Cryptography Stack Exchange
Cryptography Stack Exchange

Questions tagged [finite-field]

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 this tag. Your question may concern finite fields if you are asking about AES, block ciphers or modular arithmetic.

395 questions
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What is the main difference between finite fields and rings?

In the course I'm studying, if I've understood it right, the main difference between the two is supposed to be that finite fields have division (inverse multiplication) while rings don't. But as I remember, rings also had inverse multiplication, so…
Jobo
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Why are finite fields so important in cryptography?

I am just getting into cryptography and currently learning by trying to implement some crypto algorithms. Currently implementing the Shamir secret sharing algorithm, what I have noticed is that finite fields keep coming up. I just don't understand…
iammadab
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What is this "finite field cryptography"?

See RFC 5931 § 2.2.1 which talks about "finite field cryptography" as opposed to elliptic curve cryptography and it looks like it is describing the Diffie-Hellman protocol. But Diffie-Hellman is not a done over a finite field, so is this some…
Melab
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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 trouble getting the expected answer for some of…
Jacob Wang
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Finding the n-th root of unity in a finite field

I'm trying to find the n-th root of unity in a finite field that is given to me. n is a power of 2. The finite field has prime order. I know that if this were just normal numbers, I could find it using $e^{(2\pi ik/n)}$. I have no idea how to…
fraiser
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5
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Why does a Galois field have to have an order of $p^n$ where $p$ is prime?

I was reading about this in a cryptography book last night. I have a hunch about this, but I can't quite put my finger on it. I think this is a similar situation to an affine cipher, where the multiplication factor has to be relatively prime with…
Zen Hacker
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How Multiplication Table is generated for GF(2^2) field

I was unable to solve the multiplication table given in the book for $\mathrm{GF}(2^2)$.However, I have managed to solve the addition table. Acoording to the Book multiplication is the AND operation, but when I applied this I did not get the answer…
Sufiyan Ghori
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Itoh Tsuji algorithm

I'd like to use the Itoh-Tsujii algorithm for a dynamic substitution table, but I do not get the following line: $$r\ \gets\ (p^m - 1)\,/\,(p - 1)$$ And why can $r$ be used to calculate the multiplicative innverse of a number in a Galois field…
Metalhead
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Should tower field implementations use the x^k element representation?

I'm working on a friendly tower finite field implementation for educational purposes. The library should allow easy building of tower fields from smaller ones - a user may define $\mathbb F_q$ and then build a tower field such as $\mathbb F_q…
tk2928
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Standard basis representation of elements in binary field

In Remark B.1 from this paper it says: We assume canonical representation for binary fields $\mathbb{F}$, given by an irreducible polynomial and a primitive element $g \in \mathbb{F}$ for it (i.e., $g$ generates $\mathbb{F}^*$). We use the standard…
irakliy
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How to optimise a finite field multiplication?

I'm currently trying to optimise the finite field multiplication in $ \operatorname{GF}(2)[x]/(p)$, where $p = x^8 ⊕ x^7 ⊕ x^6 ⊕ x ⊕1 ∈ \operatorname{GF}(2)[x] $. The thing is that I have to multiply by $ 16,32,133,148,192,194 \ \ and \ \ 251 $.…
Cedric
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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 find $n$ if the answer is yes.
Helium
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homomorphic mapping from $F_{p^n}$ to $Z_{p^n}$

Is it possible to have a homomorphic mapping from $F_{p^n}$ to ${\mathbb Z}_{p^n}$ that preserves both the add and multiplication operators? Or if we relax requirement, can we have a homomorphic mapping from the multiplicative group $F_{p^n}^*$ to…
Sean
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Program to find the inverse of polynomial

Can anyone tell me how to find the inverse of a given polynomial using python programming? Ex: input given is to find the inverse of (x^2 + 1) modulo (x^4 + x + 1). the output should be : (x^3 + x + 1).
N Tirumala Rao
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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 to find $x_0,y_0$ Does the previous set of equations…
Herc11
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