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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173 views

How many field operations are needed when you compute kG in elliptic curves with a multiple additions or the double-ans-add-algorithm?

For an assignment, we have to calculate how many field computations are needed to calculate kG in an elliptic curve. They want us to show this for two different ways of calculating kG. The first way: ...
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567 views

Algebraic structures in RSA

Why do we need a field for RSA and what are the two operations in this field? Why can't we have a ring or group for example? Because in a group you also have inverse elements.
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On the hardness of addition when the elements of a field is represented by the powers of generator and possible any existant scheme

We can represent elements of a finite field $F$ in various ways polynomial basis and normal basis. There is one other; generator-based representation and this is based on the fact that the ...
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56 views

Understanding simplification steps when solving complicated equations in Galois Field

I just encountered a problem when I tried to understand a basepoint conversion from x25519 to ed25519. I can't really wrap my head around how the value of $x$ can be the stated value below? Can ...
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Question about algorithm

https://www.shoup.net/ntb/ntb-v2.pdf, page 523, algortithm IPT I don't get the for loop, since "k" isn't used anywhere inside the loop. What am i missing?
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Order of subgroups formed by Elliptic Curves with a Cofactor

In this question: Why are the lower 3 bits of curve25519/ed25519 secret keys cleared during creation? The answer indicates that the order of all points on the curve over the finite field ...
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51 views

How can we evaluate a polynomials in a group instead of a field? (verifiable secret sharing on elliptic curves)

I am trying to understand how we can have cryptographic schemes that builds on both secret sharing, which is build on top of a finite field, and bilinear maps, which are built on top of elliptic curve ...
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Get points of an Elliptic Curve defined over a Finite Field on Twisted Edwards Extended Coordinates

I'm working on a crypto library, and I need to perform some tests for the implementation of: Point Addition. Point Subtraction. Point Doubling. Scalar Mul Point. The operations are performed on ...
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91 views

Elliptic curve over prime field with high order roots of unity

Suppose I have an elliptic curve defined over a prime field $\operatorname{GF}(p)$ where $p$ is a large prime (e.g. 256-bit). Suppose also that $p = kn +1$, where $n$ is a relatively large power of $2$...
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How to calculate the order of the subgroup?

Given a curve with points over GF(p), a subgroup of prime order q and a co-factor h. How do I calculate the size of q which is ...
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99 views

Computing a sextic twist

Let $(x,y) \in E'_{\mathbb{F}_{p^2}}$ be a point of the sextic twist. I am currently trying to compute: $\psi : (x, y) \leftarrow (\mu^2x,\mu^3y)$ with $\mu \in \mathbb{F}_{p^{12}}$ the root of $(Y^...
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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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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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453 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
150 views

XOR and bilinear form property

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,...
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55 views

Probability of generating same master secret key in Identity-based Encryption

Suppose multiple servers use same IBE domain parameters (I mean same curve description parameters and field) for master secret key setup. Is there any possibility for generating the same system ...
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367 views

What is the difficulty of DLP in GF(P^Q) with a subgroup with a prime order of L

Given a finite field GF(P^Q), having a subgroup with a prime order of L (P,Q,L are all primes), how difficult is it to find the discrete log, is it related to P and Q or is it related to L, or to both....
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408 views

Elliptic curve trapdoor function without modular arithmetic?

From what I understand, an elliptic contains a set points satisfying the equation $y^2=x^3 + ax + b$ together with the point at infity. It seems clear how multiplication with a scalar and a point ...
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24 views

Pedersen commitment on binary field $GF(2^n)$

I am curious whether one can do Pedersen commitment on $GF(2^n)$. One method I thought of was to get a prime order multiplicative subgroup of $GF(2^n)$. But for efficiency and security, what would be ...
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38 views

Matrix multiplicative inverse

How to determine the inverse multiplicative matrix of the following two irreducible polynomials GF(2^8) x^8 + x^5 + x^3 + x^2 + 1 and x^8 + x^5 + x^4 + x^3 + x^2 + x +1
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38 views

Generalization of Bezout Identity for Polynomials

Let $i \in \{1,\ldots, n\}$, $f_i(x)$ be a univariate polynomial, and $g(x) = \mathsf{GCD}(f_1(x), \ldots,f_n(x))$. According to Bezout identity, there exists $a_i(x)$ such that: $$\sum_{i \in [n]}a_i(...
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How fast compute high degree power in finite field?

