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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135 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.
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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 ...
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1answer
86 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 ...
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1answer
399 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 ...
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1answer
128 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 ...
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2answers
141 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 ...
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2answers
606 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 ...
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1answer
134 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
110 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 $m(x) = 100011011$ ...
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1answer
139 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
208 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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244 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 ...
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3answers
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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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1answer
618 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 ...
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2answers
690 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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62 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 : ...
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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 ...
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1answer
509 views

Key size and finite fields in ECC (References)

So somehow I know that the key size in ECC is defined over the number of elements in a finite field or that it is almost equivalent to that (Correct me if I am wrong). However, other than on Wikipedia ...
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1answer
130 views

Using Hadamard Form of a Matrix in the Block Cipher

Definition: A matrix A of size $2^n$ is a Hadamard matrix, if has the following form $$ A= \left( \begin{array}{cc} U & V \\ V & U \end{array} \right)_{2^n\times 2^n}\, , $$ where $U$ and $V$...
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2answers
218 views

LFSR, polynomial , finite field

I'm having a hard time understanding the concept of LFSR, polynomials and finite field and how to solve exercises like picture below. Could anyone give me some pointer on where to start?
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1answer
4k views

Calculating the multiplicative inverse of a number in $GF(2^n)$ where $n > 8$

Suppose that: We have a polynomial $g(x)$ of degree $n$. $n > 8$. $q$ is the multiplicative inverse of $p$ in $G(2^n)$ modulo $g(x)$. If $p = 0$, then $q = 0$. This could be used: As a non-...
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1answer
148 views

(AES in mind) How can we show two irreducible polynomials have a bit-wise linear isomorphism

This refers to answer 2 on a similar question:Design properties of the Rijndael finite field I am unsure what many of the terms mean in this answer. I think a bit-wise linear isomorphism of ...
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1answer
207 views

Why equation Y^2=X^3 +AX+B can't work with finte field of charateristic 2?

I know that we can't define $dx/dy$ with this equation because $2y = 0$ with finite field of charateristic $2$. But with $GF(2^n)$ (has characteristic by $2$) $2=x$ not $0$. Do I misunderstand here?
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3answers
368 views

How can I get the binary form of AES's MDS matrix in MixColumns tranformation?

I need to write a procedure for calculating the MixColumns's operation result in the following form: $M*X^T,$ where $M$ is a 128x128 binary matrix, $X$ is a 128-bit vector (the state). My question ...
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2answers
2k views

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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1answer
517 views

how to choose a random secret key for ECDH

I am a beginner, I can understand the basics of ECC and elliptic curve, i can't find where I missed to understand. But I have a great doubt in ECDH regarding below. Could any of you please clarify for ...
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1answer
867 views

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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1answer
136 views

are all elements of ZpxZp in ECC definite over Zp

are all elements of ZpxZp in ECC (elliptic curve) definite over Zp ? otherwise: assume G a base point of ECC and n the order of G. why n is equal or nother to p*p ? (p a prime number). (Think to a ...
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0answers
299 views

Construction of Isomorphism between Galois Fields

I am trying to create an isomorphism between GF($2^8$) and an extension field in GF($(2^4)^2$). The finite field GF($2^8$) uses the Rijndael(AES) irreducible polynomial.In this field I have found 128 ...
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1answer
368 views

Shamir's Secret Sharing on non-prime GF

I am implementing Shamir's secret sharing scheme on arbitrary binary files. I don't intend to use this; this is a project to help me explore cryptography. In setting up the finite field arithmetic, ...
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1answer
205 views

trying to understand multiplication + reduction in binary finite fields [duplicate]

The wikipedia.org article on Finite field arithmetic provides an example of multiplying $83$ and $206$ in $\mathbb{F}_{2^8}$ with $x^8+x^4+x^3+x+1$ as the reducing polynomial (in fact it is the ...
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1answer
227 views

representing binary finite fields in ASN.1

In SEC 2: Recommended Elliptic Curve Domain Parameters two types of finite fields are utilized - $\mathbb{F}_p$ and $\mathbb{F}_{2^m}$. In the case of sect193r1, $\mathbb{F}_{2^m}$ is the finite field,...
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2answers
731 views

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 ...
4
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2answers
552 views

Elliptic Curve Isogenies, Frobenius endomorphism relation to characteristic equation

In Schoof's 1995 paper, Counting points on elliptic curves over finite fields, page 236, Proposition 6.1(i) states: Let $\mathbb{E}$ be an elliptic curve over $\mathbb{F}_p$. Suppose that its $j$-...
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0answers
97 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 ...
11
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1answer
678 views

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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0answers
252 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 ...
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1answer
2k views

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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3answers
821 views

addition on finite elliptic curves

I tried to calculate the sum of two Points on an elliptic curve in a finite field. The Curve is defined as following: $$y^2 \equiv x^3 + x \mod 257$$ So the curve parameters are $a = 1,b = 0,p = 257$...
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2answers
1k views

How to apply Pollard's Rho Method on elliptic curves to solve discrete logarithm problem in finite field?

I have ElGamal signature scheme implemented in finite field $\mathbb{F}_p$. The thing is that I need to apply Pollard's Rho Method on elliptic curve $E(\mathbb{F}_p)$ to this scheme, solve discrete ...
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1answer
243 views

Abelian groups in Elliptic curves [closed]

Do every elliptic curve defined over a prime field forms an abelian group?
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0answers
59 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 ...
2
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1answer
123 views

A Question about Irreducible Polynomials [closed]

I am doing some self-study in the area of Cryptography. I am using the Third Edition of the book "Cryptography Theory and Practice" by Douglas R. Stinson. Based upon the information on page 105, in ...
5
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2answers
569 views

LED (AES like) algorithm Decryption-Mixcolumn

I want to program decryption algorithm for the LED cipher. The lightweight block cipher LED(Jian Guo, Thomas Peyrin, Axel Poschmann, Matt Robshaw:CHES 2011). All the things is routine except the ...
5
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1answer
185 views

Can we have Rijndael s-boxes constructed using reducible polynomials?

Is it necessary for the Rijnael polynomial to be irreducible? Can we have s-boxes constructed using reducible polynomials? If not what is the mathematical property that is obstructing to do so?
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1answer
1k views

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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1answer
2k views

Understanding elliptic curve point addition over a finite field

I am new to elliptic curve cryptography as well as finite field theory. I am trying to understand point addition in affine coordinates. I understand, that for an elliptic curve $ y^{2}=x^{3}+ax+b $ ...
0
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1answer
172 views

Order of an elliptic curve defined over a prime field

I found the following algorithm to find the generator of an elliptic curve: Find the order of the curve - N. Choose any random point on the curve - P. Find the order of that point - n. Calculate co-...
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1answer
3k views

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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1answer
2k views

AES MixColumns using L and E lookup tables

I am trying to verify the multiplication by $\mathtt{02}$ in Galois Fields for MixColumns function using the L and E lookup tables. I could verify $\mathtt{D4}\cdot\mathtt{02}=\mathtt{B3}$ by manual ...