Questions tagged [discrete-logarithm]
In cryptography, a discrete logarithm is the number of times a generator of a group must be multiplied by itself to produce a known number. By choosing certain groups, the task of finding a discrete logarithm can be made intractable.
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Index Calculus for Discrete Logarithm
I'm studying the Index Calculus method for Discrete Logarithm. In the book "Introduction to Cryptography with Coding Theory" by Trappe it's told that, if $$\alpha^k\equiv \prod p_i^{a^i} \mod p$$ ...
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Why does using a prime-order subgroup in DLP improve security?
Let's consider a discrete logarithm
$\beta \equiv \alpha ^{x} \bmod \,\, p$
We can solve it using Pohlig-Hellman algorithm. And, if $p-1 = tq$ where $q$ is a large prime factor, we can avoid any ...
2
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1answer
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Weakness in Pohlig-Hellman algorithm
Let's try to solve a discrete logarithm:
$\beta \equiv \alpha ^{x} \bmod \,\, p$
using the Pohlig-Hellman algorithm. Let's suppose that $p-1=tq$, where $q$ is a large prime number. This means that ...
2
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1answer
39 views
Identity Based Encryption: Known Random Value
Let's consider a situation whereby:
Alice generates a ciphertext c from a message m using Bob’s ID.
An attacker Carol can get c from the open channel. She knows that c is
generated by using ...
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For discrete elliptic curves, can you find G, if you are given b and B?
I know you cannot find $b$ if you are given $B$ and $G$, where $B = [b]G$,
but can you find $G$ given $b$ and $B$?
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Elliptic curve discrete logarithm problem
I'd like to know what is the maximum bits of the finite field that we can solve the ECDLP in a "regular" computer in trivial time. Is there any recent data about this?
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1answer
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How does the order of Q affect the time it takes to solve ECDLP?
I use Sagemath's built-in function discrete_log() to solve ECDLP and according to the documentation it uses Pohling-Hellman algorithm to solve an ECDLP.
This is ...
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1answer
43 views
Does the discrete log assumption hold if k > p
In discrete log cryptosystems like ElGamal it is noted that the "private key" $k$ should be chosen as any element of the group $G$ i.e. $k < p-1$. Does the integrity of the cryptosystem rely on $k$ ...
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1answer
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Retrieving correct ciphertext from additive ElGamal
I have been studying additive ElGamal and I think I have the hang of it except the part where the message $M$ must be retrieved by computing the discrete log of $g^M$.
From what I've read, the ...
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1answer
60 views
How do I interpret the CADO-NFS output for discrete logarithm calculation in GF(p)?
I'm using CADO-NFS to calculate discrete logarithm in a finite field GF(p). The problem is when I type
...
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1answer
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Attacking any one in many public keys
The problem of finding private key from public key is typically studied in the one-key setup: what's the expected cost of breaking one key (e.g. by factoring a public modulus, or solving a discrete ...
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Group in the context of elliptic curve crypto [duplicate]
I understand that the discrete log problem is defined as
$G^y \bmod p = x$
Speaking generally, $G$ here is a generator for the group zp*, where $G$ is able to ...
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Pollard's Kangaroo: How random does $f$ have to be?
I'm implementing Pollard's kangaroo algorithm as described here.
Wikipedia's description of the protocol says that you should have "a pseudorandom map $f:G\rightarrow S$."
Does anyone know what ...
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55 views
DLP instances for given number of congruences
Let $S$ be the set of $k$ congruences of the type $g_{i}^{x_i} \equiv h_i \pmod{n_i}, \quad \forall i,1\leq i \leq k$
What would be the best way to describe such set $S$ with a term in literature. ...
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1answer
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structure in modular arithmetic
I was playing around with modular arithmetic when I noticed something.
\begin{align}
g^k &= a \bmod n\\
g^{k+0.5(n-1)} &= b \bmod n
\end{align}
Then $a + b = n$, so you can also write
$$g^{...
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1answer
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Questions regarding random values in Schnorr authentication
In Chapter 21.3 of Schneier, Applied Cryptography I read the following about the Schnorr Authentication Protocol:
To generate a key pair, first choose two primes, $p$ and $q$, such that $q$ is a ...
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1answer
175 views
Simultaneous Discrete Logarithm with Small Exponent
Given two distinct safe primes $p_1 = 2 \cdot q_1 + 1$, $p_2 = 2 \cdot q_2 +1$, consider the following two instances of the discrete logarithm problem with the same unknown exponent $x$.
$$ g_1^x \...
