Skip to main content
New
Stack Overflow Jobs powered by Indeed: A job site that puts thousands of tech jobs at your fingertips (U.S. only). Search jobs

# 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.

56 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
18 votes
0 answers
514 views

### The aftermath and considerations of the new record of 30750-Bit Binary Field Discrete Logarithm - 2020

Granger et al. recently published a paper about breaking a record for discrete logarithm on the Binary field Computation of a 30 750-Bit Binary Field Discrete Logarithm, Robert Granger and Thorsten ...
• 48.7k
13 votes
0 answers
497 views

### 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. ...
• 46.1k
10 votes
0 answers
479 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 ...
• 872
9 votes
0 answers
251 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 ...
• 2,927
4 votes
0 answers
128 views

### Security of ECDLP using elliptic curves over an extension field

It is known that, for an elliptic curves $E$ defined over a prime field $\mathbb{F}_p$ such that $E(\mathbb{F}_p)$ is a prime number, the best algorithms (beside some specific cases) for solving the ...
• 141
4 votes
0 answers
246 views

• 243
2 votes
0 answers
28 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 ...
• 273
2 votes
0 answers
119 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 ...
• 21
1 vote
0 answers
39 views

### I want to find the Zero Value Points on SECP256R1 curve... Is there an alternative to Chien's method of finding roots over large Finite Fields?

This PDF explains that on certain elliptic curves, there exists ZVP (Zero Value Points) that cause zero value registers during the scalar-to-point multiplication (i.e during the double operation or ...
• 11
1 vote
0 answers
41 views

1 vote
1 answer
183 views

### Multi party computation over ring and fields

I am recently reading about multi party computation and its various existing protocols. From what I understand, all the arithmetic operations are performed over a field or a ring such that when two ...
1 vote
0 answers
44 views

1 vote
0 answers
50 views

### Algorithm that solves a system of linear equations over finite fields when a parameter is needed

I was reading Kipnis' and Shamir's paper on Cryptanalysis of the HFE Public Key Cryptosystem by Relinearisation and I wanted to implement the example at the end in Octave without using any additional ...
• 11
1 vote
0 answers
82 views

### Proving a function in $\operatorname{GF}(2^n)$ is differentially k-uniform

I want to show that $F(x) = x^{-1}$ in $\operatorname{GF}(2^{n})$ is differentially 4-uniform for even $n$, and is differentially 2-uniform for odd $n$, without looking at the Differential ...
• 75
1 vote
0 answers
30 views

### A finite group with a threshold functionality

I am trying to find a generator of a finite group that its powers devides the group into two parts. For example look at the last row of this table that shows the powers of 10 in the group Z_19. You ...
1 vote
0 answers
244 views

### Multiplication in Tower Field $GF(2^4)^2$

I'm currently reading an article which deals with "Squeezing Polynomial Masking into Tower Fields " for performing an efficient multiplication of elements in $GF(2^8)$. Thereby it is ...
1 vote
0 answers
48 views

### Determining order of the group for an elliptic curve defined over a finite field

I need to find out the order of the group for an elliptic curve. See the image for the question. The inequality condition after simplification leads to $-2 \sqrt{p} \leq m \leq2 \sqrt{p}.$ Also, the ...
1 vote
0 answers
87 views

### Looking for a group where multiplicative inverse is hard but calculating multiple pairwise inverses is easy

I have an equation that comes up in trying to do an Attribute Based Encryption scheme, where users have certificates that sign off on attributes that are also watermarked. I perform a division to ...
• 349
1 vote
0 answers
58 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 ?
• 71
1 vote
0 answers
55 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 ...
• 1,343
1 vote
0 answers
119 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 ...
1 vote
0 answers
404 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 ...
• 629
1 vote
0 answers
478 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 ...
• 11
1 vote
0 answers
70 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 : ...
• 121
1 vote
0 answers
77 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 ...
1 vote
0 answers
83 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 ...
• 101
1 vote
0 answers
181 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 $$P(X)=\sum_{i=0}^{r-1}\sum_{j=0}^{r-1}p_{ij}x^{q^i+q^j} \in K[X]$$ ...
• 2,711
0 votes
1 answer
69 views

### wrting algorithm for torsion group elements

Yesterday,I took an exam. There are two questions I received very low points. I will write the first question in this post. The question says let $E:y^2:x^3+kx+1$ in GF(p) be an elliptic curve where p ...
0 votes
0 answers
71 views

### Why do we use elliptic curves instead of just the discrete logarithm problem?

We have a cyclic field Fp where p is a prime number, a generator g, and an order n. A generator is an element such that $g^n=1$. A random number x has been chosen as the private key, selected from the ...
• 101
0 votes
0 answers
50 views

### How to calculate Cardinality of an Elliptic curve over $\mathbb Z_{23}$

How to calculate Cardinality of an Elliptic curve over $\mathbb Z_{23}$. $E: y^2 = x^3 + x + 1$ defined over $\mathbb Z_{23}$.
0 votes
0 answers
70 views

### AES encryption step: calculate the byte substitution process

I am a self learner and tried to learn AES, so I have a book written in my native language. It has some practical examples about finite fields and AES. It has a question that asks me to calculate the ...
0 votes
0 answers
76 views

### AES SBOX Masking in Python

I am trying to create an AES Sbox masking based on this paper. In the paper, they tried to mask the sbox described in this other paper. The inversion is been done in GF(4). I know that it is mostly ...
• 1
0 votes
0 answers
32 views

0 votes
0 answers
45 views

### How to Solve Discret Log Computation Problem

Given $g^a$ in $Z_p$, it is hard to get the solution $a$. Everyone says yes, I wonder why it is hard. Could anyone give a specific mathematical way to show it is indeed hard?