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A Set of Three Unsolved (?) Problems in Number Theory

A Set of Three Unsolved (?) Problems in Number Theory. N. Rama Murthy Scientist ‘F’ CAIR DRDO-IISc Workshop on Mathematical Engineering 9 June 2007. Description of the Problems. Let p be an odd prime of the form 4 k + 1 or 4 k + 3 Problem No. 1

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A Set of Three Unsolved (?) Problems in Number Theory

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  1. A Set of Three Unsolved (?) Problems in Number Theory N. Rama Murthy Scientist ‘F’ CAIR DRDO-IISc Workshop on Mathematical Engineering 9 June 2007

  2. Description of the Problems Let p be an odd prime of the form 4k + 1 or 4k + 3 Problem No. 1 Given any positive integer n such that 0 < n < p, is it possible to develop a fast algorithm for testing if n is a quadratic residue (QR) in GF(p)? e.g. when p = 11, QRs : 1,3,4,5,9 QNRs : 2,6,7,8,10

  3. Problem Description (Cont’d) Problem No. 2 Is it possible to develop a fast algorithm to identify an integer n (0 < n < p ) such that both n and (n2 + 1) mod p are quadratic residues in GF(p)? e.g. when p = 11, both n = 5 and (52 + 1) mod 11 = 4 are QRs in GF(11)

  4. Problem Description (Cont’d) Problem No. 3 Is it possible to develop a fast algorithm to quickly identify an integer n that is a QNR (0 < n < p ) such that (n + 1) mod p is a QR in GF(p)? e.g. when p = 11, n = 2 is a QNR and (2 + 1) mod 11 = 3 is a QR in GF(11) Solutions to the above three problems have great potential for application in symmetric key cryptography

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