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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.

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How Significant is the New Quasi-Polynomial-Time Attack on Fixed Characteristic Discrete Logarithms?

There is a new paper by Kleinjung and Wesolowski on eprint that claims and proves a new attack on the discrete logarithm problem in finite fixed characteristic fields in quasi-polynomial time. ...
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136 views

GCM with reversed poly

These slides talk about how GCM can be sped up if one uses $x^{128}+x^{127}+x^{126}+x^{121}+1$ as the reduction polynomial instead of $x^{128}+x^7+x^2+x^1+1$. When one is doing that one needs to ...
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205 views

Share Conversion between Different Finite Fields

Let us have any linear secret sharing scheme (LSSS) that works on some field $Z_{p}$, where p is some prime or a power of a prime e.g., Shamir Secret Sharing, Additive secret Sharing. The problem at ...
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88 views

Group Rings on Cryptography

Let $R[G]$ or $RG$ be the group ring where $R=F_q$ and $G$ is any group. Let $Dim(V)=\vert G \vert$. It's clear that $V$ has $\vert R \vert^{\vert G \vert}$ distinct $\vert G \vert$-tuples. This ...
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109 views

How to map the points of an elliptic curve cyclic group to $\mathbb{Z}_q$ using a hash function?

Let $E$ be an elliptic curve defined over $\mathrm{GF}(q)$, where $q=p^r$. Let $G$ be a cyclic group of points of $E$. Then how we can map points of $G$ into $\mathbb{Z}_q$ using a hash function.
3
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180 views

Homomorphic encryption over finite fields

I'm curious on the following question: let $\mathbb{F}_{2^n}$ be a finite field which is an extension of $\mathbb{F}_2$ with order of $n$, is there an encoding scheme $e:=\mathbb{F}_{2^n}\rightarrow \...
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99 views

Solving not so much overdetermined system of multivariate polynomial equations

I'm studying algorithms solving multivariate equations. I'm stuck in solving overdetermined set of quadratic equations. Concretely, with the number $n$ of variables, the number of equations is $m=\...
2
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0answers
250 views

Construction of Isomorphism between Galois Fields

I am trying to create an isomorphism between GF($2^8$) and an extension field in GF($(2^4)^2$). The finite field GF($2^8$) uses the Rijndael(AES) irreducible polynomial.In this field I have found 128 ...
2
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0answers
94 views

How many Affine function can be made from $4 \times 4$ and $8 \times 8$ S-boxes?

The nonlinearity of an S-Box is defined as the non-linearity of its vectorial Boolean Function. Let $F$ be the hamming distance between the set of all non-constant linear combinations of component ...
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37 views

MULalpha and DIValpha operations in SNOW 3G

I'm not able to appreciate the importance of mul alpha and div alpha operations in feedback polynomial of LFSR in SNOW 3G. What problems or weakness do they help in mitigating and how ?
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1answer
32 views

Cost model for different curve models

Is there a cost model for each curve model and their conversions? For example: Take the curve models: Projective, Completed, Extended, Affine. Is there a table which shows how many multiplications, ...
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37 views

Efficient fields over arithmetic circuits

What sort of fields is efficient over an arithmetic circuit? Efficient meaning that given a field $\mathbb{F}_p$, reduction (modulo) does not require many multiplications and preferably inversion was ...
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169 views

How to use Galois LFSR to find multiplicative inverses

My question is how can a Galois Linear Feedback Shift Register be used to discover multiplicative inverses of polynomials? This is a homework assignment. Here is a list of things I did before asking ...
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0answers
14 views

Legendre conditions on the factors of the fundamental negative discriminant to minimize the 2-Sylow subgroup of the class group

If we know the prime factorisation of the fundamental negative discriminant $\Delta_K$, say $\Delta_K=p_1\cdot p_2 \cdots p_n$, then we are guaranteed that at least $2^{n-1}\mid h_K$, the class number ...
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199 views

Expressing a given linear transformation in Galois Field GF(256) in terms of another linear transformation with a different reduction polynomial

Before giving a better and detailed description of what I ask, let me first tell why I need what I am looking for: Intel processors already provide instructions (AES-NI) for very efficient AES ...
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0answers
60 views

Grøstl MixBytes Python implementation

I am trying to find an efficient way to implement the Grøstl matrix multiplication on python3. So far I have managed to get this result : ...
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0answers
58 views

Efficient proof of linear subspace membership

I am trying to find an efficient method of doing the following: Let $U$ be a $d$-dimensional linear subspace of $\mathbb{F}_q^n$. I want a protocol between a trusted party $T$, a prover $P$ and a ...
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0answers
71 views

Using quadratic residue to learn the sign of a field element

Given $x' \in \{-x, x\} \bmod q$ (where $q$ could be any prime of my choice), $s$ is a random element in the field, $y = x'\cdot s$ and $y' = \pm\sqrt{(x\cdot s)^2}\bmod q$ (i.e., both solutions to ...
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169 views

Can you provide an example in relation to Hidden Field Equations Multivariate?

I'm reading about Hidden Field Equations Multivariate scheme. My lecture states that the central map is a univariate polynomial $$P(X)=\sum_{i=0}^{r-1}\sum_{j=0}^{r-1}p_{ij}x^{q^i+q^j} \in K[X]$$ ...
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20 views

What is the difference between these pairings classifications?

I know the basic definitions of bilinear groups. For example, there is a bilinear pairing that uses elliptic curves and has the following properties: For $G_1$, and $G_2$ are cyclic groups of prime ...
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54 views

How can a node establish pairwise shared key with other nodes using its own polynomial share together with other's public values?

A server has a symmetric bivariate polynomial $ F(x, y) = \sum_{{i,j}=0}^{t-1}a_{i,j}x^iy^j$ $\in GF(p)[X, Y] $ of degree $t-1$. For simpliciy, $ F(x, y) = a_{0,0}+a_{1,0} x+a_{0,1}y+ a_{1,1}xy$ mod ...
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91 views

Numerical Algebraic geometry in the finite fields

Does the numerical algebraic geometry method work in the finite fields? I am working on this method to find a solution for a low-degree proximity testing problem. Would you please guide me how they ...
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305 views

Cube root modulo prime

I make research about big numbers in finite fields and I need to calculate a cube root modulo prime P for the number N: ...
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276 views

Multiplicative inverse ($17^{-1} \mod 31$)?

So. Sorry for bothering you with such a simple question, but I can't really get this done. It's just an exam question in which I need to use CRT in order to calculate the RSA signature of a msg m=101(...
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158 views

Discrete log in Galois Extension Field

I was reading 'Pinocchio Coin' paper by Danezis et al. where they have said, "If we use the efficient pairing groups of Pinocchio, computing discrete logarithms in the exponent field $\mathbb{F}_p$ ...
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1answer
125 views

How to find roots of equation $f(x)=0 \pmod p $, where $p$ is prime number?

$f(x)$is any nth degree equation $n>0$, how to find roots of $f(x)$ over prime modulo.