Many forms of cryptography are based on the following discrete log problem. Choose a cyclic group G (produced by one generator g) with finitely many elements. Alice and Bob want to set up a secret communication and need a secret key for that, which becomes a group element from G. They set up the following exchange (the so-called Diffie-Hellman protocol): 1. Alice chooses a secret group element a and sends unsecured g^a to Bob 2.Bob chooses a secret group element b and sends unsecured g^b to Alice 3. Alice calculates (g^b )^a and keeps this group element secret 4.Bob calculates (g^a )^b and keeps this group element secret Since a cyclic group is abelian, Alice and Bob now share the secret key (g^b )^a=(g^a )^b. The blue group elements are publicly known, the red elements are secret. The discrete log problem presented to an eavesdropper is computing say a from the publicly known data G,g,g^a. The naming of the problem comes from the solution a= glog(ga ) which is generally difficult to find. Assignment Choose as group G=(Z/pZ)^× the group of invertible numbers modulo prime p=23 and verify that g=5 is a generator. Calculate the key that Alice and Bob share if a=4 and b=3 and show that this key is equal to 18 "mod" 23. please if able write explanations with the taken steps, I have no clue how to approach this, thank you in advance.
Many forms of cryptography are based on the following discrete log problem. Choose a cyclic group G (produced by one generator g) with finitely many elements. Alice and Bob want to set up a secret communication and need a secret key for that, which becomes a group element from G. They set up the following exchange (the so-called Diffie-Hellman protocol):
1. Alice chooses a secret group element a and sends unsecured g^a to Bob
2.Bob chooses a secret group element b and sends unsecured g^b to Alice
3. Alice calculates (g^b )^a and keeps this group element secret
4.Bob calculates (g^a )^b and keeps this group element secret
Since a cyclic group is abelian, Alice and Bob now share the secret key (g^b )^a=(g^a )^b. The blue group elements are publicly known, the red elements are secret. The discrete log problem presented to an eavesdropper is computing say a from the publicly known data G,g,g^a. The naming of the problem comes from the solution a= glog(ga ) which is generally difficult to find.
Assignment
Choose as group G=(Z/pZ)^× the group of invertible numbers modulo prime p=23 and verify that g=5 is a generator. Calculate the key that Alice and Bob share if a=4 and b=3 and show that this key is equal to 18 "mod" 23.
please if able write explanations with the taken steps, I have no clue how to approach this, thank you in advance.
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