I'm self-studying NTRU encrypt, and I read following in the book "an introduction to mathematical cryptography" -Chapter 6.10-remark 6.50 (Jeffrey hoffstein, Jill Pipher, Joseph H. silverman):
NTRU is an example of a probabilistic cryptosystem, since a single plaintext m(x) has many different encryptions p*h(x)*r(x)+m(x) corresponding to different choices of the ephemeral key r(x). (In here, h(x) is public key, m(x) is a message, and p is a public parameter. r(x) is randomly chosen in the encryption phase.)
It would be a bad idea to send the same message twice using different ephemeral keys (i.e. r(x)), just as it would be bad to use the same ephemeral key to send two different plaintexts. The standard solution to this danger is to generate the ephemeral key as a hash of the plaintext.
My question is:
why is it dangerous to send the same message with two different random ephemeral keys?
And why does using the hash of the plaintext keep security?
I read a paper, "Authenticating Privately over Public Wi-Fi Hotspots", and they generate a query vector $v_i$ like in the below figure. (for Private Information Retrieval). And the query consists of lots of (NTRU-encrypted) ciphertexts of '$0$', and an attacker absolutely knows it. Is it safe?