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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 ...

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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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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
32 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
48 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
43 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 ...
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2answers
111 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 ...
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1answer
79 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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21 views

Elgamal verification scheme?

According to Elgamal signature scheme; The verification process as following: the verifier determine whether or not the following condition apply $g^h(m)$=$y^r$$r^s$ mod p. where: $y$ is the public ...
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1answer
47 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
65 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
40 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 ...
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50 views

Understanding creation of distortion map

I'm trying to implement a distortion map but I have a problem. I know the basics and I read some questions like this one and asked some questions here. If I have an elliptic curve $E : y^2 = x^3 + 1$ ...
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1answer
69 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
61 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: ...
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0answers
60 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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1answer
288 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 ...
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1answer
66 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
137 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
73 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
76 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
89 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
41 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
66 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
77 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
74 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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129 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
2k 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
137 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
162 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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48 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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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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176 views

Is there any benefit to this technique for calculating a multiplicative inverse?

I know very little about cryptography, but many years ago I came up with a relatively simple method for calculating modular multiplicative inverses of binary numbers. My goal was to reduce the ...
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1answer
178 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
94 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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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
597 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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84 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
184 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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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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458 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
216 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
485 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
81 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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125 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
173 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
82 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
174 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
335 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 ...
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2answers
217 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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67 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 ...