Related papers: Birthday attack to discrete logarithm
For $q$ a prime power, the discrete logarithm problem (DLP) in $\mathbb{F}_{q}$ consists in finding, for any $g \in \mathbb{F}_{q}^{\times}$ and $h \in \langle g \rangle$, an integer $x$ such that $g^x = h$. We present an algorithm for…
In this note, we describe a simple approach to obtain a differentially private algorithm for k-clustering with nearly the same multiplicative factor as any non-private counterpart at the cost of a large polynomial additive error. The…
We present a new algorithm for locating a small cluster of points with differential privacy [Dwork, McSherry, Nissim, and Smith, 2006]. Our algorithm has implications to private data exploration, clustering, and removal of outliers.…
In this paper homomorphic cryptosystems are designed for the first time over any finite group. Applying Barrington's construction we produce for any boolean circuit of the logarithmic depth its encrypted simulation of a polynomial size over…
In this short note, we develop a novel idea of a bilinear cryptosystem using the discrete logarithm problem in matrices. These matrices come from a linear representation of a finite $p$-group of class 2. We discuss an example at the end.
The ability to hide information from unauthorized individuals has been a prevalent issue over the years. Countless algorithms such as DES, AES and SHA have been developed. These algorithms depend on varying key length and key management…
Auditing mechanisms for differential privacy use probabilistic means to empirically estimate the privacy level of an algorithm. For private machine learning, existing auditing mechanisms are tight: the empirical privacy estimate (nearly)…
We describe a novel type of weak cryptographic private key that can exist in any discrete logarithm based public-key cryptosystem set in a group of prime order $p$ where $p-1$ has small divisors. Unlike the weak private keys based on…
Differential privacy is a recent notion of privacy for statistical databases that provides rigorous, meaningful confidentiality guarantees, even in the presence of an attacker with access to arbitrary side information. We show that for a…
Consider updates arriving online in which the $t$th input is $(i_t,d_t)$, where $i_t$'s are thought of as IDs of users. Informally, a randomized function $f$ is {\em differentially private} with respect to the IDs if the probability…
We heuristically show that Shor's algorithm for computing general discrete logarithms achieves an expected success probability of approximately 60% to 82% in a single run when modified to enable efficient implementation with the…
Private computation is a generalization of private information retrieval, in which a user is able to compute a function on a distributed dataset without revealing the identity of that function to the servers. In this paper it is shown that…
Cryptographic systems are derived using units in group rings. Combinations of types of units in group rings give units not of any particular type. This includes cases of taking powers of units and products of such powers and adds the…
A new proof is given for the correctness of the powers of two descent method for computing discrete logarithms. The result is slightly stronger than the original work, but more importantly we provide a unified geometric argument,…
We construct three public key knapsack cryptosystems. Standard knapsack cryptosystems hide easy instances of the knapsack problem and have been broken. The systems considered in the article face this problem: They hide a random (possibly…
Distributions on integers are ubiquitous in probabilistic modeling but remain challenging for many of today's probabilistic programming languages (PPLs). The core challenge comes from discrete structure: many of today's PPL inference…
Local differential privacy (LDP) can provide each user with strong privacy guarantees under untrusted data curators while ensuring accurate statistics derived from privatized data. Due to its powerfulness, LDP has been widely adopted to…
One of the possible generalizations of the discrete logarithm problem to arbitrary groups is the so-called conjugacy search problem (sometimes erroneously called just the conjugacy problem): given two elements a, b of a group G and the…
We consider differentially private counting when each data point consists of $d$ bits satisfying a partial order. Our main technical contribution is a problem-specific $K$-norm mechanism that runs in time $O(d^2)$. Experiments show that,…
Public-key cryptosystems are suggested based on invariants of groups. We give also an overview of the known cryptosystems which involve groups.