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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What does Shor's algorithm tell us about the complexity class of RSA and the DLP?

If quantum computers operate in BQP and (using Shor's algorithm) they are able to factor large integers and break the discrete log problem, what does that tell us about the complexity class of these ...
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Are analog quantum computers a threat to RSA and DLP?

We already know that D-WAVE's "quantum computers" can't really run the Shor's algorithm, because the way they're built doesn't qualify them as universal quantum computers. Now researchers actually ...
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Using Shor's algorithm to solve the discrete logarithm problem

I have read about Shor's algorithm and my understanding is that it can be used to factor large numbers efficiently. Can Shor's algorithm, though, be used to solve this problem: Find the number $e$ ...
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Why $p-1$ needs large factors in discrete logarithm?

In discrete logarithm over cyclic group $\Bbb Z_p$ where $p$ is a prime or $p=q^n$ a prime power it is desired that $p-1$ needs to have large factors except for $2$. What is the consequence even if ...
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How does the Number Field Sieve find the target number for Diffie-Hellman?

I have read some papers relating to the Number Field Sieve, but I could not figure out how this algorithm helps in Logjam, or even what is meant by the number field. What is this? What is meant by ...
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How to calculate the exponent in modular exponentiation?

I have a problem when calculating power in modular, $a^b \bmod c = d$. where we can know values of $a$, $c$ and d, but we don't know values of $b$. example : $29^b \bmod 1024 = 365$. So, how can I ...
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Fixed primes in discrete logarithm

The discrete logarithm problem is to find $z$ when the inputs are $g,h,p$ where $g^z\bmod p$. Supposing if you fix $p$ then does the problem become any easier to attacks and is there a $(\log p)^c$ ...
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77 views

How secure is this logarithmic encryption algorithm?

How secure would the following logarithmic encryption algorithm be when tested under the same conditions as high end encryption algorithms (AES, RSA etc...). Note: For smaller text the text will be ...
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67 views

Calculating the discrete logarithm

I'm given a prime number $p = 1217$ I'm also given the following equations: $$ 40 \equiv \log2 \pmod{64} \\ 63 \equiv \log3 \pmod{64} \\ 13 \equiv \log5 \pmod{64} \\ 13 \equiv \log2 \pmod{19} \\ 10 \...
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Software timing attack using Kocher method

What's the minimum number of random sample points needed in Kocher's timing attack, so that we can determine enough valid measurements of $A_{i,r}$ and $D_{i,r}$? I'm working from this paper: Volker ...
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Sigma protocol: witness hiding

I am working on an assignment and I am stuck with the last part of proving witness hiding for the protocol. I have previously proved it is witness indistinguishable, and it has q (primer number ...
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89 views

Is the Discrete logarithm problem suitable for this pairing scheme?

Let $Ans$ be the product of two pairings : $e(g,h)^{k} \times e(g,h)^{r}=Ans$ If everybody knows only $[g,h,e(g,h)^{k}]$ but $[r,Ans]$ is not known. In the discrete logarithm problem, the user knows ...
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207 views

Discrete logarithm problem is easy in a cyclic group of order a power of two

Let $G=\langle g\rangle$ be a cyclic group of order $2^{k}$ and let $h\in G$. I have read that it is easy to find $\log _{g} h$, but I haven't been able to figure out how. Do you know why this can be ...
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Is original DSA a TEGTSS-I scheme?

Brickell et al. define TEGTSS-I scheme in paper "Design validations for discrete logarithm based signature schemes" In this paper original DSA is generalized as DSA-I variant where $r = g^k \bmod p \...
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134 views

What is the difference between discrete logarithm and logarithm? [closed]

Why discrete logarithm? Is discrete logarithm is part of logarithm? I do not understand the difference between these two concepts.
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56 views

How to compute the discrete logarithm of Diffie-Hellman with a composite modulus?

Imagine $n = pq$ with $p-1 = 2 p_1 p_2$ and $q-1 = 2 q_1 q_2$. I can compute the discrete logarithm of $y = g^x \pmod{n}$ by computing the discrete logarithm of $y$ modulo $p$ and $q$. But then how ...
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Is the complexity of Pollard rho for discrete logartihm really the modulus?

so I'm reading everywhere that the Pollard Rho for Dlog's complexity of $g^a \pmod{n}$ is $\mathcal O(\sqrt{n})$. Shouldn't it be $\mathcal{O}(\sqrt{q})$ with $q$ the order of $g$?
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Why do the subexponential algoriths for the DLP not work for the ECDLP?

