Hatem M. Bahig

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An addition chain for a natural number n is a sequence 1=a 0<a 1< . . . <a r =n of numbers such that for each 0<i≤r, a i =a j +a k 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(More)
Let (n = pq, e) be an RSA public key with private exponent d = n<sup>delta</sup>, where p and q are large primes of the same bit size. Suppose that p<sub>o</sub> ges radicn be an approximation of p with |p - po| les 1/8n<sup>alpha</sup>, alpha les 1/2. Using continued fractions, we show that the system is insecure if delta &lt; 1-alpha/2. Our result is(More)
An addition sequence problem is given a set of numbers X = {n 1, n 2, . . . , n m }, what is the minimal number of additions needed to compute all m numbers starting from 1? This problem is NP-complete. In this paper, we present a branch and bound algorithm to generate an addition sequence with a minimal number of elements for a set X by using a new(More)
— An addition chain for a natural number n is a sequence 1 = a 0 < a 1 <. .. < a r = n of numbers such that for each 0 < i ≤ r, a i = a j + a k for some 0 ≤ k ≤ j < i. Thurber [9] introduced the function NMC(n) which denotes the number of minimal addition chains for a number n. Thurber calculated NMC(n) for some classes of n, such as when n has one or two(More)
We present a new optimal deterministic parallel algorithm for merging two sorted arrays A = (a<sub>0</sub>, a<sub>1</sub>, ... , a<sub>n1</sub> - 1) and B = (b<sub>0</sub>, b<sub>1</sub>, ... , b<sub>n2</sub> - 1) of integers. The elements of two sorted arrays are drawn from a domain of integers [0, n - 1], where n = Max(n<sub>1</sub>, n<sub>2</sub>). The(More)