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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9
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1answer
280 views

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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1answer
119 views

Why only non-prime order fields have small subgroup attacks?

Why don't prime-order curves have small subgroup attacks? It seems that I can choose a Generator such that it has a small order, maybe 2 points, and so an attack could generate all of the points in ...
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138 views

For an elliptic curve, what is the difference between the base field modulus $Q$ and subgroup $r$

What is the difference between the basefield modulus $Q$ and a subgroup of prime order $r$? They are all fields, but what is their relevance to the curve they are defined upon? How does this relate ...
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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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1answer
155 views

find additive inverse of modular arithmetic [closed]

For a set to be called as a ring, it should have the following properties closed commutative associative Identity existence Inverse existence but how is Z7 a ring, as there aren't any inverse ...
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2answers
1k views

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 ...
5
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1answer
571 views

Why doesn't the GCM spec use a more efficient multiplication algorithm?

NIST SP 800-38D § 6.3 Multiplication Operation on Blocks describes a multiplication algorithm that, in my testing, appears to be a good amount slower then algorithm 2.40 (arbitrary reduction ...
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1answer
83 views

GHASH with a finite field multiplication algorithm in reverse order

NIST SP 800-38D § 6.4 GHASH Function describes the GHASH algorithm thusly: Prerequisites: block $H$, the hash subkey. Input: bit string $X$ such that len($X$) = $128m$ for some positive ...
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1answer
50 views

modular reduction algorithm over $F_{2^m}$ doesn't seem to work when order of polynomial being reduced is small

I was considering algorithm 2.40 (arbitrary reduction polynomials) in the Guide to Elliptic Curve Cryptography and... it doesn't appear to work when the order of the polynomial you're trying to ...
3
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1answer
94 views

Is Finite Field Multiplication Distributive? Moving Affine Transform in AES

In AES the output of the SubBytes step is equal to: $a_{0-15} = d*c_{0-15}^{-1}+b$ where $d$ is a constant 8x8 matrix and b is a constant 8x1 matrix both in $GF(2)$. The inversion is done in $GF(2^...
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1answer
73 views

Multiplication and squaring the binary polynomials

I have tried to calculate $trace$ of a coordinate $X$ of EC in binary representation. Before that I tried to pre-calculate traces of the various bits of $X$ using formula: $$Tr(X) = Tr(\sum_{i = 0}^{...
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57 views

Pre-computing a log and exp table for a primitive polynomial in $GF(2^8)$ [duplicate]

I'm new on the topic of finite fields, specifically $GF(2^8)$. I've come across the information that it's possible to implement multiplication using logarithm and exponential tables. But how are ...
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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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2answers
226 views

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 ...
2
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1answer
94 views

Decoding a message on elliptic curve

Let's say I have an elliptic curve $E$ $y^2=x^3 + 486662x^2 + x$ over a prime field $GF(2^{255} - 19)$. My algorithm for computing $E(m)$ is as follows: I take the bits 1 through 32 of the message ...
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2answers
100 views

Finding Nonlinear boolean functions

Let $\mathbb{F}_2=\{0,1\}$ be the field with two elements. I wonder if there is any known algorithm/construction that, given any $n\geq 1$, returns a boolean function $f:\mathbb{F}^n_2\rightarrow \...
5
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1answer
831 views

Finding the n-th root of unity in a finite field

I'm trying to find the n-th root of unity in a finite field that is given to me. n is a power of 2. The finite field has prime ...
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1answer
183 views

How Elliptic-Curve affects the Server Key Exchange parameters

In Finite Field DHE, the server sends the following parameters in the server key exchange message: $p$: prime $g$: group $g^b$: the server's public DH key In DHE_RSA (non anonymous DHE), the server ...
0
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1answer
48 views

Given paramaters of an Edward's curve and x, determine a y value if it exists

I'm making a demonstration cryptosystem using ECC ElGamal. I've currently got a working implementation of Edward's Curve operations and a basic ElGamal implementation (Encrypts only points on the ...
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2answers
193 views

How to calculate Trace function for a point on an elliptic curve

I encountered trouble with calculating Tr (trace function) for points on an elliptic curve in polynomial basis ( $GF(2^m), m = 431$). Maybe there are any assumptions that can simplify and allow ...
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0answers
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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1answer
108 views

questions about modular reduction algorithm over $F_{2^m}$

So I'm trying to understand algorithm 2.40 (arbitrary reduction polynomials) from the Guide to Elliptic Curve Cryptography and have some questions. The very first sentence of this section says this: ...
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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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1answer
55 views

choices for k in binary finite field modular reduction algorithm

In the Guide to Elliptic Curve Cryptography there's this algorithm: My question is... what is $k$? Is it just some random value we pick? If so are some numbers better than others?
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2answers
113 views

powers of g in $GF(256)$

The finite field $GF(256)$ is usually implemented $mod$ 0x11b to keep the numbers inside that field. I understand that 0x11b was ...
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1answer
106 views

Galois Field multiplication instead of Diffie Hellmans discrete logarithm

I am wondering if the inversion of multiplication of polynomials is equally hard as the discrete logarithm problem used for key exchange. Or are there algorithms that weaken such an usage. I ...
2
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2answers
185 views

