You can put anything into the public exponent without weakening the
modulus. Some time ago, it was popular to do it in a way that the
base64 representation displays a message. This should also be possible
with malicious bitstreams.

Regarding the key posted by Hanno:
My intuition tells me that if the product of two primes has a weak
pattern like that, then it means that each of the prime factors
probably has an even weaker pattern. (Or it is not a product of two
primes in the first place, which makes it even weaker.) However, I did
not test that hypothesis.

Regards
Stephan

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