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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Is there a simple zero knowledge proof of $x$ for $b=x^x\pmod p$?

We have a multiplicative cyclic group $G$ which is a subgroup of $(\mathbb{Z}/n\mathbb{Z})∗$. There are two parties, Alice and Bob: If: Alice knows: $b$ and $x$ such that $x^x = b$; Bob knows: $b$. ...
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Reliability of a single-pass deniable authentication protocol?

I look for one-pass deniable authentication protocol with a short message payload for my project and find a solution: ...
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88 views

Is it possible to determine or estimate the period for Blum-Micali PRG?

The Blum-Micali is a cryptographically secure pseudorandom number generator. The construction (from wikipedia): Let $p$ be an odd prime, and let $g$ be a primitive root modulo $p$. Let $x_0$ be a ...
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138 views

Why is “multiplying” $g^x$ and $g^y$ not possible?

The computational Diffie-Hellman problem states that for a cyclic group $G$ of order $p$ and a generator $g$, it is hard to find the value $g^{xy}$ given only $g^x$ and $g^y$ (but easy if either $x$ ...
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Discrete log problem, when we have many examples

Suppose I have many instances of the discrete log problem, all using the same unknown exponent. Is this problem easier than the standard discrete log problem? Oh, heck, I should be more precise. ...
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why it is not possible to find inverse of its own in diffie-hellman protocol? [duplicate]

In DH-Protocol bob calculate $R=g^b$, R as public parameter and Alice calculate $S=g^a$ & common key($K=g^{ab}$) using value $R^a$ then bob calculate common key $K= S^b=g^{ab}$ It can be ...
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75 views

Subexponential algorithms for DLP in $\mathbb{Z}_s \times \mathbb{Z}_t$

Consider the accepted answer to the question: Why are elliptic curves better than cyclic groups? It seems to suggest there are subexponential algorithms (i.e., algorithms with running time $$ ...
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Finding an x such that xP = (11,44) on an elliptic curve

Given the elliptic curve $$E:y^2 = x^3+17x+5 \mod 59$$ with point $P = (4,14)$, how do I find $x$ such that compute $x\cdot P = (11,44)$ Is there a mathematical method to compute $x$, or do I ...
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1k views

How to calculate the time it'll take to crack RSA or DH?

Sometimes the easiest way to describe security of a type of cryptography is to say that "the time it takes to solve for an x-bit key would be y years". How would one go about doing such a calculation ...
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1answer
147 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 ...
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68 views

Generating a valid signature on El-Gamal without knowing the private key

Suppose we are given $p$, the large prime, $g$ which is the primitive root for $p$, $b$ which is calculated as $b=g^x$ mod $p$ where $x$ is the private key and $0<x<p-1$. Also suppose we know ...
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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 ...
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274 views

Safe generator for ElGamal signature

What are the properties a generator $g$ should have to be secure for ElGamal signatures (original scheme)? I am aware that it is poorly chosen and not secure when $g|p-1$ or $g^{-1}|p-1$, where $p$ ...
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1answer
242 views

How to protect from Silver–Pohlig–Hellman algorithm

I read that Silver–Pohlig–Hellman algorithm solves the discrete logarithm with prime module $p$ in $O(\log^2(p))$ if $p-1$ is a smooth number. This seems pretty fatal for cryptography, since it is a ...
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Finding if exponent share is present in dlog instance [closed]

Let $g$ be a group generator of prime order $q$. Suppose we are given two elements $g^y$ and $x_1$. Can we find out if $y=x_1+x_2$ for some $x_2$? Thanks
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1answer
118 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 ...
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71 views

Generating Diffie-Hellman parameters efficiently

I am working on an Android project for school and I am supposed to do a DHKE (Diffie Hellman Key Exchange). Everything works well. The problem is that it takes a lot of time (really a lot) to ...
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186 views

Why can ECC key sizes be smaller than RSA keys for similar security?

I understand how ECC is based on the discrete log problem and RSA on integer factorization. I've read several references that show how a solution to either of these problems can typically be adapted ...
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Hardness of finding mutual discrete logarithms of small generators in $\mathbb{Z}_p$

Suppose you want to select a prime $p$ such that finding e.g. $log_2(3)$ in $\mathbb{Z}_p$ is expected to be either at least as hard as the general Discrete Logarithm Problem in $\mathbb{Z}_p$, or at ...
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Factorization or discrete logarithm is difficult for an attacker?

I have read that difficulty in breaking many algorithms are based either on Factorization or discrete logarithm. I am reading about schemes that are similar to RSA which make use of integer ...
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What is the relation between Discrete Log, Computational Diffie-Hellman and Decisional Diffie-Hellman?

How are the three problems Discrete Logarithm, Computational Diffie-Hellman and Decisional Diffie-Hellman related? From my understanding, since the Discrete Log (DL) Problem is considered hard, then ...
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118 views

Recovering the random number r

For a padded message, M, using the El Gamal encryption schema, how can we determine the random number $r$, when we are given $p$, the prime number, $g$ which is the primitive root of $p$, $b$ and $x$ ...
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Why is the discrete logarithm problem assumed to be hard?

This might be a quite stupid question: since a naive brute force algorithm to solve the discrete logarithm problem will only take O(n) time for a group G with order ...
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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 ...
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Is there a practical zero-knowledge proof for this special discrete log equation?