Elliptic curve cryptography is much more secure for the same parameters because attacks that work on the DLP do not work on the ECDLP. Why do the attacks fail in the latter case?
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How can prime factor of $q-1$ divide prime $q$?

In this paper - "A Subexponential Algorithm for the Discrete Logarithm Problem", author mentions (page 56), For each $p_l^{e_l} | q$ proceed.... $p_l^{e_l}$ is one of the prime factors of $q-1$...
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56 views

What is base of $\log$ in Subexponential Algo for DLP?

I am currently going through this paper - "A Subexponential Algorithm for the Discrete Logarithm Problem" by Leonard Adleman. On page 56, author mentions that Dixon's algorithm - Asymptotically Fast ...
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Discrete logarithm hash function Exercises

I have a problem with this exercise: Let $G$ be a group of order a prime $q$ and let $g, h$, be two randomly selected elements of $G$, with $g,h\ne 1$. Consider the following hash function on ...
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The difficulty of computing discrete logs

I understand that in Diffie-Hellman it should be hard to compute $a$ given $g$ and $g^a$. In computational Diffie-Hellman, it appears to be hard to compute $(g^{ab})$ from $g^a$ and $g^b$. As for ...
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Recover secret $x$ when $c\equiv m^x \pmod p$ with public $p$ (modified)

Given an encryption system where $c\equiv m^x \pmod p$, $p$ is a known prime, 1. Is it possible to recover $x$ with a known plaintext attack? Given $(p,\text{factorization of }\varphi(p),m,c)$ 2. Is ...
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Convert messages to elliptic curve points [duplicate]

Let $E$ be an elliptic curve; $\alpha,\beta$ two points of $E$; and $a$ a private key such that $\beta=a\cdot\alpha$. We choose random integer $k$ and plain text $x\in E$. Encryption and decryption ...
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468 views

How is El Gamal different from Diffie Hellman Key Exchange [duplicate]

I am Reading RSA and Public-Key Cryptography by Mollin and I can't make out how El Gamal is different from Diffie Hellman Key Exchange. Any thoughts?
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Do I understand (below) why Q = dP is easy while finding d is hard

As we all know for discussion of Dual_EC_DBRG, the point on an elliptic curve Q can be calculated from P and some (large) integer d $Q = dP$ And we know that knowledge of Q and P is not sufficient ...
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Diffie-Hellman: *assumed* largest prime factor of order of $g$?

I've read in several places: some of my confusions about DDH assumption - comment by Thomas Pornin on his own post Is there a key length definition for DH or DHE? - the answer by Tom Leek that ...
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Generalization of the DL-assumption in bilinear group pair

When thinking about a pairing-based cryptographic scheme, I encountered the following problem. Let $e \colon G_1, G_2 \to G_T$ be a Type 3 pairing. Then: Given $P, zP \in G_1$ and $Q, zQ \in G_2$, ...
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Elliptic Curve ElGamal and DSA - smooth group order and element of large prime order

In regular ElGamal and DSA, we choose large primes $p$ and $q$ such that $p\equiv 1\pmod{q}$, and a group element $g$ of order $q$ by computing $a^{(p-1)/q}$ for some random $a$. This is to prevent ...
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Cost of attack on DSA with attack on DLP

Are there any (recent) estimates of cost of attack on DSA by solving the discrete logarithm? I'm especially interested in attacks that use Pollard's rho algorithm. Are there any optimized ...
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Given $g,g^t$ in a cyclic group of order $pq$, is it hard to compute $g^{t^{-1}}$?

Suppose we have a group $G$ cyclic of order $pq$ , where $p,q$ are primes. Let $g$ be a generator of $G$ and $t\in \mathbb{Z}_{pq}$. Having $g$ and $g^t$, it seems to be very hard to find $g^{t^{-1}}$,...
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RSA & DH at risk due to math advances, will this eventually affect elliptic curves too?

