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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715 views

Need help understanding math behind Rijndael S-Box

in Rijndael SubBytes() step all bytes of input block are substituted based on a lookup table S-Box. S-Box is initialized by taking all elements of $GF(2^8)$, ...
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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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1answer
289 views

Cryptographic properties of field multiplication

While reading about AES-GCM, I discovered there is a multiplication over $\operatorname{GF}(2^{128}$). My question is about its cryptographic properties, such as: Take a random element $X$ from $\...
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2answers
139 views

Why use prime number $q$ such $q$| $(p-1)$ in discrete logarithm based schemes?

In discrete logarithm based schemes on finite field we have a prime number $q$ that divides $p-1$ and $q$ is to specify a subgroup with the order $q$. But why do we do that? Why do not we work on the ...
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1answer
79 views

How does Montgomery reduction work?

I want to reduce a multi-precision integer $x$ modulo a prime $p$, very fast. Performing the traditional Euclidean division for only calculating the modulo, is inefficient and modular reduction is at ...
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428 views

Distribution of the Difference of Uniformly Random Elements

In the search to decision reduction of 'On Ideal Lattices and Learning with Errors over Rings', the authors implicitly use the fact that the difference of distinct, uniformly random elements of a (...
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1answer
85 views

Complexity of Gaussian Elimination over a Finite Field

I read somewhere that the complexity of solving a Linear $n\times n$ system over a Finite Field $\Bbb F_q$ using Gaussian Elimination is $\mathcal{O}(n^3)$ operations in $\Bbb F_q$. What's the role of ...
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1answer
188 views

Multiplicative inverse in ${GF}(2^4)$

I want to create a $4\times4$ multiplicative inverse table in $GF(2^4)$. The primitive polynomial given is $P(x)= x^4+x+1$ (NOTE: the values in the table need to be in hexadecimal format, hence I'll ...
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The aftermath and considerations of the new record of 30750-Bit Binary Field Discrete Logarithm - 2020

Granger et al. recently published a paper about breaking a record for discrete logarithm on the Binary field Computation of a 30 750-Bit Binary Field Discrete Logarithm, Robert Granger and Thorsten ...
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2answers
390 views

Constant time multiplication in GF(2^8)

I am trying to implement AES in C; I would like to make it resistant to side-channel attacks but I can't implement the multiplication in constant time. My current code: ...
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1answer
506 views

How do client and server agree on the values of p and g during DHE exchange?

How do the communicating parties using Finite Field DHE agree on the values of the $p$ and $g$ variables? Are their values fixed for each DHE group? Reading through the TLS 1.3 RFC (8446), the client ...
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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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1answer
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In a secret image sharing scheme, what are the advantages of Galois field $\mathrm{GF}(p^m)$ over finite field $ F_p$?

Shamir's Secret Sharing Scheme uses arithmetic in a finite field of prime order. In a secret image sharing scheme, what are the advantages of Galois field $\mathrm{GF}(p^m)$ over finite field $ F_p$?
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1answer
125 views

Itoh Tsuji algorithm

I'd like to use the Itoh-Tsujii algorithm for a dynamic substitution table, but I do not get the following line: $$r\ \gets\ (p^m - 1)\,/\,(p - 1)$$ And why can $r$ be used to calculate the ...
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1answer
41 views

RSA assumption and relationship given by generating elemts of a Cayley graph

I have read a very interesting description of computation related to the RSA group as follows. "By the Chinese remainder theorem, we have that: $$(\mathbb{Z}/pq\mathbb{Z})^* \cong (\mathbb{Z}/p\...
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1answer
102 views

Is there a multiplicative group of integers modulo p in which the discrete logarithm is easy?

The complexity of computing discrete logarithms in a multiplicative group modulo a prime $p$ is assumed to be sub-exponential time. The complexity is determined by $q$, the biggest factor of the group ...
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Determining order of the group for an elliptic curve defined over a finite field

I need to find out the order of the group for an elliptic curve. See the image for the question. The inequality condition after simplification leads to $-2 \sqrt{p} \leq m \leq2 \sqrt{p}.$ Also, the ...
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1answer
111 views

Distinguishing between a Polynomial and a Laurent Polynomial

Let $f(x) \in \mathbb{Z}_p[x]$ (for a prime $p \gg d$) be a polynomial of degree $d$, and let $g(x)$ be a Laurent polynomial with the same degree and only the first negative exponent term ($g(x) = \...
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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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1answer
84 views

Diffie-Hellman with Galois field

I Google around and can't find any page mentioning Diffie-Hellman with Galois field $GF(p^n)$ with $n>1$. Is there a reason for this? For example, wouldn't Diffie-Hellman with $GF(2^n)$ be ...
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1answer
76 views

Relationship between generating elements given by cycles in Cayley graph

The strong RSA assumption is that the following problem is hard to solve. "Given a randomly chosen RSA modulus $n$ and a random $z \in \mathbb{Z}_n^*$, find $r>1$ and $y \in \mathbb{Z}_n^*$ such ...
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1answer
28 views

Decoding in Reed solomon codes

I have code encoded in GF(7) with primitive 5 Сf(4,1,0,4,5,5). (last four symbols is redundancy) While decoding using DFT we use formula $$ С_k=N^{-1}*c(z^{-kj}) $$ example: $$ C_1 = c(5^{-1*j})/6 ...
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1answer
190 views

Prime fields vs non-prime fields

I was watching this class about AES in this LINK and I was trying to grasp the concept of prime fields, which is a finite field with prime order $p$. The non-prime field part (order is $p^n$) is ...
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1answer
91 views

Inversion in $GF(2^{10})$ Using Composite Fields

I'm designing a circuit that uses many $GF(2^{10})$ inverters. Normally for this sort of thing I use lookup tables. (Itoh-Tsujii is not efficient for these smaller fields.) This application is for ...
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33 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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1answer
76 views

How to make this cipher strong?