We have a multiplicative cyclic group $G$ with generators $g$ and $h$, as in El Gamal. Assume $G$ is a subgroup of $(\mathbb{Z}/n\mathbb{Z})^*$. There are two parties, Alice and Bob: Alice knows: ...
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93 views

Pollard’s Rho Method

I can't get my head around Pollard’s Rho Method for solving discrate log problem I have read in a book: The basic idea is to pseudorandomly generate group elements of the form α^i · β^j ...
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143 views

Attacks against El Gamal private key

El Gamal encryption involves picking $(p,g,b)$ which is our public key. We compute $b=a^x$ $mod$ $p$. Here, $x$ is the private key which we don't know. What are some efficient and strong algorithms ...
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iterated discrete log problem

Consider the following problem: given $g_1 \ldots g_i,h_1 \ldots h_i \in G$, $\forall i$ find $x_i$ such that $g_i^{x_i}=h_i$ For $i=1$ this is the discrete log problem and is assumed to to have ...
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121 views

Difference between Pedersen commitment and commitment based on ElGamal

Does any of you know what is the difference between the Pedersen commitment and the commitment that uses the ElGamal encryption scheme? For the sake of completeness, I recall what both of them look ...
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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 ...
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Are there security issues with discrete logarithm keys not being uniformly distributed?

Generally, algorithms based on discrete logarithm specify that private keys are chosen as scalars between 1 and the order of the group (denoted $q$ here). For instance IEEE P1363 and FIPS 186-3 both ...
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Is it possible to decide the base of a discrete logarithm?

Given $R$, a prime $p$ and two bases $g_1$ and $g_2$, is it possible to decide if $R = g_1^r$ mod $p$ or $R = g_2^r$ mod $p$ without knowing $r$?
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Are there asymmetric cryptographic algorithms that are not based on integer factorization and discrete logarithm?

In the computer security class (in which cryptography is a big chapter) that I took, I remembered the professor said about current asymmetric cryptography algorithms are based on integer factorization ...
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Reuse of a DH / ECDH public key

I was wondering whether it is safe to use the same DH or ECDH key pair in more than one key agreement, particularly if these public keys are in a public registry. These public keys could be used by ...
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206 views

Decrypting without using the private key

Let $g$ be a generator of a multiplicative group $G$ of order $q$, $x$ be a private key, and $h=g^x$ be a public key of an exponential ElGamal cryptosystem. Given a ciphertext $c$ produced as the ...
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178 views

Relationship between Elliptic Curve Discrete Log, Integer Discrete Log, and Integer Factorization

I am trying to look into a relation between the following three problems which are widely used to build public crypto systems: Integer Discrete log Elliptic Curve Discrete log Integer Factorization ...
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263 views

Perfect zero knowledge for the Schnorr protocol?

Can somebody explain (or point to a reference) why the Schnorr protocol cannot be proved zero knowledge?
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181 views

some of my confusions about DDH assumption

The wiki defines the decisional Diffie–Hellman assumption as follows: Decisional Diffie–Hellman assumption Consider a (multiplicative) cyclic group $G$ of order $q$, and with generator $g$. The DDH ...
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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 ...
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135 views

Proof of correctness of a homomorphic ElGamal sum

Let's suppose we are using the exponential ElGamal as a public-key encryption scheme, so that we encrypt $g^m$ instead of $m$, for some generator $g$. Let $x$ be the private key, and $h=g^x$ be the ...
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Could this be a valid variation of the Schnorr protocol?

The Schnorr protocol is a 3-steps proof of knowledge of a discrete logarithm, whose interactive version works as follows. Let $p$ and $q$ be two public primes, such that $q \mid (p-1)$, and let $G$ ...
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Why is the Pedersen commitment computationally binding?

This is how the Pedersen commitment seems to work: Let $p$ and $q$ be large primes such that $q \mid (p-1)$, let $g$ be a generator of the order-$q$ subgroup of $Z_p^{\star}$. Let $a$ be a random ...
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Finite fields in elliptic curve

I have an elliptic curve defined over finite field where $S_1=aP$ . Is it valid to say that $S_1P$ can also be computed. $P$ is the generator of the group. What my real question is that. Should '$a$' ...
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Discrete log problem with modulus prime

I am a bit confused on the hardness of the discrete logarithm problem. Does it become intractale only when it is mod n, where n is a large composite number (Like RSA key). What about if it is mod a ...
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What is the difference between Shor's algorithm for factoring and Shor's algorithm for logarithm

There is a paper from Peter W. Shor from 1994: http://www.csee.wvu.edu/~xinl/library/papers/comp/shor_focs1994.pdf "Algorithms for Quantum Computation: Discrete Logarithms and Factoring", and I have a ...
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221 views

Finding where I am in a linear recurrence relation

Suppose I have a linear recurrence relation $$a(n) = c_1 a(n-1) + \dots + c_k a(n-k) + d,$$ where the constants $c_1,\dots,c_k,d$ are given and the initial values $a(0),\dots,a(k-1)$ are given as ...
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391 views

Reduction of Integer factorization to Discrete logarithm problem

I was reading Eric Bach paper entitles "Discrete logarithms and factoring", in which he states the following reductions: solving the integer factorization problem suffices to solve the discrete ...
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Hardness of CDH in different groups

What is the difference of the CDH problem in different groups? In particular, given a group $\mathbb{G}_1$ of order $q$ that is a subgroup of $\mathbb{Z}_q^*$, $q$ prime, and another group ...
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194 views

Efficiency of finding sub group order vs factorization

Suppose you got a prime $p = 2\mathbb\Pi_{i=0}^{n-1}q_i+1$, where $2^{k-1} \lt q_i \lt 2^k$ for some $k$ and all $0 \le i \lt n$, and that you also got a generator $g$ of one of the prime order sub ...
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Do Gap-CDH groups exist?

A Gap-CDH group is such that, given group elements $g, a = g^x, b = g^y$, it is hard to compute $g^{xy}$, but, given a group element $c$, easy to verify if $c = g^{xy}$. While such groups have been ...