I was looking into the predictions by some researchers that RSA and Diffie-Hellman may not be secure in the next few years due to advances in math and being able to calculate the discrete logarithm ...
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Bilinear map assumpion

Is there an assumption that says from a tag $k\cdot e(g,g_1)^{rx}$ ($k,r$ are secret) it is difficult to forge it with some x': $k\cdot e(g,g_1)^{rx'}$, as long as you cannot solve DL in $\mathbb{G}_1$...
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How to find the integer multiplicand from given two points?

I have two points on an elliptic curve $P(x_1,y_1)$ and $Q(x_2,y_2)$ and a scalar value $x$, where $P=x \cdot Q$. What is the best way that I could figure out the value of $x$? Given that I know all ...
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How many bits of an exponent are leaked when doing a powmod?

How many bits of the exponent $x$ are leaked when you calculate and reveal $g^x \pmod p$ for some generator $g$ of $\mathbb Z^∗_p$? The low bit of $x$ is obviously leaked: the low bit equals $1-\frac{...
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258 views

How to prove knowledge of discrete logarithm in a product?

Definitions Suppose I have two large safe primes $p$ and $q$, and a composite number $N=pq$. I have $G$, a large cyclic subgroup of $\mathbb{Z}^{*}_{N}$; $g$ and $h$ are generators of $G$. I commit ...
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Discrete Logarithm problem with inverse

Let $\mathbb G$ be a cyclic group of order $q$. The Discrete Logarithm Problem (DLP) is, given $g, g^x \in \mathbb G$, to compute $x \in \mathbb Z_q $. I'm interested to know if there is a known ...
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State level “Weak Diffie-Hellman” working for SRP too?

I've read about the "Weak Diffie-Hellman" attack (paper, website), where a resourceful entity like a state can pre-compute values for known primes to aid solving the discrete logarithm problem for ...
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Is it possible to generate backdoored DH parameters?

I know it has been already asked and answered whether it's possible to generate weak DH parameters. But "recentely" we experienced the Logjam attack, which makes use of the pre-computation ...
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Testing PRNG quality from ECC public keys?

Having a large set of ECC public keys $P_i = n_iB$ on a fixed curve $E$ over a prime field, is there a way to determine if coefficients $n_i$ were generated using a bad PRNG? In other words, can a ...
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Reduction to cdp,dl or cdh?

Assuming a randomize encoding scheme that takes as input a secret key $\mathsf{sk} \in \mathbb{Z}_p$ for a large prime number $p$. Then the algorithm outputs $g^{\mathsf{sk}x}, x \in \mathbb{Z}_p^*$. ...
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Brute forcing the secret key in Elgamal encryption

Crypto noob here, I am attempting to do this programming challenge. I do not have the secret key that is used to decrypt the message. However, the key is small enough for a brute force approach. I am ...
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Degenerate discrete logarithm in binary field

Given a field $\mathbb{F}_{2^n}$, are there any choices of primitive element $g$ that make the discrete logarithm easier for that generator? That is, are there any degenerate cases? For example, if I ...
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El-Gamal and Lines on Planes

I've been thinking about a geometric picture for El-Gamal. The idea is to understand the set $\{(my^{x},g^x) \mid x \in Z_p\}$ (the set of encryption of $m$ for fixed $g$ and $y$) by taking the $\...
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294 views

Discrete Logarithm (512 bits numbers) - Find exponent parameter

I'm trying to resolve a discrete logarithm equation: $$y = g^x \bmod p$$ Every parameter is a 512-bit number. I know the values for $g$, $y$ and $p$ and I need to find the $x$ value. Finally, I know ...
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Solving discrete logarithm when p is not a safe prime

If you have the cyclic group of integers modulo $p$, where $p$ is not a safe prime, as well as a generator $g$ with which for all factors $q$ of $(p-1)$, $g^{(p-1)/q} \ne 1$, This answer says that ...
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Is $(a,g^{ab})$ computationally indistinguishable from $(a, g^c)$?

From wikipedia, the DDH assumption says,given a cyclic group $G$ of order $q$ with generator $g$, $(g^a, g^b, g^{ab})$ looks like $(g^a, g^b, g^c)$ where $a,b,c$ are randomly and independently chosen ...
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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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Logjam on Elliptic Curves?

I think we're all aware of the Logjam attack. From now on we know that re-using primes for DH is a bad idea. But we also say that elliptic curves are safe from the attack (relying on the NFS), ...