Question

Show how an attacker can conduct a man-in-the-middle attack when Alice and Bob perform the Diffe-Hellman protocol.

Answer 1

for example suppose eve was listening to alic and bob and agree upon the prime number to use for their key exchange to take place in the near future and the prime number they choose p =11 with a primitive root of alpha = 2 and eve uses the attack algorithm = (p-1)/2 ie (11-1)/2 =5 =q

Next eve runs 5 through primality test and the result is prime and this situation represents the initial case with the prime number being of p = 2q +1 ie p = 2*5 +1 ie p = 11 therefore 11= 2*5+1

Next eve must intercept the number alice attempts to send the bob suppose alice chooses x = 6 and attempts to send ( alpha )^x mod p ie

                                  = 2^6 mod 11

                                  =64 mod 11

                                   =9

             therefore A -> E : 9

Then eve intercepts the communication then takes  $\alpha$ ^x ( 9) and raises it to the q power i.e

                           = ( $\alpha$ ^x)^q = (2⁶)⁵ mod 11

Eve must also intercept the number bob ie attempting to send to alice suppose bob chooses y=8 and attempts to send

$\alpha$ ⁴ = 2⁸ mod 11 =

    = 256 mod 11

=3

B -> E : 3

eve again intercepts the communication and  $\alpha$ ⁴ ( 3) and raises it to the q power ie (  $\alpha$ ^y) ^q = (2⁸)⁵mod 11

eve then sends the result to the intended receipients

E -> B : 9^qmod 11

E->A : 3^qmod 11

Alice and Bob both finishes the key agreement by raising the receieved number to their private keys x and y respectively and arrive the same number the seckret key i.e

                               ( $\alpha$ ^yq)^x = ( $\alpha$ ^xq)^y

as result of the theory discussed above without any knowledge of x and y and eve knows the ony possible keys 1 and 5 and eve must wait for the message to be sent between alice and bob and try both keys and figure out which one is being used and she can then evesdrop and alice and bob secret communication has been compromised.

In this way the attacker can conduct a man in middle attack when alica and bob perform the diffe- hellman protocol as he can knew the secret key