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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What is so special about elliptic curves?

There seems to be sources like this, this also, and some introductions that discuss elliptic curves in general and how they're used. But what I'd like to know is why these particular curves are so ...
16
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3answers
1k views

How robust is discrete logarithm in $GF(2^n)$?

"Normal" discrete logarithm based cryptosystems (DSA, Diffie-Hellman, ElGamal) work in the finite field of integers modulo a big prime p. However, there exist other finite fields out there, in ...
14
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1answer
570 views

Do recent announcements about solving the DLP in $GF(2^{6120})$ apply to schemes proposed for cryptographic use?

A recent paper by Göloğlu, Granger, McGuire, and Zumbrägel: Solving a 6120-bit DLP on a Desktop Computer seems to "demonstrate a practical DLP break in the finite field of $2^{6120}$ elements, using ...
12
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5answers
6k views

Galois fields in cryptography

I don't really understand Galois fields, but I've noticed they're used a lot in crypto. I tried to read into them, but quickly got lost in the mess of heiroglyphs and alien terms. I understand they're ...
12
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2answers
2k views

Design properties of the Rijndael finite field

So we've already had a question on replacing the Rijndael S-Box. My question is - can we use a different finite field other than the one given by $x^8 + x^4 + x^3 + x + 1$ in $GF(2^8)$. In other ...
12
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1answer
440 views

Security of pairing-based cryptography over binary fields regarding new attacks

In the last week, the discrete logarithm problem was broken for the binary fields $\mathbb{F}_{2^{(14 \times 127)}}$ and $\mathbb{F}_{2^{(27 \times 73)}}$. Pairing-based cryptography using binary ...
12
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1answer
591 views

Choice of multiplication polynomial in Rijndael s-box affine mapping

The Rijndael specification details the design choices for the s-box in section 7.2. They describe the choice of affine mapping as follows: We have chosen an affine mapping that has a very simple ...
10
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3answers
412 views

Mapping between subgroups and the integers

This question is a companion to the equivalent question on elliptic curves. Preliminaries Diffie-Hellman, Elgamal, DSA, etc. are examples of protocols that work in the integers modulus a large prime ...
10
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2answers
380 views

How to determine the order of an elliptic curve group from its parameters?

Let $\quad E:\; y^2 = x^3 + ax + b \quad$ be an elliptic curve defined over a finite field $\mathbb F_q$ where $q = p^n$, $a,b \in \mathbb F_q$ and $p \neq 2, 3$. By Hasse's theorem we know that the ...
8
votes
1answer
1k views

Necessity for finite field arithmetic and the prime number p in Shamir's Secret Sharing Scheme

Shamir's original paper (PDF, 197kb) describing a threshold secret sharing scheme states: To make this claim more precise, we use modular arithmetic instead of real arithmetic. The set of ...
8
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1answer
198 views

Solving hard problems in $\mathbb Z_{p}^{*}$ when $\mathbb p$ is close to $\mathbb 2^{n}$

Suppose, for some security parameter $n$ you choose a prime $p$ such that $p = 2^n+c$ for some relatively small $|c| < 2^m << 2^n$. I have seen such primes being called Pseudo-Mersenne Primes ...
7
votes
0answers
79 views

Default algorithm for scalar multiplication of elliptic curve points by the MIRACL Library

What is the default algorithm used by the MIRACL-Library [1] for elliptic curve cryptography systems to perform scalar-point multiplication with curves of Weierstrass form satisfying the equation : ...
6
votes
2answers
282 views

Secure degree reduction for Shamir's secret sharing

I understand the basic Shamir Secret Sharing protocol, and when two shares are multiplied, the degree of the polynomial increases. I've seen in a number of papers a reference to a degree reduction ...
6
votes
1answer
7k views

Multiplication/Division in Galois Field (2^8)

I'm attempting to implement multiplication and division in $GF(2^8)$ using log and exponential tables. I'm using the exponent of 3 as my generator, using instructions from here. However I'm having ...
6
votes
1answer
339 views

Why use variable p, q, g for Diffie-Hellman?

In the book Cryptographic Engineering, it is said that fixing p, q, g for a key negotiation protocol based on DH is a bad idea (page 228 1st ed). But allowing for flexible p, q and g requires a lot ...
5
votes
1answer
1k views

AES mixcolumn stage

I'm studying AES, and am having problems with the "mixcolumn" stage. I read about finite fields, but I still don't know. How do I construct $GF(2^8)$? ...
5
votes
1answer
399 views

What do recent announcements about solving the DLP in $GF(2^{6120})$ mean for RSA

After just reading the post Do recent announcements about solving the DLP in $GF(2^{6120})$ apply to schemes proposed for cryptographic use? I was a bit confused. DSA, ElGamal and others are based on ...
4
votes
3answers
229 views

Algorithm for computing square roots in $GF(2^n)$

Short question: is there an algorithm for efficiently computing square roots in $\mathbb{F}_{2^n}$?
4
votes
2answers
193 views

In a group, is it hard to calculate the base $g$ given $g^a$ and $a$?

