# 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 to perform addition and multiplication in Python in finite field (GF) F_{2^2}

I am trying to code something using NMDS matrices. I need to find the minimum distance of a code. How can I do the same?
1 vote
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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 ...
232 views

### 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 ...
1 vote
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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....
1 vote
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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 ...
1 vote
150 views

### 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 ...
135 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
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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 ...
1 vote
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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 ...
1 vote
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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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### Why can't RSA signatures be forged algebraically?

Compute $n = pq$ where p and q are prime. Fix $e$ to be coprime to $\phi(n)$. Compute $d = e^{-1} \pmod n$ and verify $ed \equiv \phi(n) \pmod n$. We sign the (hash of) a message with $s = h^{d}$. A ...
1 vote