3 research outputs found
Lower bounds on the lengths of double-base representations
A double-base representation of an integer n is an expression n = n_1 + ... +
n_r, where the n_i are (positive or negative) integers that are divisible by no
primes other than 2 or 3; the length of the representation is the number r of
terms. It is known that there is a constant a > 0 such that every integer n has
a double-base representation of length at most a log n / log log n. We show
that there is a constant c > 0 such that there are infinitely many integers n
whose shortest double-base representations have length greater than c log n /
(log log n log log log n).
Our methods allow us to find the smallest positive integers with no
double-base representations of several lengths. In particular, we show that 103
is the smallest positive integer with no double-base representation of length
2, that 4985 is the smallest positive integer with no double-base
representation of length 3, that 641687 is the smallest positive integer with
no double-base representation of length 4, and that 326552783 is the smallest
positive integer with no double-base representation of length 5.Comment: 8 pages, LaTeX. Added DOIs for most references; corrected a minor
error in arithmetic; made small copy-editing changes. To appear in Proc.
Amer. Math. So
Multi-Base Representations of Integers: Asymptotic Enumeration and Central Limit Theorems
In a multi-base representation of an integer (in contrast to, for example,
the binary or decimal representation) the base (or radix) is replaced by
products of powers of single bases. The resulting numeral system has desirable
properties for fast arithmetic. It is usually redundant, which means that each
integer can have multiple different digit expansions, so the natural question
for the number of representations arises. In this paper, we provide a general
asymptotic formula for the number of such multi-base representations of a
positive integer . Moreover, we prove central limit theorems for the sum of
digits, the Hamming weight (number of non-zero digits, which is a measure of
efficiency) and the occurrences of a fixed digits in a random representation
Fast Generation of RSA Keys using Smooth Integers
Primality generation is the cornerstone of several essential cryptographic
systems. The problem has been a subject of deep investigations, but there is
still a substantial room for improvements. Typically, the algorithms used have
two parts trial divisions aimed at eliminating numbers with small prime factors
and primality tests based on an easy-to-compute statement that is valid for
primes and invalid for composites. In this paper, we will showcase a technique
that will eliminate the first phase of the primality testing algorithms. The
computational simulations show a reduction of the primality generation time by
about 30% in the case of 1024-bit RSA key pairs. This can be particularly
beneficial in the case of decentralized environments for shared RSA keys as the
initial trial division part of the key generation algorithms can be avoided at
no cost. This also significantly reduces the communication complexity. Another
essential contribution of the paper is the introduction of a new one-way
function that is computationally simpler than the existing ones used in
public-key cryptography. This function can be used to create new random number
generators, and it also could be potentially used for designing entirely new
public-key encryption systems.Comment: This paper contains 11 pages and 8 tables, in IEEE Transactions on
Computer