1. Solve the following Discrete Logarithm Problem via Shank’s Algorithm:
37 x  12295 modulo 79839983
Here is the description of Shank’s algorithm:
Let n  79839983 , a  12295 , b  37 , m  n  1  8935 .

1

Computer a  62577284 mod n , a  14339825 mod n .
Then computer the following groups:
(1) b, ba 1 , ba 2 ,, ba  m
m
2
(2) a m , a 2m ,, a m
Until we find some 0  i, j  m , such that ba i  a mj mod n . In this problem, we
find that i  5010 , j  6623 . Then b  a mj i mod n .
Therefore, the solution is x  mj  i  59181515
2. Solve the following Discrete Logarithm Problem via Pollard Rho Algorithm:
79 x  4341 modulo 39839983
Here is the link of the description of Pollard’s Pho algorithm.
http://en.wikipedia.org/wiki/Pollard%27s_rho_algorithm_for_logarithms
When the loop stops, we have the following values.
x  X  23290281 , a  1545129 , A  21963305 , b  38350184 , B  28070868
Then solution of the problem is x  11687164 .
3. Solve the following Discrete Logarithm Problem via Pohlig-Hellman Algorithm:
17 x  5099 modulo 19839997
Here is the description of Pohig-Hellman algorithm.
Let n  19839997 , g  17 , e  5099 . Then
 (n)  n  1  19839996  2 2  32  11 50101  4  9  11 50101 is a smooth
number. Let p1  4 , p2  9 , p3  11 , p4  50101. Then solve the following
equations e ( n ) p  ( g  ( n ) p ) b mod n , i  1,2,3,4 . Since 0  bi  pi , we can find all
bi ’s by coding. In this problem, b1  1 , b2  4 , b3  5 , b4  41273 .
i
i
i
Now we use the Chinese Remainder Theorem to solve the system of equations
x  bi mod n , i  1,2,3,4
Finally, the solution is x  8257837
4. Solve the following Discrete Logarithm Problem via Index Calculus:
83x  12295 modulo 89839993
Here is the link of the description of Index Calculus algorithm.
http://en.wikipedia.org/wiki/Index_calculus_algorithm
The solution of the problem is x  52282748 .