Here is the exercise:
Suppose that $(p,p_1,g_1,a)=(107, 53,4,13)$ is the public key for a digital signature and $(52,47),(29,7)$ are the signatures of the messages $m=15$ and $m'=12$, correspondingly. If $k,k'=2 k$ are the daily keys that have been used then find out the private key : $h$
What I did:
According to the statement we can make two congruences:
$47k\equiv15+52h\mod53$ (1)
$14k\equiv12+29h\mod53$ (2)
Multiplying the (1) by $14$, the (2) by $-47$ and sum together, I get $$52h\equiv17\mod53$$ or $$h\equiv36\mod53$$
But if $h=36$, then it should also be $$a\equiv4^h\mod107$$ or $$13\equiv4^{36}\mod107$$, which is wrong...
Is the exercise correct or am I making a mistake? Thanks.
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