Discrete Logarithms 1 Let p 53047 Verify that L38576 1234

Discrete Logarithms

1. Let p = 53047. Verify that L₃(8576) = 1234.

2. Let p = 31. Evaluate L₃(24).

Solution

1. p =53047

L3(8576)

Now according L3(8576) =1234

base = 3, a= 8576

suppose Logb(a) = x

then discrete logorithm is logb(a) = n mod p

so now a = b^n (mod p)

substituting values => 8576 = 3^1234 (mod 53047)

=> 8576 = 8576

=> LHS = RHS HENCE Proved.

2 . p=31 so L3(24)

base = 3 , p = 31

so 24 = 3^n mod(31)

=> apply log

=> log 3(24) = n mod(31)

=> log(24) / log(3) = n mod(31)

=> 0.477121 = n mod(31)

=> 1/n = (1/0.477121) mod(31) Just exchange the n to other side and (0.477121) of the side

=> 1/n = 2.095904 mod(31) = 2.095904

=> 1/n = 2.095904

=> 1/2.095904 = n

=> n = 0.4777121