The discrete logarithm modulo a composite hides 0(n) Bits

The Discrete Logarithm Modulo a Composite Hides O(n) Bits

In this paper we consider the one-way function f g;N (X) = g X (modN), where N is a Blum integer. We prove that under the commonly assumed intractability of factoring Blum integers, all its bits are individually hard, and the lower as well as upper halves of them are simultaneously hard. As a result, f g;N can be used in eecient pseudo-random bit generators and multi-bit commitment schemes, whe...

The Discrete Logarithm Hides O(log n) Bits

The main result of this paper is that obtaining any information about the $O(log |p|)$ " most significant " bits of $x$, given $g^x(mod p)$, even with a tiny advantage over guessing, is equivalent to computing discrete logarithms $mod p$.

The Composite Discrete Logarithm and Secure Authentication

For the two last decades, electronic authentication has been an important topic. The first applications were digital signatures to mimic handwritten signatures for digital documents. Then, Chaum wanted to create an electronic version of money, with similar properties, namely bank certification and users’ anonymity. Therefore, he proposed the concept of blind signatures. For all those problems, ...

The discrete logarithm problem modulo one: cryptanalysing the Ariffin-Abu cryptosystem

The paper provides a cryptanalysis the AAβ-cryptosystem recently proposed by Ariffin and Abu. The scheme is in essence a key agreement scheme whose security is based on a discrete logarithm problem in the infinite (additive) group R/Z (the reals modulo 1). The paper breaks the AAβ-cryptosystem (in a passive adversary model) by showing that this discrete logarithm problem can be efficiently solv...

Problem 3 : The Discrete Logarithm