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.

230 questions
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Galois fields in cryptography

I don't really understand Galois fields, but I've noticed they're used a lot in crypto. I tried to read into them, but quickly got lost in the mess of heiroglyphs and alien terms. I understand they're ...
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What is so special about elliptic curves?

There seems to be sources like this, this also, and some introductions that discuss elliptic curves in general and how they're used. But what I'd like to know is why these particular curves are so ...
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How robust is discrete logarithm in $GF(2^n)$?

"Normal" discrete logarithm based cryptosystems (DSA, Diffie-Hellman, ElGamal) work in the finite field of integers modulo a big prime $p$. However, there exist other finite fields out there, in ...
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Choice of multiplication polynomial in Rijndael s-box affine mapping

The Rijndael specification details the design choices for the s-box in section 7.2. They describe the choice of affine mapping as follows: We have chosen an affine mapping that has a very simple ...
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How to determine the order of an elliptic curve group from its parameters?

Let $\quad E:\; y^2 = x^3 + ax + b \quad$ be an elliptic curve defined over a finite field $\mathbb F_q$ where $q = p^n$, $a,b \in \mathbb F_q$ and $p \neq 2, 3$. By Hasse's theorem we know that the ...
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Design properties of the Rijndael finite field?

So we've already had a question on replacing the Rijndael S-Box. My question is - can we use a different finite field other than the one given by $x^8 + x^4 + x^3 + x + 1$ in $GF(2^8)$. In other words,...
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Are there any (asymmetric) cryptographic primitives not relying on arithmetic over prime fields and/or finite fields?

Trying to figure out if any (asymmetric) cryptographic primitives exists, which do not rely on arithmetic over a prime field and/or arithmetic over a finite field, some people might get lost in ...
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Do recent announcements about solving the DLP in $GF(2^{6120})$ apply to schemes proposed for cryptographic use?

A recent paper by Göloğlu, Granger, McGuire, and Zumbrägel: Solving a 6120-bit DLP on a Desktop Computer seems to "demonstrate a practical DLP break in the finite field of $2^{6120}$ elements, using ...
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Necessity for finite field arithmetic and the prime number p in Shamir's Secret Sharing Scheme

Shamir's original paper (PDF, 197kb) describing a threshold secret sharing scheme states: To make this claim more precise, we use modular arithmetic instead of real arithmetic. The set of ...
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Security of pairing-based cryptography over binary fields regarding new attacks

In the last week, the discrete logarithm problem was broken for the binary fields $\mathbb{F}_{2^{(14 \times 127)}}$ and $\mathbb{F}_{2^{(27 \times 73)}}$. Pairing-based cryptography using binary ...
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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 ...
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Multiplicative inverse in $\operatorname{GF}(2^8)$?

I know how to do multiplication over ${\rm GF}(2^8)$: ...
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Mapping between subgroups and the integers

This question is a companion to the equivalent question on elliptic curves. Preliminaries Diffie-Hellman, Elgamal, DSA, etc. are examples of protocols that work in the integers modulus a large prime ...
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Why are elliptic curves over a field of characteristic 2 or 3 insecure?

The following is a quotation from my cryptography course: Recent results on the discrete logarithm raise big concerns on the security of elliptic curves over a binary field. What are these ...
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How was the GCM polynomial found?

As far as I understand, there is no general way to enumerate irreducible polynomials in a particular finite field, which are similar in nature to prime numbers over the integers. The GCM mode finite ...
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how does BearSSL's GCM modular reduction work?

BearSSL (in src/hash/ghash_ctmul.c) seems to be doing a modular reduction that I don't completely understand. Here's the code: ...
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Advantage of $\operatorname{GF}(2^8)$ over $\mathbb Z/2^8\mathbb Z$ in AES/Rijndael

The Galois Field is used in the mixColumns step of the Rijndael-Algorithm. Over $\operatorname{GF}(2^8)$ (irreducible polynomial: $x^8 + x^4 + x^3 + x + 1$), the ...
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Is the additive discrete Logarithm problem always easy in Fields?

While thinking about additive DH key exchanges, I somehow had the idea that additive DH key exchange may always be easy to break, if we are in a field. So here's (directly) the question: In any ...
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Why are elliptic curves constructed using prime fields and not composite fields?

