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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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 ...
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648 views

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 safe primes $p=2^k \pm s$ with $s$ small less recommandable than others as a discrete log modulus?

I take the definition of safe prime as: a prime $p$ is safe when $(p-1)/2$ is prime. Safe primes of appropriate size are the standard choice for the modulus of cryptosystems related to the discrete ...
11
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385 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 ...
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161 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 ...
8
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331 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 ...
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What should be the size of a Diffie-Hellman private key?

I'm implementing the SRP-6 protocol, which relies on discrete logarithms for it's security (essentially Diffie-Hellman). The RFC documents state: The private values $a$ and $b$ SHOULD be at least ...
7
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518 views

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 ...
7
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319 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 ...
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1answer
419 views

Why are elliptic curve variants of RSA “chiefly of academic interest”?

Yesterday I was thinking about elliptic curve variants of popular protocols/algorithms (ECDH, ECES[1], etc) and the thought occured that I had never seen an elliptic curve variant of RSA. My ...
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Would the ability to efficiently find Discrete Logs have any impact on the security of RSA?

This answer makes the claim that the Discrete Log problem and RSA are independent from a security perspective. RSA labs makes a similar statement: The discrete logarithm problem bears the same ...
6
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172 views

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 ...
6
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142 views

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 ...
6
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1answer
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 ...
6
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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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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 ...
5
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270 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 ...
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219 views

Why would this method of discrete logarithm finding not work?

Say we do know $b$ but not $k$, and are given $g$ such that $g\equiv b^k\pmod p$. And say there exist factors $E = e + m'p$ ($e \equiv b^i \bmod p$) and $F = f + m''p$ ($f \equiv b^j \bmod p$) of $g$. ...
5
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426 views

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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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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273 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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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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313 views

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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121 views

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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1answer
128 views

How much do we trust KEA1 Assumption?

Let $$(g,h=g^s,q)$$ be a tuple such that $g$ is a generator for a group $\mathbb{G}$ of ord $q$ and $s$ is uniformly random in $\mathbb{Z}_q$. The KEA1 Assumption saies that for any adversary ...
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How to practically find solutions to a discrete logarithm?

Are there any ongoing or current practical attempts to solve instances of the discrete logarithm problem of the order of magnitude used in cryptographic applications, for example with a 256 bit ...
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250 views

A discrete-log-like problem, with matrices: given $A^k x$, find $k$

Let $p$ be a large prime; we will work in $GF(p)$. Let $A$ be a $n\times n$ matrix. Also, let $x$ be a $n$-vector and $k$ a positive integer. Suppose we are given $p$, $A$, $x$, and $y$. The goal ...
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How can I solve the discrete logarithm modulo 2q+1 if I can solve it in the subgroup of order q?

As part of my cryptography course I came across an exercise that neither me or my friends could figure out. The problem statement is as follows: Let $p$ be a large prime of the form $p = 2q + 1$ ...
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262 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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261 views

Trying to better understand the failure of the Index Calculus for ECDLP

So I'm going to give you guys my understanding and then if you would be so kind as to tell me where I'm off the mark (hopefully I'm not completely wrong). So basically the index calculus for the ...
4
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1answer
96 views

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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302 views

An Elliptic curve cryptography implementation which can be terminated

I'd like to have an implementation of elliptic curve cryptography along the lines of secp256k1 which is secure until some information is published after which it is broken. One idea would be to use ...
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RSA security assumptions - does breaking the DLP also break RSA? [duplicate]

Possible Duplicate: Would the ability to efficiently find Discrete Logs have any impact on the security of RSA? I'm wondering if breaking the DLP, that is the basis for ElGamal and DSA, ...
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125 views

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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210 views

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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1answer
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Why is the discrete log problem easy when the exponent comes from a binomial distribution?

I read in http://epubs.surrey.ac.uk/7219/2/esorics06.pdf that in exponential El Gamal the discrete log problem for recovering $m$ from $g^m$ can be made tractable when $m$ is drawn from a binomial ...
3
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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 ...
3
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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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How hard are discrete logarithms problems in $\mathbb Z^{*}_{n}$ and $\mathbb Z^{*}_{n^2}$, where $n$ is the RSA $n=pq$

Use the notations form the Wikipedia article Paillier Cryptosystem , assume that the chipertext $c$ and $c^{\lambda} \mod n^2$ are both given, is it possible to compute $\lambda$ easily?
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Why can't I break ElGamal encryption by brute-forcing the secret exponent?

I am doing a course on cryptography on coursera and one of the topics covered was the ElGamal Encryption system. I am using the terms as defined in Wikipedia. Alice publishes $g$ and $g^x$. ...
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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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Probability that an attacker wins the discrete logarithm game when exponents are drawn from a subset

Suppose $g$ is a generator of an order $p$ cyclic group in which discrete logarithm is hard and $p$ is a prime (i.e., given $g^x$ for a random $x \in \{0,1,\ldots, p-1\}$, it is hard to recover $x$ ...
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Chosen ciphertext insecurity in an ElGamal variant

I'm trying to prove something and if I can show that there is a simple way to calculate $(g^a \bmod p)^k$ if I know both $g^k \bmod p$ and $g^a \bmod p$, then (I think) it will help me prove it, but ...
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1answer
185 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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How to test if a number is a primitive root?

How to test if a number is a primitive root, assuming the modulus is a prime? And if not? Is it not enough if the number is relatively prime to the modulus or prime?
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187 views

Is there a cumulative commitment scheme?

For a certain application I need a commitment scheme where each user could make a commitment, and a single verification operation could verify all the commitments simultaneously, faster than single ...
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1answer
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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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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186 views

Stream ciphers based on discrete logs

Blum Blum Shub is a stream cipher that is provably reducible to the difficulty of factoring integers. I'm wondering whether there is a similar construction for discrete logs? For example, I could ...
2
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ECC algorithm pollard's $\rho$ complexity

One of the methods to break a ECDLP is Pollard's rho algorithm. When ECDLP is defined over a finite field $F_p$, and given a relation $S=w.T$, where S and T are a member of $F_p$. Then ECDLP is to ...