Hatem M. Bahig

Improved Generation of Minimal Addition Chains

An addition chain for a natural number n is a sequence 1=a0<a1< . . . <ar=n of numbers such that for each 0<i≤r, ai=aj+ak for some 0≤k≤j<i. An improvement by a factor of 2 in the generation of all minimal (or one) addition chains is achieved by finding sufficient conditions for star steps, computing what we will call nonstar lower bound in a minimal addition and omitting the sorting step.

Cryptanalysis of multi-prime RSA with small prime difference

We show that the attack of de Weger on RSA using continued fractions extends to Multi-Prime RSA. Let (n,e) be a Multi-Prime RSA public-key with private key d, where n=p1p2⋯pr is a product of r distinct balanced (roughly of the same bit size) primes, and p1 < p2 < … < pr. We show that if pr−p1=nα, 0 < α≤1/r, r≥3 and

A new RSA vulnerability using continued fractions

2d^2+1<\frac{n^{2/r - \alpha}}{6r},

On a generalization of addition chains: Addition-multiplication chains

then Multi-Prime RSA is insecure.

New Attacks on the RSA Cryptosystem

Let (n = pq, e) be an RSA public key with private exponent d = ndelta, where p and q are large primes of the same bit size. Suppose that po ges radicn be an approximation of p with |p - po| les 1/8nalpha, alpha les 1/2. Using continued fractions, we show that the system is insecure if delta < 1-alpha/2. Our result is deterministic polynomial time and an extension of Coppersmiths result on a factorization.

Merging on PRAM

We show that some notations and facts on addition chains can be generalized to addition-multiplication chains. In other words, we show that addition-multiplication chains resemble addition chains in many aspects.

A fast GPU-based hybrid algorithm for addition chains

This paper presents three new attacks on the RSA cryptosystem. The first two attacks work when k RSA public keys (N i ,e i ) are such that there exist k relations of the shape e i x − y i φ(N i ) = z i or of the shape e i x i − yφ(N i ) = z i where N i = p i q i , φ(N i ) = (p i − 1)(q i − 1) and the parameters x, x i , y, y i , z i are suitably small in terms of the prime factors of the moduli. We show that our attacks enable us to simultaneously factor the k RSA moduli N i . The third attack works when the prime factors p and q of the modulus N = pq share an amount of their least significant bits (LSBs) in the presence of two decryption exponents d 1 and d 2 sharing an amount of their most significant bits (MSBs). The three attacks improve the bounds of some former attacks that make RSA insecure.

Binary addition chain on EREW PRAM

Abstract Many data sets to be merged consist of (1) n distinct keys and (2) the keys are serial numbers. Merging such data can be represented as merging two sorted arrays A=(a0, a1, …, an1−1) and B=(b0, b1, …, bn2−1) of records such that (1) the keys are distinct; (2) 0≤ai · key, bj · key < n, ∀0≤i<n1 and 0≤j<n2, where n=n1+n2 and key is the primary key of the record. We present an optimal deterministic merging algorithm on EREW PRAM in O(n⁄p) time, where p is the number of processors and 1≤p≤n. The algorithm uses linear number of space. We also extend the algorithm to capture repeated keys such that the repetition of the key is at most constant.

Interactive visualization applets for modular exponentiation using addition chains

A graphics processing unit (GPU) has been widely used to accelerate discrete optimization problems. In this paper, we introduce a novel hybrid parallel algorithm to generate a shortest addition chain for a positive integer e. The main idea of the proposed algorithm is to divide the search tree into a sequence of three subtrees: top, middle, and bottom. The top subtree works using a branch and bound depth first strategy. The middle subtree works using a branch and bound breadth first strategy, while the bottom subtree works using a parallel (GPU) branch and bound depth first strategy. Our experimental results show that, compared to the fastest sequential algorithm for generating a shortest addition chain, we improve the generation by about 70% using one GPU (NVIDIA GeForce GTX 770). For generating all shortest addition chains, the percentage of the improvement is about 50%.

Interactive visualization system for DES

An addition chain for a natural number x of n bits is a sequence of numbers a0, a1, ..., al, such that a0 = 1, al = x, and ak = ai+aj with 0 ≥ i, j > k ≥ l. The addition chain problem is what is the minimal number of additions needed to compute x starting from 1? In this paper, we present a new parallel algorithm to generate a short addition chain for x. The algorithm has running time O(log2 n) using polynomial number processors under EREW PRAM (exclusive read exclusive write parallel random access machine). The algorithm is faster than previous algorithms and is based on binary method.