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I didn't follow all of the question, but let's take a small example. Suppose $X$ is uniformly distributed in $\{1,2,3\}$, so that $\Pr[X = x] = 1/3$ for any $x \in \{1,2,3\}$. If we define \begin{equation*} f(1) = 2, \qquad f(2) = 3, \qquad f(3) = 2, \end{equation*} then we have \begin{align*} \Pr[f(X) = 1] &= 0, \\ \Pr[f(X) = 2] &= \...


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I was told that you can determine the private key of an RSA encryption with the public key. Were they joshing me or can it be done? Yes, it can be done. What you have not been told is that to factor a public key (usually hundreds of digits) to find the private key, requires a time exponential in the length of the public key, therefore even a supercomputer ...


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It's simply due to a hash function acting (within it's block size $n$) as:- You'll notice that there is no possible 'A' output as there was a collision at 'C'. It's a version of the pigeon hole principle meaning that when two birds occupy one hole, you must have an empty hole remaining. Some of the output bins of a pseudo random function (hash extractor) ...


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Problem summary: in textbook RSA, it is given $N$, $\phi(N)$, and a ciphertext $c$. It is wanted the plaintext message $m$ and a private exponent $d$. If $e$ or $m$ was random, that would be infeasible. But usually, $e$ is small thus guessable, and $m$ is highly redundant/recognizable. Thus we can try to compute $$\begin{align} d_e&=e^{-1}\bmod\phi(N)\\ ...


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..how does this formula $(aG+bG) = (a+b)G$ work in ECDSA? Perfectly well. It follows from the definition of $kG$ as $\overbrace{G+\cdots+G}^{k\text{ times}}$, associativity and commutativity of point addition. Notice that operator $+$ in $(aG+bG)$ and $G+\cdots+G$ is elliptic curve point addition, while operator $+$ in $(a+b)$ is addition in $\Bbb Z$ (...


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The relevant security property of $\operatorname{Poly1305}_r$ is that it has bounded difference probability—that is, for any distinct messages $x \ne y$ of up to $L$ bytes, and any difference $\delta$, $$\Pr[\operatorname{Poly1305}_r(x) - \operatorname{Poly1305}_r(y) = \delta] \leq 8\lceil L/16\rceil/2^{106},$$ under random choice of $r$. (Here the ...


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