y_0}{x_1-x_0}$$ $$M_{2}=\frac{y_2-y_0}{x_2-x_0}$$$M_1, M_2, x_1, y_1, x_2, y_2,$are known and they are chosen from a$GF(2^m)$. I want ... 1answer 54 views ### Question about block erasure codes I have a question about linear block erasure codes that are described in this paper. I briefly describe the idea behind the linear erasure codes and then I ask my question. Given a set$d=\langle x_i ...
I'm trying to understand the notation used in the literature for Pairing-based cryptography. I know (and I hope I've understood it well) from Wikipedia that $\mathbb{Z}_p$ is the finite field of ...