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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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38 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
35 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
43 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
99 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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0answers
45 views

Why does using infinite fields leak secrets in Shamirs Secret Sharing Scheme? [duplicate]

There is a similar question here, but the answers as I understand them basically say (1) you can leak the parity of the secret and (2) you can run into over/underflow issues as well as floating point ...
5
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2answers
115 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
36 views

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
45 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
64 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
53 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
45 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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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
66 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
40 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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81 views

Group Rings on Cryptography

Let $R[G]$ or $RG$ be the group ring where $R=F_q$ and $G$ is any group. Let $Dim(V)=\vert G \vert$. It's clear that $V$ has $\vert R \vert^{\vert G \vert}$ distinct $\vert G \vert$-tuples. This ...
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0answers
94 views

Solving not so much overdetermined system of multivariate polynomial equations

I'm studying algorithms solving multivariate equations. I'm stuck in solving overdetermined set of quadratic equations. Concretely, with the number $n$ of variables, the number of equations is $m=\...
2
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1answer
82 views

Are there any security risks in using Elliptic Curves defined over fields $\mathbf{F}_{p^n}$ where $n>1$

I've recently been studying elliptic curves, and I've found that most of the current implementations use fields $\mathbf{Z_p}$ or in some cases $\mathbf{F}_{2^n}$. All the reasons I've seen for not ...
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0answers
20 views

What is the difference between these pairings classifications?

I know the basic definitions of bilinear groups. For example, there is a bilinear pairing that uses elliptic curves and has the following properties: For $G_1$, and $G_2$ are cyclic groups of prime ...
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2answers
258 views

How are these AES MixColumn multiplication tables calculated?

I'm using the mul2, mul3, mul9, mul11, mul13 and mul14 tables for the MixColumn and InvMixColumn steps in AES-128. However, I got these off some Github repository, and now I'm looking for an actual ...
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1answer
55 views

Is inversion always cheap with Twisted Edwards curves?

I'm reading on Jubjub, which is planned for the next upgrade of Zcash. It is based on a Twisted Edwards curve with parameters $a = -1$ and $d = −(10240/10241)$. The reading says Jubjub does not need ...
2
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2answers
131 views

Why aren't Complex number fields used in cryptography?

Most of the cryptography is limited to use of real integer fields. Complex numbers are seldom used. Is there any reason for it? Wouldn't complex numbers provide more structure? If there is already ...
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0answers
56 views

How do I find an efficient sqrt algorithm for primes? [closed]

For example, If my prime was congruent to 3 mod 5. Is there a list where I can find for these primes which do not take the form 3 mod 4? Edit: Excluding the general tonelli-shanks algorithm
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1answer
56 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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1answer
39 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
78 views

Is there a concept of embedding degree for non-pairing based elliptic curves?

From this post, I learned the concept of embedding degree. Intuitively, if embedding degree of an elliptic curve $E(F_p)$ is $k$, it means there is a way to transform points in $E(F_p)$ to $F_{p^k}$. ...
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1answer
54 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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1answer
51 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 $n=2^{32}$. ...
2
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1answer
100 views

Standard basis representation of elements in binary field

In Remark B.1 from this paper it says: We assume canonical representation for binary fields $\mathbb{F}$, given by an irreducible polynomial and a primitive element $g \in \mathbb{F}$ for it (i.e., ...
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0answers
55 views

How to find multiplicative inverse of a hexadecimal number in the finite field GF(2^8)? [duplicate]

I am new to cryptography. As I was reading a note regarding S-box construction, I found a step mentioning multiplicative inverse of a number in the finite field GF(2^8). I couldn't understand it!! Can ...
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1answer
51 views

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 ...
1
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1answer
31 views

Cost model for different curve models

Is there a cost model for each curve model and their conversions? For example: Take the curve models: Projective, Completed, Extended, Affine. Is there a table which shows how many multiplications, ...
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0answers
37 views

Efficient fields over arithmetic circuits

What sort of fields is efficient over an arithmetic circuit? Efficient meaning that given a field $\mathbb{F}_p$, reduction (modulo) does not require many multiplications and preferably inversion was ...
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1answer
62 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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1answer
40 views

ECDSA over $\mathbb{F}_{p^n}$ for $n>1$. How to calculate $r$ and $s$

I'm having some trouble understanding how to calculate $r$ and $s$ as specified in the wikipedia page for ECDSA (https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm) We can see ...
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1answer
71 views

Is the One Time Pad secure in additive Rings?