We need find y for some x on elliptical curve in cases: decompressing public key restoring public key from signature for given message hash, r and s. This algorithm uses power twice: first it is y1 =...
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45 views

Cryptographic properties of field multiplication (Continued)

This question follows-up from this question/comment. Suppose, you are given $X \odot Y$, where $X~(\neq 0) \in \operatorname{GF}(2^{128})$ is random, $Y~(\neq 0) \in \operatorname{GF}(2^{128})$ is ...
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Finding the relationship given by the generating elements of a non-abelian group (for a maximum case) where they correspond to finding cycles

Let $G=(\mathbb{Z}_p \times \mathbb{Z}_p) \rtimes_{\phi} \mathbb{Z}_q$, where $p$ and $q$ are odd distinct primes. Let $G$ be generated by the elements $s=(g_1,e_q)$ and $t=(e_p,g_3)$, $g_1,e_p \in \...
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A related question to “relationship between generating elements given by cycles in Cayley graphs”

I am writing this question with reference to the post at Relationship between generating elements given by cycles in Cayley graph When considering the generating elements $g_qg_p$, does it have the ...
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35 views

How to Solve Discret Log Computation Problem

Given $g^a$ in $Z_p$, it is hard to get the solution $a$. Everyone says yes, I wonder why it is hard. Could anyone give a specific mathematical way to show it is indeed hard?
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51 views

Difference in elliptic curve order and finite field size [duplicate]

Must the prime finite field, Fp, an elliptic curve is defined over always have a greater number of elements than the cardinality of an elliptic curve. For example, If I have ...
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1answer
107 views

Is the following non-interactive zero-knowledge set membership protocol provable secure?

Given the following Zero-knowledge set-membership protocol https://infoscience.epfl.ch/record/128718/files/CCS08.pdf]. That is given in the following steps (Please refer to page 9). The Verifier -...
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45 views

Explanation of Gallant-Lambert-Vanstone method / Endomorphism speedups [duplicate]

Can someone explain how the Gallant-Lambert-Vanstone method works (or which literature explains it)? It is also unclear to me how the Frobenius endomorphism can be used in some cases for a speedup. ...
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54 views

How can a node establish pairwise shared key with other nodes using its own polynomial share together with other's public values?

A server has a symmetric bivariate polynomial $ F(x, y) = \sum_{{i,j}=0}^{t-1}a_{i,j}x^iy^j$ $\in GF(p)[X, Y] $ of degree $t-1$. For simpliciy, $ F(x, y) = a_{0,0}+a_{1,0} x+a_{0,1}y+ a_{1,1}xy$ mod ...
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60 views

Prove I know a value v in a set s.t. K = H(v) [duplicate]

Is it possible to prove that I know a value v in a finite set, such that the hash of the value v is K. Where v is private and K is public
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1answer
119 views

Do the elliptic curves over prime fields must always contain prime number of elements (prime order)?

I have gone through one example where i saw a curve defined over some prime number containing non-prime order.
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372 views

Multiplicative inverse ($17^{-1} \mod 31$)?

So. Sorry for bothering you with such a simple question, but I can't really get this done. It's just an exam question in which I need to use CRT in order to calculate the RSA signature of a msg m=101(...
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209 views

Discrete log in Galois Extension Field

I was reading 'Pinocchio Coin' paper by Danezis et al. where they have said, "If we use the efficient pairing groups of Pinocchio, computing discrete logarithms in the exponent field $\mathbb{F}_p$ ...
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1answer
975 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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1answer
246 views

Abelian groups in Elliptic curves [closed]

Do every elliptic curve defined over a prime field forms an abelian group?
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1answer
104 views

How do I generate the multiplication table for GF(3^2)?

I understood, how this works for arbitrary n and p = 2, but I am struggling with higher prime numbers as a base. In the following, I wanted to use the irreducible polynomial ...
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1answer
83 views

Finding a prime field with n-th roots of unity

How can I find the smallest prime $p$, such that field $GF(p)$ has $n$-th roots of unity? For example, I know that for $p=2^{256} - 351 \times 2^{32} + 1$ there exit roots of unity for $n=2^{32}$. ...
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2answers
233 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
605 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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1answer
127 views

How to find roots of equation $f(x)=0 \pmod p $, where $p$ is prime number?

$f(x)$is any nth degree equation $n>0$, how to find roots of $f(x)$ over prime modulo.
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3answers
831 views

How to reverse this hash function?

I have a function that takes an $m$ byte inputs $x_i$ and maps it a 32 byte outputs $y_j$. The hash function is defined as: $$y_j = \sum_{i=1}^{m} (x_i)^{i-1} \pmod {127}$$ The input is restricted ...
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1answer
147 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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1answer
71 views

Given a point $c$ in a field $Z_p$. Can we get another value $c^{'}$ such that $\left(c^{\prime}-c\right)$ is invertible in $Z_p$?

If we have a point in a field $c$. Can we get another value $c^{'}$ such that $\left(c^{\prime}-c\right)$ is invertible in $Z_p$ ?
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1answer
220 views

Computing inverses in a binary field

Please suggest how can i solve the below question What is the inverse of {03} in GF (2^8) with the irreducible polynomial x8+x4+x3+x+1?

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