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Not really, for example if your language has $m$ letters, and the letters are independent identically distributed then $H(X)=H(X_1,\ldots,X_N)=N H(X_1)$ where $H(X_1)$ is is the entropy of the single letter distribution $(p(1),\ldots,p(m)),$ where $P[X_k=u]=P[X_1=u]=p(u)$ for any $k\geq 1.$
This is because for independent random variables
$$
H(X_1,\ldots,...
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called asymmetric self encryption
No, not really, you just made up that term.
might seem an unusual choice in situations where a key exchange is not required
No it doesn't, although commonly a hybrid cryptography is used, especially for EC based cryptography.
an attacker apparently needs both to have the public key and brute-force the passphrase to ...
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You're describing methods of solving LWE via reduction to SVP. In particular:
Sieving and Enumeration are methods of solving exact SVP
Basis reduction is a method of solving approximate SVP
There are additionally ways of solving LWE directly (the classic example is the Arora-Ge attack, which works when the noise distribution is too concentrated).
Daniele ...
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In a "fair trade" paper at Bitcoin workshop Financial Crypto 2016, a snark-proof was introduced to assure that a message (a Sudoku solution) could be properly decrypted.
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