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# PublicKey - MA 187 Garsia CRYPTOGRAPHY PUBLIC KEY 1 A...

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MA 187: Garsia CRYPTOGRAPHY PUBLIC KEY may 27, 2009 1 A Public Key Interchange System Primitive Roots Given a prime p , an integer q is said ot be a primitive root mod p if the numbers q 1 , q 2 , q 3 , . . . , q p 1 are all distinct mod p . For instance 2 is a primitive root mod 13 since successive 12 powers of 2 ( mod 13) are s: 1 2 3 4 5 6 7 8 9 10 11 12 2 s : 2 4 8 3 6 12 11 9 5 10 7 1 The possession of a primitive root enables us to solve several congruence problems with the greatest of ease. For instance suppose we wish to ±nd the solution of the congruence 11 x 9 ( mod 13) (1) Looking at the above table we see that 11 2 7 9 2 8 ( mod 13) this given, setting x = 2 y , equation (1) can be rewritten in the form 2 7+ y 2 8 ( mod 13) (2) Note now that by Fermats’s theorem we have 2 12 1 ( mod 13) this implies that for any a we have 2 a 2 12 a 2 a +12 a 2 12 1 ( mod 13) Using this with a = 7 in 2 gives that x 2 y 2 5 2 8 2 5+8 2 1 2 ( mod 13) The calculation we have carried out in this example should remind us precisely of what we usually do when solving equations such as (1) using a logarithm table. Indeed, the table avove gives us precisely the ”logarithms” of the different integers ( mod

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PublicKey - MA 187 Garsia CRYPTOGRAPHY PUBLIC KEY 1 A...

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