Discrete logarithm, that is: calculate $a$ given $g$ and $g^a$, is assumed to be a hard problem in some groups. Is it also hard to calculate $g$ given $g^a$ and $a$?
4
votes
2answers
1k views

Why does Shamir's Secret Sharing Scheme need a finite field?

I read ampersand's question "Necessity for finite field arithmetic and the prime number p in Shamir's Secret Sharing Scheme", where he asked why Shamir's Secret Sharing Scheme uses arithmetic in a ...
4
votes
2answers
196 views

What is this “finite field cryptography”?

See RFC 5931 § 2.2.1 which talks about "finite field cryptography" as opposed to elliptic curve cryptography and it looks like it is describing the Diffie-Hellman protocol. But Diffie-Hellman is not a ...
4
votes
3answers
566 views

Does RSA operate over a Finite Field (Galois Field)?

Is it correct to say that RSA operates over a Finite Field (Galois Field)? In this case GF(p)? I do understant that the modulo in RSA is not itself a prime number, but all the operations ...
4
votes
3answers
140 views

Calculating $\mathbb F_{p^2}$-rational points of an elliptic curve defined over $\mathbb F_p$

How can I calculate points on an elliptic curve defined over $\mathbb F_p$, for example $y^2 \equiv x^3 + 1 \pmod p$, with coordinates in $\mathbb F_{p^2}$? (points might have complex number format in ...
4
votes
1answer
668 views

Understanding Feldman's VSS with a simple example

I'm trying to understand Feldman's VSS Scheme. The basic idea of that scheme is that one uses Shamir secret sharing to share a secret and commitments of the coefficients of the polynomial to allow the ...
4
votes
1answer
307 views

Best choice of finite field for AES on a 4-bit microcontroller?

As the finite field of $GF(2^8)$ are isomorphic to $GF((2^4)^2)$, $GF((2^2)^4)$ and $GF(((2^2)^2)^2)$, which of the fields is best suited and most efficient for 4-bit MCU and why? Would it be ...
3
votes
3answers
1k views

Complexity of arithmetic in a finite field?

I am wondering what the complexities are of adding/subtracting and muliplying/dividing numbers in a finite field $\mathbb{F}_q$. I need it to understand an article I am reading. Thank you
3
votes
1answer
56 views

Do $v_1=\alpha\cdot r_1$ and $v_2=\alpha\cdot r_2$ leak information about $\alpha$

Please consider we have finite field $\mathbb{F}_p$ for large prime number $p$. We have a fixed field element $\alpha$. By $r_i\leftarrow \mathbb{F}_p$ we mean we pick $r_i$ uniformly random from the ...
3
votes
2answers
141 views

What is the difference between the standard representants of $\mathbb Z/q\mathbb Z$?

The symbol $\mathbb Z/q\mathbb Z$ (given that $q$ is prime) represents the prime field $\mathbb Z_q$. Basically, the elements of this field are represented by $\{0, 1, \dots, q-1\}$, let's call this ...
3
votes
2answers
847 views

Additive ElGamal cryptosystem using a finite field

I'm trying to implement a modified version of the ElGamal cryptosystem as specified by Cramer et al. in "A secure and optimally efficient multi-authority election scheme", which possesses additive ...
3
votes
2answers
204 views

How to use the Extended Euclidean algorithm to invert a finite field element?

I was doing some practice on cryptography (I'm new to this topic) and was wondering what the following question even means or what it is asking me to find. I do know how to do Extended Euclidean ...
3
votes
2answers
114 views

Implementation of ECC over binary field

I am supposed to implement ECC over binary field (in C++) for equations of the type - $y^2 + xy = x^3 + ax + b$, as my project. I wish to include the following features : The user will enter a prime ...
3
votes
1answer
111 views

Does $i^n=j^n$ for $i, j \in GF(2^q)$ and $i \neq j$ for some $n<2^q-1$

Let $i, j \in GF(2^q)$ and $i \neq j$ and $i,j\neq0$. Is that possible that $i^n=j^n$ for some $n$ such that $0 < n < 2^q-1$? I am looking for a proof if the answer is no, or for a method to ...
3
votes
2answers
213 views

Choice of reduction polynomial in Whirlpool's internal cipher

Whirlpool is an interesting little hash function in the Miyaguchi-Preneel family. In my mind, it's most interesting feature is the design of internal cipher W, where the distinction between key and ...
3
votes
2answers
489 views