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1answer
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Proof Dlog is hard in generic group model
I want to know a proof for why the dlog problem is hard in the generic group model. But i can't find any resources online. Can someone please provide me a link or an explanation?
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1answer
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Discrete logarithm problem particularly hard for Schnorr groups?
The Wikipedia article on Discrete Logarithm just states without source:
In some cases (e.g. large prime order subgroups of groups ($\mathbb{Z}_p)^×$) there is not only no efficient algorithm known ...
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1answer
122 views
Finding tetration in a multiplicative group modulo p
I have a variant on the discrete logarithm problem, involving finding tetration in a multiplicative cyclic group of integers modulo a large prime $p$:
$$a = x^x \mod p$$
Where $a$ and $p$ are known, ...
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2answers
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Why are some group representations much easier to compute discrete logarithm for? [duplicate]
The multiplicative group mod $p$ is isometric to the additive group mod $p-1$, yet computing discrete logarithms in the additive group is easy and completing discrete logarithms in the multiplicative ...
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1answer
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Does knowing that the exponent is in a certain range help solving discrete log?
given:
$c=g^i \bmod P$
$g$ generator for group with group size $\varphi(P)$
$g,P,\varphi(P)$,c is known by the attacker
He wants to know $i$.
Now the attacker also knows $j,k$ with $j<i<k$
$k-j$...
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1answer
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How do discrete logarithm with modulo a prime and a non-prime compare?
Let
$c_N = g^i \mod N$ and $N=p \cdot q$
and
$c_P = f^j \mod P$ and $P$ a prime
We assume $N,P$ has the same bit-length. $P$ is the best type of prime you can choose (e.g. safe prime). $N$ is a ...
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1answer
91 views
Elliptic Curve Discrete Log in a Composite Ring
Elliptic curves are usually defined over prime rings (fields), but what if we chose a ring of composite order? Let $n = pq$ for $p,q$ large primes. Say I have elliptic curve $y^2 = x^3 + ax + b$ over ...
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1answer
73 views
Elliptic curve with prime subgroup equal to field size
I am aware that when the equation $\#E(\mathbb{Z}_p) = p$ holds for prime $p$, the elliptic curve is called "anomalous" and is insecure do to "Smart's attack".
Consider the similar case that $E(\...
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1answer
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Understanding the Definition of Most Significant Bits in the Hidden Number Problem
Boneh's and Venkatesan's "Hardness of computing the most significant bits of secret keys in Diffie-Hellman and related schemes" defines the Hidden Number Problem (HNP). The HNP shows that computing ...
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2answers
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Bulletproof - Is it possible to do range proof only from public key?
is it possible to proof x in range (2^1,2^64) from xG=Q with bulletproof or something else only from Q? I’m new in this subject, so thanks before.
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Subexponential algorithms that apply only one of factoring and discrete logarithm?
Shor (quantum polynomial), Number Field Sieve (subexponential), Pollard rho (square root) all have both factoring and discrete logarithm over $\mathbb F_p^*$ variants.
What are the subexponential ...
3
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1answer
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Why is the NIST recommendation for the private key exponent so much smaller than the modulus?
NIST recommends a 256-bit private key exponent for DLP with a 3072-bit modulus. This question answered how the modulus was chosen/calculated, however, why isn't the private key size closer to the ...
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Verify the discrete log of ECDSA is in range
Is it possible to verify the discrete log in elliptic curve is within range without uncovering it? I need to verify that $x$ is within $1$, $2^{64}$ for $xG=P$.
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How can I prove that the plaintext of an elgamal ciphertext is the discrete log of an element?
Is there any (efficient) method to prove that the plaintext of an ElGamal ciphertext is the discrete log of an element?
In the scenario I concerned, I have an El Gamal key pair $(pk, sk) = (g^y, y)$....
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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
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How is the recommended NIST modulus for DLP chosen/calculated?
NIST recommends a 256-bit private key exponent for DLP with a 3072-bit modulus. From this answer it appears that the range of private key numbers is derived by calculating a prime modulus via $2⋅p$ ...
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1answer
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Pollard's Kangaroo— What is the failure probability (assuming random functions)?
I'm reading Pollard's paper on solving the discrete log problem, i.e. to find $x$ given $y = g^x$, where $g$ is a generator of the group.
He has a Kangaroo Algorithm (page 4) which allows you, if you ...
3
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1answer
130 views
What can be said about the self-power map on groups based on DLP?
Introduction
I've been playing with group representation theory some time, concretely representing groups as permutation groups (Cayley's theorem), where the group $G$ has an embedding into the ...
