The Exchange

The Shared Secret

The public values cross the wire, then each side raises the other value to its own secret. Both ladders land on the same toy shared value.

highlighted = computed this step

Why the same secret appears twice

Alice receives B and raises it to her secret a. Bob receives A and raises it to his secret b.

Ba=AbB^a=A^b
Public DH exchangeOnly public values cross the wire.Public values on the wireAlicekeeps acomputes sharedBobkeeps bcomputes sharedA seq=8 public valueB seq=19 public value

Only public values cross

The flow shows A=8 crossing to Bob and B=19 crossing to Alice.

A=8,B=19A=8,\quad B=19
Public DH exchangeOnly public values cross the wire.Public values on the wireAlicekeeps acomputes sharedBobkeeps bcomputes sharedA seq=8 public valueB seq=19 public value

Alice computes the shared value

Alice computes B to the a power modulo p, which gives 2.

196 mod 23=219^{6}\bmod{}23=2
Alice computes sharedThe modular exponentiation ladder is recomputed from exact DH inputs.Alice computes shared - 19^6 mod 23stepbitpriorsquaremultiplyresult01111919111916552052skip2

Bob computes the same value

Bob computes A to the b power modulo p, and he also gets 2.

815 mod 23=28^{15}\bmod{}23=2
Bob computes sharedThe modular exponentiation ladder is recomputed from exact DH inputs.Bob computes shared - 8^15 mod 23stepbitpriorsquaremultiplyresult01118811818662161312123112622

Summary

Both sides land on shared value 2. NOTE: toy-modulus; no-side-channel; no-production; never-roll-your-own. Exact arithmetic on a deliberately tiny toy modulus; real DH uses 2048+ bit groups inside authenticated reviewed protocols.

Ba=Ab=2B^a=A^b=2
Shared secret checkDiffie-Hellman exchange values recomputed from the toy modulus.Shared secret check - shared=2quantityvaluep23g5a (secret)6A=g^a mod p8b (secret)15B=g^b mod p19shared=B^a=A^b2A^b mod p2