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8

The LWE assumption I think we should start from the LWE assumption. Let $n$ and $q$ be integers and let $\chi$ be a distribution over $\mathbb{Z}_q$. We often take $\chi$ as a Gaussian with small variance. (We take an error $e$ from this distribution $\chi$ and assume that $|e| \ll q$.) The LWE assumption states that any efficient adversary cannot ...


4

First, on the difference between perfect security and semantic security. Both definitions concern confidentiality, so let us first define what confidentiality means. Note first that an adversary as some a priori knowledge of the message. We can capture that by e.g. having the adversary choose two messages and then flipping a fair coin to decide which one to ...


3

If an attacker can choose the points $P_i$, than this system is not semantically secure. For example, they may choose $P_2=2P_1$, and the corresponding encryption $Q_2$ would be equal to $2Q_1$. If the points are chosen at random, this system is semantically secure if decisional Diffie-Hellman assumption holds for the curve. This assumption is presumed to ...


2

Not always, it depends on the particular encryption scheme. Strictly speaking, the proofs only say that breaking indistinguishability is equivalent to breaking the hardness assumption they are based on. There are some cryptosystems, like Rabin's, where the security of the key is equivalent to the security of the ciphertexts, i.e. factoring <=> key ...


2

The initial notion of semantic security from Goldwasser and Micali has been shown to be euqivalent to what we call today indistinguishability under chosen plaintext attacks (IND-CPA). Yes that's only security against a passive adversary and actually the weakest reasonable security notion that we use today. The authors of the second paper you link seem to ...


1

For perfect secrecy: $$number\_ of\_keys >= number\_of\_cipher >= number\_of\_plaintext$$ According to Shannon's perfect secrecy theorem: let, $$number\_ of\_keys = number\_of\_cipher = number\_of\_plaintext$$ then we have perfect secrecy if and only if: each key is used with same probability, and for each (plain,cipher) pair there is unique ...



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