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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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### Is there any reference about the half-trace when m is even in F(2^m)

There is a algorithm listed in D.1.6, Algorithm 3, it seems that it is used to solve the quadratic equation when $m$ is even in $F(2^m)$. However, I can not find any reference about this algorithm, as ...
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### Problem with efficiency of projective coordinates in Elliptic Curve arithmetic

Ok sort of long post incoming. Will go slow to make it as clear as possible I'm trying to build a C library for Elliptic Curve Arithmetic. Since the idea is to learn from the process, I decided to ...
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### How do you securely implement a finite field?

I'm not sure if this question belongs here or to StackOverflow. Please flag it if not. I'm trying to implement a standalone library for finite field arithmetic of prime and prime power order as a way ...
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### 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 ...
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### 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 ...
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### Near Maximum Distance Separable Code

I want to find the minimum distance of an $[8,4]$ Near MDS code over a finite field F_4 (NMDS Code is a type of linear code). I want to know which programming language has a built-in function that ...
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### Is there an algebra group (or ring) in which computing the inverse element is hard without some trapdoor information?

Specifically, I want an algebra group $G$ (or ring $R$) features: Given elements $g,h\in G$ (or $R$ ), computing $g\cdot h \in G$ (or $R$ ) is easy. Given an element $g \in G$ (or $R$ ), finding the ...
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### Elliptic curves over extension fields

I'm trying to understand which benefits can using of extension fields in elliptic curve cryptography bring over prime fields. Popular curves like secp256k1, curve25519, secp384r1 are defined over a ...
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### Convert multiplication in $GF(2^{128})$ to bitwise AND?

Suppose we have $GF(2^{128})=F_2[x]/(x^{128}+x^7+x^2+x+1)$ and $a,b,c \in GF(2^{128})$ with $a*b=c$, where * is multiplication in $GF(2^{128})$. Could we convert ...
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### Query about arithmetic finite fields

I was working on implementing shamir secret sharing in GF(2^256), According to my knowledge multiplication in a finite field is defined as mul(a,b) = (a*b)%mod, where mod is the irreducible polynomial....
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### learning with errors

If I talk about efficiency of system of learning with error, is it it fine for q to be composite in Z_q, the ring of integers. As when q would not be prime, Z_q will not be field anymore, won't it ...
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### On the bit security of elliptic curves

My understanding is that an elliptic curve $E$ over a finite field $\mathbf{F}_q$ has a bit security of $\sqrt{q}$ assuming Pollard rho or Baby-step giant-step. In this thread, it is explained that ...
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### Discrete log hardness when secret is multiplied by public value

Given y = g ^ x is discrete log hard on some finite field, is y = g ^ (kx) also equally secure if the value ...
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### 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 ...
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### How to write monomials in $GF(2^n)$ as a system of equations in $GF(2)$

Let $F = GF(2^n)$ and $P(x) = x^e, P : F \rightarrow F$ be a monomial of degree $e$. How to write each bit of the output of $P$ as a function of input bits? In other words, how to write it as a system ...
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### How does big Galois groups yield better security in NTRU Prime?

I'm still kinda new to Galois theory so I apologize if this question is very obvious to some people. Basically I'm reading this paper by the NTRU Prime team and in section 2.5 it's explaining how ...
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### Feldman Verifiable secret sharing verify [closed]

I recently started to learn about Shamir secret sharing and Feldman's VSS Scheme. I know the concepts But I can't figure out how it works. mostly because many of modulates are with "p" and ...
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### Are you aware of cryptographic contexts (e.g., post-quantum) in which a square root $\sqrt{\cdot}$ must be computed in constant time?

Let $\mathbb{F}_q$ be a finite field of odd characteristic. I know that a constant-time implementation of the square root extraction $\sqrt{\cdot} \in \mathbb{F}_q$ is used in the context of hashing ...
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### Data fingerprint using multiple multilinear polynomials

Related to this question. I'm trying to find a way to use this fingerprint system without a second pre-image attack. Assume I have a set of elements $V = [v_0, v_1, v_2]$ in $\mathbb{F}_p$. Assume the ...
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### If SNARKs generally work in finite fields, how are non integer values handled - say fixed point decimal numbers?

In Vitalik Buterin's write-up on SNARKs Quadratic Arithmetic Programs: from Zero to Hero, he writes Note that the above is a simplification; “in the real world”, the addition, multiplication, ...
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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 ...
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### Galois field problem in Cryptography [closed]

This problem is related to Fields in Cryptography, My Question is why there is no multiplicative inverse for 2, isn't it 0.5?? or matters are diffrent if it was related to galois field ? I don't quite ...
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### Data fingerprint using polynomial and Schwartz-Zippel Lemma

I'm working on a protocol and am looking for a way to fingerprint a set of elements. All elements are evenly distributed across a finite field that is integers modulus $2^{256}$. Assume I have a set ...
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### multiplicative inverse computations on binary galois fields yield partial result when sampled

I want to compute the multiplicative inverse of 0x2 over $GF(2^{233})$ in hardware. To do so, I compute $a^{-1} = (a^{2^{m-1}-1})^{2}$. Here's the result of that ...
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### Division, scope finite fields polynomials in general vs. f.f. polynomials in ECC [closed]

A cryptography course covered among others following questions: arithmetic of polynomials over $GF(2^m)$ fields - polynomials division elliptic curves over field $GF(2^m)$ In scope of former point ...
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