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In a DHKE scheme with the a prime p = 467. an element g = 4. The element 4 has order 233 and generates thus a subgroup with 233 elements. after Computing the common key for A. a=400, b=134 B. a=167, b=134 the session keys are identical = 161
Can someone tell me the answer as to why the key is the same for both schemes given different values for a ?
Since $\gcd(2,467)=1$, one can observe using Fermat's little theorem that $4^{233}\pmod{467}=2^{466}\pmod{467}=1$.
Thus,
$$g^{a_1}=4^{400}=4^{233+167}\equiv\underbrace{1\cdot4^{167}}_{=g^{a_2}}\pmod{467}=89.$$
This results into identical session keys (shared secrets) $S=(g^a)^b=89^{134}\equiv161\pmod{467}$.
