# 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.

241 questions
Filter by
Sorted by
Tagged with
395 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 ...
710 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 ...
114 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 ...
226 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 ...
284 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, ...
130 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 ...
202 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,...
528 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 ...
426 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 $j$-...
89 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 ...
528 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 ...
190 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 ...
1k 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 ...
622 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 $a = 1,b = 0,p = 257$...
829 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 ...
223 views

### Abelian groups in Elliptic curves [closed]

Do every elliptic curve defined over a prime field forms an abelian group?
56 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 ...
115 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 ...
401 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 ...
180 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?
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 ...
1k 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 $y^{2}=x^{3}+ax+b$ ...
144 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-...
2k 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 ...
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 ...
576 views

267 views

### Generating Diffie-Hellman parameters

I'm trying to implement a diffie-hellman key exchange in c++, and I'm struggling with my missing understanding of math / group theory. Let's say I found a large prime number p - how can I find a ...
807 views

### 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 ...
487 views

### Short Weierstrass equation is non-singular for not 2 or 3 characteristic

Consider a field $K$ of characteristic $p \neq 2,3$. Consider a curve $E$ over $K$ defined by the equation $y^2 = x^3 + ax + b$. How can I show that: $E$ is not an elliptic curve (it is not ...
477 views

### How to perform the modular reduce of Rijndael's finite field

I am trying to understand how to calculate the modular reduction of Rijndael's finite field. The example on this page says that {53} • {CA} = {01}, because ...
3k views

### Find the generators of multiplicative group of units efficiently?

Say you're give some prime numbers $p_{1},p_{2},p_{3}, p = 2p_{1}p_{2}p_{3} + 1$ (which is assumed to be also prime) and a list of numbers $L$ and you're asked to find the generators of the ...
123 views

### How to find irreducible polynomial for Barreto-Naehrig curves?

As described in this paper(section 3) to implement pairing on Barreto-Naehrig curves. The prime in their case is $p=82434016654300679721217353503190038836571781811386228921167322412819029493183$ and ...
594 views

### Shamir Secret Sharing GF(p) or GF(2^8)

I'm implementing Shamir's Secret Sharing Scheme, but I've hit a conceptual roadblock. In Shamir's paper "How To Share A Secret" he creates his shares an a finite field of order p, where p is some ...
### In what sense addition modulo $n$ ($n>2$) isn't linear in the field $\mathbb{F}_2$?
I've been reading the Reason why “XOR” is a linear operation, but ordinary “addition” isn’t? question, in which one of the answers states that addition modulo $n$ ($n>2$) is linear in $\mathbb{Z}_n$...
I know that $XOR$ is equivalent to modular addition in the field $\mathbb{F}_2 = \{0,1\}$ (is it right?), and thus should satisfy the following property of a bilinear form: \oplus(u+v,w) = \oplus(u,...