AES alternate equation for the S-Box affine transformation

The Wikipedia article for the AES S-Box gives an alternate equation for the affine part of the S-Box transformation: $$b_{out} = (b_{in} \times 31_d) \operatorname{mod} 257_d \oplus 99_d$$ It is not ...
2
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1answer
259 views

Questions about the Curve25519-donna implementation

I'm trying to understand the implementation of the following function: Please note questions in comments. ...
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1answer
66 views

Computations in extended finite field p^2

I would like to construct a distortion map from a point $\in \mathbb{F}_p$ to $\mathbb{F}_{p^2}$. If I have an elliptic curve $Y^2 = X^3 + 1$ over $\mathbb{F}_p$ and a distortion map $\phi(x,y) \...
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1answer
124 views

How to use Frobenius for finding Square Roots in $GF(2^m)$

Given a polynomial $x$ with degree $n$ in $GF(2^m)$, $1 < n < m$, will any generator of $GF(2^m)$ suffice when applying the Frobenius automorphism to determine the square root of $x$ as ...
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1answer
72 views

How to handle points in extended finite field

Following the response to my previous question, I would like to know if you could give me some information or give me a link on how to perform arithmetic operations once I changed a point from the ...
3
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1answer
292 views

How to optimise a finite field multiplication?

I'm currently trying to optimise the finite field multiplication in $ \operatorname{GF}(2)[x]/(p)$, where $p = x^8 ⊕ x^7 ⊕ x^6 ⊕ x ⊕1 ∈ \operatorname{GF}(2)[x] $. The thing is that I have to multiply ...
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0answers
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: ...
3
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0answers
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.
5
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1answer
391 views

Why does a Galois field have to have an order of $p^n$ where $p$ is prime?

I was reading about this in a cryptography book last night. I have a hunch about this, but I can't quite put my finger on it. I think this is a similar situation to an affine cipher, where the ...
1
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1answer
81 views

What hash functions can be (efficiently) computed over GF(2^m)?

Given an arithmetic circuit over a finite field of characteristic 2, what families of cryptographic hash functions can be efficiently computed with this circuit? Can standard hash functions be ...
0
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1answer
323 views

Using the encryption matrix from AES, how do you compute the decryption matrix?

So I don't want the answer but somewhere to start with this problem, first I want to know if my logic and thinking is on the right path before I dive right into computing the decryption matrix so here ...
1
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1answer
110 views

do koblitz curves over $\mathbb{F}_{P}$ as generalized in SEC2 always have $a$ as 0?

I reviewed all the curves in http://www.secg.org/SEC2-Ver-1.0.pdf . All the secp*k* curves have the $a$ parameters as 0 and those are the only ones with the $a$ as 0. Is this a defining requirement ...
0
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2answers
109 views

Difference between $F_2^n$ and $\Bbb F_2^n$ for a field

I am confused between the notation $F_2^n$ and $\Bbb F_2^n$ for a field in regards to codes. I thought that $F_2^n$ and $\Bbb F_2^n$ were both fields composed by codes of length n and entries in mod ...
0
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2answers
495 views

How does wNAF work with prime finite fields?

According to wikipedia, in the precomputation step of the w-ary non-adjacent form (wNAF) point multiplication method you do $d \bmod 2$ and, later, $d \gets \frac{d}2$. The mod operation doesn't make ...
2
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1answer
108 views

AES S-Box: Possible options for constant to calculate S-Box values

To calculate the values of S-Box in AES, I came across a lot of resources where constant {63} was chosen. It is said that {63} satisfies the condition of S-Box that it should not have any fixed points ...
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1answer
106 views

AES S-Box: How is value for 01 mapped to 7c?

If irreducible polynomial $m(x) = x^8+x^4+x^3+x+1$ is chosen, or even for any other value, the multiplicative inverse will not exist for $01$, as $0000 0001$ will perfectly divide $m(x) = 100011011$ ...
3
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1answer
99 views

Primitive root in a finite field

Wen-Her Yang and Shiuh-Pyng Shieh proposed two password authentication schemes by employing smart cards, one is timestamp-based and the other one is nonce-based. Their scheme consists of 3 phases: ...
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1answer
176 views

How do we reduce the multiplications in the AES mix column layer using $x^4 +1$

I recently learned AES uses $x^4 +1$ to reduce the multiplications in the MixCol layer. However, I used $p(x) = x^8 + x^4 + x^3 + x + 1$ not knowing it was the wrong polynomial and got the correct ...
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0answers
204 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
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3answers
3k views

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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1answer
394 views

Elliptic Curve - Divide by 2

Can anyone tell me the specific equations and steps for dividing a point on an elliptic curve by 2? For instance, I have the point $(P_x, P_y)$, and I would like to find the point $(R_x, R_y)$ which ...
4
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2answers
505 views

Verify Points on curve secp256k1

I am trying to verify whether or not these points are on the secp256k1 curve. I am finding several points included below. (I have verified 2*G, 8*G and 10*G with the pycoin script) My Questions are: ...
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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 : ...
20
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5answers
2k views

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 ...