A Note on Addition Chains

نویسندگان

  • Maurice MIGNOTTE
  • Amadou TALL
چکیده

Abstract Addition chains give a very easy way of computing xn knowing x and n. The fact of having a minimal addition chain for an integer n gives the least number of multiplications needed to compute xn. In this paper, we will present the binary method which is optimal for any integer of Hamming weight 1 or 2. We will show that if n has k digits in its binary expansion and the minimal length of all addition chains for n is k + 1, then n has 2 as Hamming weight and we can then deduce that there exists a minimal addition chain obtained by the binary method for n. Finally, we will show that there are exactly 4 kinds of addition chains possible for those kinds of integers The binary method is also optimal for integers of Hamming weigh 3 but we will show that the converse is false in this case.

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تاریخ انتشار 2011