Suppose I have an arbitrary 256 bit number $m$ another secret number $k$ of the same bit length, and then I multiply them both modulo a 256 bit prime number $p$ to get $c$ as follows: $$ c = (m\cdot k)...
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69 views

Looking for a group where multiplicative inverse is hard but calculating multiple pairwise inverses is easy

I have an equation that comes up in trying to do an Attribute Based Encryption scheme, where users have certificates that sign off on attributes that are also watermarked. I perform a division to ...
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1answer
72 views

Finding subgroup in elliptic curve over finite field $ \mathbb{F}_{11}$

For elliptic curve $ y^2 = x^3 +3x+7$ I found the finite group $ E(\mathbb{F}_{11})= \left\{ \mathcal{O}, (1,0),(5,2),(5,9),(8,2),(8,9),(9,2),(9,9),(10,5),(10,6) \right\}$. I have to find a ...
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1answer
151 views

In a shamir secret sharing scheme, why can't the secret be a middle coefficient and only the first or the last?

In the shamir secret sharing scheme, the Secret s is set as the constant in the equation $ y_p = s+ \sum_{i=0}^{i = t-1} a_i * x_p^i$ s can only be the constant term or the last coefficient or the ...
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1answer
269 views

Diffie-Hellman key exchange protocol with finite field GF(2^5)

In a Diffie-Hellman key exchange protocol, the system parameters are given as follows: finite field $GF(2^5)$ defined with irreducible polynomial $f(x) = x^5 + x^2 + 1$ and primitive element $\alpha= ...
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1answer
55 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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2answers
82 views

Size of $E$ over $\mathbb{F}_p$ contains $p+1$ points

I am struggling to prove this claim: I proved that the map $x\mapsto x^3+1$ is a bijection from $\mathbb{F}_p$ to itself if we have that $p\equiv 2\bmod{3}$. We have to use this fact to prove that ...
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2answers
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Generating Private Keys [closed]

First of all I want to say that I have exactly 0 knowledge how I can write something in Python. But I have some knowledge in math especially finite fields. That's why I want to learn more in writing ...
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1answer
44 views

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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1answer
159 views

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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1answer
135 views

Choice of finite fields for use in elliptic curves

this is maybe a basic question but I'm trying to better understand elliptic curve cryptography at a fundamental level. I understand that a finite field is required in order to define a boundary for ...
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1answer
185 views

Why does curve25519 use a cofactor of 8?

This cofactor (as I understand it) effectively discards valid points that satisfy the curve equation over the finite field. Why would one wish to reduce the number of possible private keys, it seems ...
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1answer
78 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
49 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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0answers
50 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
124 views

Why is $S(y) = y^{2^8-2} = y^{254}$ a one-to-one function and a permutation on $GF(2^8)$

I'm taking a course on cryptography and I have some confusions concerning the materials in our notes. Say we have the field $GF(2^8)$, we create a substitution algorithm (the S Box in AES) so we need ...
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2answers
285 views

The definition and origin of Schnorr groups?

Wanting to write something on Schnorr groups in a publication I realised how hard it is to find anything citable about them on the internet. Who can help me with the following questions? What (and ...
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0answers
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MULalpha and DIValpha operations in SNOW 3G

I'm not able to appreciate the importance of mul alpha and div alpha operations in feedback polynomial of LFSR in SNOW 3G. What problems or weakness do they help in mitigating and how ?
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1answer
72 views

What does MULx operation in SNOW 3G correspond to?

According to the spec, MULx operation is defined as - MULx maps 16 bits to 8 bits. Let V and c be 8-bit input values. Then MULx is defined: If the leftmost (i.e. the most significant) bit of V ...
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1answer
227 views

Computational Complexity: ECC multiplication vs Modular multiplication

How does performing scalar multiplication on an elliptic curve compare to exponentiation in a multiplicative group modulo a prime? I.e. on a given elliptic curve of size $|t|$, what's the complexity ...
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1answer
101 views

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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0answers
50 views

Constructing Low-Density Parity-Check Codes of length $n$ and minimum distance = $\delta n$ over $GF(q)$? [closed]

I am looking for a way to construct an LDPC (Low-Density Parity-Check) Code $C$ of length $n$ and minimum distance $d_C$ that scales linearly to $n$, meaning $d_C = \delta n$ for $\delta \in (0,1)$. ...
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0answers
392 views

How Significant is the New Quasi-Polynomial-Time Attack on Fixed Characteristic Discrete Logarithms?

There is a new paper by Kleinjung and Wesolowski on eprint that claims and proves a new attack on the discrete logarithm problem in finite fixed characteristic fields in quasi-polynomial time. ...
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1answer
88 views

Explanation of trace function $\operatorname{Tr}_m(x) = x^{2^{m}} \oplus x$

The following statement is from a paper (Partitions in the S-Box of Streebog and Kuznyechik) about S-Boxes: For all $ x \in \operatorname{GF}(2^{n})$, it holds that $x^{2^{n}} \oplus x = 0$. If $...
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1answer
106 views

Does any problem arise when the order of an elliptic curve is equal to its prime field modulus? [duplicate]

Regarding cryptographic schemes in elliptic curve cryptography, is there a problem with having the order of an elliptic curve being equal to its prime field modulus? That is, an elliptic curve where $...

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