Solving Quadratic equations in Galois Field (2^163)

Hello I am working on implementing a message to elliptic curve point mapping hardware circuit I have done some research and found out the koblitz mapping method: I will be using a field of binary ...
3
votes
2answers
57 views

Non primitive lfsr sequence

Given a non-primitive LFSR sequence (i.e number of states is less than $2^n - 1$); how do we find out the the characteristic polynomial? Will Berlekamp-Massey algorithm work in this case? for ...
3
votes
1answer
74 views

Schnorr signature security level as compared to AES/RSA

As per the Schnorr's original paper (1991), The Security Complexity $2^t$: We wish to choose the parameters $p$, $q$ so that forging a signature or an authentication requires about $2^t$ steps by ...
3
votes
1answer
61 views

converting finite field elements to octet strings

I need to convert elements of the finite field $GF(p^k)$, where $p$ is an odd prime, to octet strings. To be more precise, I want to include elliptic curve points over $GF(p^2)$ in a Subject Public ...
3
votes
1answer
673 views

Why does FIPS 186-4 require specific sizes for keys?

In FIPS 186-4, page 32, about FFC crypto it is required that the length of $p$ will be exactly 1024 bit and the length of $q$ will be exactly 160 bit. Why is the requirement not stated in terms of ...
3
votes
1answer
178 views

Sextic twist optimization of BN pairing - cubic root extraction required?

I found the following paper really interesting: http://www.researchgate.net/publication/220378229_A_family_of_implementation-friendly_BN_elliptic_curves/file/79e4150b3a773beecd.pdf It allows ...
3
votes
1answer
110 views

Fast reduction in $GF(2^{128})$ using x86 `PCLMULQDQ`

Modern x86 CPUs support the PCLMULQDQ instruction, which does an XOR-multiply of two 64-bit numbers instead of an add-multiply (i.e. typical arithmetic ...
3
votes
0answers
35 views

Seeking an implementation of the Satoh algorithm for elliptical curve point counting

I would be very grateful if someone has an implementation of the Satoh algorithm (Fast Elliptic Curve Point Counting). Can someone point me to practical algorithm implementations or provide some ...
2
votes
3answers
157 views

How does $g$ being a generator imply Diffie-Hellman's correctness?

In the Diffie-Hellman protocol for key exchange over an unsecured channel, we choose $p$ and $g$, where $g$ is a generator of $\mathbf{Z}_p^*$. However I want to know why this assumption makes the ...
2
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3answers
296 views

Generating bilinear pairing parameters - running time of finding member of p-torsion group

Update: Question completely rephrased. I want to create the parameters for a bilinear pairing (the Tate pairing in this case). In case you're interested I'm following this thesis, specifically the ...
2
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1answer
552 views

Solve a system of non linear equations over GF

I have the following set of equations: $$M_{1}=\frac{y_1-y_0}{x_1-x_0}$$ $$M_{2}=\frac{y_2-y_0}{x_2-x_0}$$ $M_1, M_2, x_1, y_1, x_2, y_2,$ are known and they are chosen from a $GF(2^m)$. I want ...
2
votes
1answer
907 views

How Multiplication Table is generated for GF(2^2) field

I was unable to solve the multiplication table given in the book for $\mathrm{GF}(2^2)$.However, I have managed to solve the addition table. Acoording to the Book multiplication is the AND operation, ...
2
votes
1answer
54 views

Question about block erasure codes

I have a question about linear block erasure codes that are described in this paper. I briefly describe the idea behind the linear erasure codes and then I ask my question. Given a set $d=\langle x_i ...
2
votes
1answer
140 views

Cryptographic Arithmetic Toolbox/Software [closed]

This term I have many cryptography courses treating finite fields. I was wondering if there is any good software that could help doing basic operations in galois fields etc. I already googled but ...
1
vote
1answer
231 views

Choosing finite field size in Shamir's Secret Sharing Scheme

The Wikipedia article on Shamir's Secret Sharing says to that to have information theoretical security the splitting algorithm should be evaluated using finite field arithmetic on the field ...
1
vote
1answer
449 views

Why do we use 1024 / 160 bit primes in DSA?

I am looking at DSA's parameter generation and don't understand why for $p$ a 1024 bit prime is needed if $q$ is chosen as a $160$ bit prime. I thought that the security of DSA relates on the discrete ...
1
vote
1answer
369 views

Solving a discrete logarithm using GDlog

I am trying to calculate an $x$, such that $t = g^x \pmod p$ in order to crack a weak ElGamal encryption for university. I found GDlog, but I cant figure out how I can use the input to calculate my ...