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1answer
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Bit-strength of discrete logarithm for a group of integers modulo a safe prime
Preliminaries
Let $p$ be a safe prime number.
Let $\mathbb{Z}_p^*$ be the multiplicative group of integers modulo $p$.
We have $\mathbb{Z}_p = \{\,a \in \mathbb{Z} \mid 1 \le a \lt p\,\}$ .
Let $g \...
3
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1answer
179 views
What information does $g^x$ reveal about $x$?
Let $p$ be a large prime number. Let $G$ be a subgroup of $\mathbb{Z}_p^*$ with order $q$ - again a large prime. Let $g$ be a generator of $G$.
Consider the following standard protocol for ...
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2answers
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Why do they use elliptic curve instead of circle or other simpler curves? [duplicate]
I am curious why people use elliptic curve in cryptography. I know the main requirement is DLP, but elliptic curve is not the only curve with such property. Some of curves seem to be even simpler.
As ...
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1answer
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What makes the Discrete Logarithm Problem hard?
I am missing a crucial piece of the maths behind the DLP, and I'm hoping someone can give me a really dumbed down answer..
If $h=g^x \bmod p$ and we're working in the group $Z^*_p$, why can I not ...
2
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1answer
63 views
How to determine if $\{n \cdot g^a \mod P\}$ and $\{m \cdot g^a \mod P\}$ generate the same sets? (set size < $P-1$)
given some examples $k_{n_i},k_{m_i}$ out of each value set:
$k_{n_i} \in \{n \cdot g^a \mod P\, \forall a \in \mathbb{N}<P\}=G_m$
$k_{m_i} \in \{m \cdot g^a \mod P, \forall a \in \mathbb{N}<P\}...
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Is there a fast way to solve $k = n \cdot g^a \mod P$? (get $a$ for unknown $n$)
Would a factor besides the normal discrete logarithm problem increase or decrease the solving time?
$k = n \cdot g^a \mod P$
with given $k,g,P$ and the knowledge $P= 2 \cdot N \cdot f+1$, while $...
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Is there any property of the product you can predict before using $n$-times generator $g$ $\mod P$? Can any $n$'th element have a certain property?
Given a value $v$ which is in same group as the generator $g$ modulo prime $P$. The group size is a prime $s$.
$v = g^a \mod P$
Only known values are $v,g,P,s$. Some (possible) computation of other ...
0
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1answer
34 views
Which is the fastest way to find a member of a subgroup with known size modulo prime $P$, with know factorization of $P-1$?->$x$ with $x^s \mod P = 1$
Assuming you know the factorization of used prime $P-1$
$P-1 = s \cdot f_2\cdot f_3...f_i$
Now you want to find a member of a subgroup $\mathbb{Z}_s$.
This means any $x$ with
$x^s \equiv 1 \mod P $
...
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1answer
321 views
How safe is a prime with $P=2 \cdot Q \cdot R \cdot S \cdot t+1$ for discrete logarithm? How to enhance/compare?
To get some certain properties for my use case I need a prime $P$ which has the form:
$P=2\cdot Q \cdot R \cdot S \cdot t+1$
with $Q,R,S,t$ primes as well.
Why that form - Use case
Together with ...
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1answer
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why a group used in cipher based on DLP must be Abelian group?
I can't understand it because $(g^x)^y=(g^y)^x$ in nonabelian group too.
thank you very much for read my question
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Is there a way to determine the number of subgroups (with size $s$) while computing $A^b \bmod P$? Constructing a $P$ with $n$ times size $s$?
If you compute $A^b \bmod P$ for all $b$ the set of results $R$ depend at $A$ (and $P$).
$R = \{A^b \bmod P, \forall \space b \in \mathbb{N}\}$
In case $R$ contain all numbers from 1 to $P-1$, it ...
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1answer
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Could Diffie-Hellman protocol serve as a zero-knowledge proof of knowledge of discrete logarithm?
The Schnorr identification protocol is widely recognized as the simplest ZKPoK of the discrete logarithm (more clearly, Honest-Verifier ZKPoK).
However, I can't figure out why this simple protocol, ...
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1answer
55 views
Does a different exponent and base but same key help to resolve discrete logarithm?
E.g.:
$k = N^a \mod P$
The attacker knows the prime $P$ and $N$, which is also a prime and
(1.) prime root of $P$ or
(2.) has a cycle size of $s$, so $1 = N^s \mod P$, (and for $\forall s'<s$, $...