Let's assume all operations are done on $\mathbb{Z}_p$ where $p$ is a large non-prime number. To mask a value $a$, we do the following: Pick a uniformly random value: $r$, from the ring. Do as ...
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1answer
96 views

What is the inverse function of the finite field GF($2^3$) and GF($2^4$)?

I am currently reading a paper Cryptanalysis of a Theorem Decomposing the Only Known Solution to the Big APN Problem. In this paper, they mention that they used $I$ which is the inverse of the finite ...
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0answers
37 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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0answers
53 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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1answer
63 views

What is the difference between the 23 bi-affine and the 39 fully quadratic equations of the rijndael sbox?

In Cryptanalysis of Block Ciphers with Overdefined Systems of Equations Nicolas Courtois and Josef Pieprzyk define 23 so called bi-affine equations (in Appendix A of the paper) between the input x and ...
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1answer
359 views

Relationship between Fibonacci LFSR and Galois LFSR

I'm studying about LFSR and have some troubles understanding LFSRs. For Galois LFSR, it is clear that LFSR just multiplies $x$, the primitive element of $GF(2^n)$, so that it makes all the elements ...
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1answer
129 views

Does a Finite Field of 36 elements exist? [duplicate]

I'm having a little trouble understanding the finite fields theory, so I'm sorry if my question would seem a little stupid. I wanted to know if a finite field of 36 elements could exist. Basically, I ...
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1answer
90 views

Inverse of an element in a RSA group

Consider a RSA group $Z_N$ for $N=pq$, where $p,q$ are large prime numbers. Under strong RSA assumption, can an adversary efficiently compute the inverse of a random element $z$ from $Z_N$ without ...
4
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1answer
84 views

Difficulty of forging MACs based on linear functions over $GF\left(2^n\right)$

This is a homework question, therefore I'm not expecting full solutions, just general guidance. I want to build a one-time MAC using universal hashing. I defined my hash functions as: $h_{a,b}:\...
3
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1answer
139 views

Elliptic curves on finite fields

I've been reading: https://github.com/bellaj/Blockchain/blob/6bffb47afae6a2a70903a26d215484cf8ff03859/ecdsa_bitcoin.pdf On page 22 it shows an eliptic curve over F17. I have added the orange lines ...
5
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2answers
118 views

Why does libSTARK use binary fields as opposed to prime fields for zk-SNARKs?

zk-STARKs make use of FRI for low degree testing of polynomials. The zk-STARKs paper states on page 11: we stress that ZK-STARK could also operate over prime fields but we have not realized this ...
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1answer
93 views

What is the difference between subtracting the modulus from a scalar field element and reducing it?

When implementing a Field element, we define the necessary operations on the data structure. One function that I see is a "scalar reduce" function, which effectively reduces a random scalar so that ...
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0answers
58 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
274 views

how does BearSSL's GCM modular reduction work?

BearSSL (in src/hash/ghash_ctmul.c) seems to be doing a modular reduction that I don't completely understand. Here's the code: ...
0
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1answer
114 views

Why only non-prime order fields have small subgroup attacks?

Why don't prime-order curves have small subgroup attacks? It seems that I can choose a Generator such that it has a small order, maybe 2 points, and so an attack could generate all of the points in ...
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1answer
130 views

For an elliptic curve, what is the difference between the base field modulus $Q$ and subgroup $r$

What is the difference between the basefield modulus $Q$ and a subgroup of prime order $r$? They are all fields, but what is their relevance to the curve they are defined upon? How does this relate ...