I come across this: Numbers mod composite number does not form a field rather it forms a ring and every number has a multiplicative inverse under integer mod prime Maybe these are the reasons ...
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Complexity of arithmetic in a finite field?

I am wondering what the complexities are of adding/subtracting and muliplying/dividing numbers in a finite field $\mathbb{F}_q$. I need it to understand an article I am reading.
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Secure degree reduction for Shamir's secret sharing

I understand the basic Shamir Secret Sharing protocol, and when two shares are multiplied, the degree of the polynomial increases. I've seen in a number of papers a reference to a degree reduction ...
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Why does Shamir's Secret Sharing Scheme need a finite field?

I read ampersand's question "Necessity for finite field arithmetic and the prime number p in Shamir's Secret Sharing Scheme", where he asked why Shamir's Secret Sharing Scheme uses arithmetic in a ...
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Solving hard problems in $\mathbb Z_{p}^{*}$ when $\mathbb p$ is close to $\mathbb 2^{n}$

Suppose, for some security parameter $n$ you choose a prime $p$ such that $p = 2^n+c$ for some relatively small $|c| < 2^m << 2^n$. I have seen such primes being called Pseudo-Mersenne Primes ...
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Why is x^8 + x^4 + x^3 + x + 1 used in AES's Rcon?

I am not familiar with field theory so please bear with me if this is obvious to you. I was wondering why this particular reducing polynomial $x^8+x^4+x^3+x+1$ is picked for AES' Rcon. Can't it be ...
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Should we use IANA groups 14 (MODP), 25, and 26 (ECP)?

By looking at SonicWall Knowledge Base article Key exchange (DH) Groups Supported - Site to Site VPN: It appears that our firewall supports DH group 25, and 26. Almost everywhere I've seen, they've ...
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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 ...
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Is it necessary for the Rijndael polynomial to be primitive?

I am working on selecting a S-box for my Cipher (Similar to AES). I found out there are 30 irreducible polynomials and over 16 primitive polynomials of degree 8. Is it necessary to choose a primitive ...
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What is the importance of Rcon in Rjindael's key expansion from a security prespective?

I do not see why the Rcon function is important, it looks like a waste of cycles. $$\operatorname{Rcon}(i) = 2^{i-1} \bmod p(x)$$ is in $\operatorname{GF}(2^8)$, ...
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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 ...
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AES mixcolumn stage

I'm studying AES, and am having problems with the "mixcolumn" stage. I read about finite fields, but I still don't know. How do I construct $GF(2^8)$? ...
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When to use safe prime or Schnorr group

Protocols that use $\mathbb{Z}_{p}^*$ arithmetic often choose $p$ to be a safe prime ($p = 2q + 1$, for prime $q$) or to have the Schnorr group form ($p = rq + 1$, for prime $q$). I understand that ...
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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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Algorithm for computing square roots in $GF(2^n)$

Short question: is there an algorithm for efficiently computing square roots in $\mathbb{F}_{2^n}$?
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How to calculate AES logarithm table?

I would like to know how to find multiplicative inverses in $\mathrm{GF}(2^8)$. I know how to multiply two elements of $\mathrm{GF}(2^8)$ (for example, I know that ...
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Understanding Feldman's VSS with a simple example

I'm trying to understand Feldman's VSS Scheme. The basic idea of that scheme is that one uses Shamir secret sharing to share a secret and commitments of the coefficients of the polynomial to allow the ...
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Endomorphism ring of a Elliptic Curve and $j$ invariant

I am reading Schoof's 1995 paper, Counting points on elliptic curves over finite fields, page 236, Proposition 6.1(i). I am trying to understand page 238 (second paragraph) of the proof: if the ...
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How to find the order of a generator on an elliptic curve?

I was looking out to find optimum generator for an elliptic curve $E$ over a prime field $\mathbb F_p$. I found the following algorithm: Choose random point $P$ on the curve. Find the order of a ...
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Algorithm for computing modular inverse in MPC

Is there any known algorithm for calculating $a^{-1} \pmod{q}$, where $q < p$ and $F_{p}$ is the prime field of the MPC, in a linear secret sharing scheme ? I have tried using the standard ...