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It's easier to take the questions out of order. what can be said about the distribution of exponents a+b mod p when b is taken at random from Zp, but a is fixed? If $b$ is a uniform independent random value from the range $[0, p-1]$, that is, the probability of each possible value is $1/p$, and $b$ is distributed independently from $a$, then $a+b \bmod p$...


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