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9

With the definitions that a function $F$ is collision-resistant when a [computationally bounded] adversary can't [with sizable odds] exhibit any $(a,b)$ with $a\ne b$ and $F(a)=F(b)$; first-preimage-resistant when, given $f$ determined as $F(a)$ for an unknown random $a$, a [computationally bounded] adversary can't [with sizable odds] exhibit any $b$ with ...


8

The difference is in the choice of $m_1$. In the first case (second preimage resistance), the attacker is handed a fixed $m_1$ to which he has to find a different $m_2$ with equal hash. In particular, he can't choose $m_1$. In the second case (collision resistance), the attacker can freely choose both messages $m_1$ and $m_2$, with the only requirement ...


4

Take a function $H:\mathbb S\to\{0,1\}^k$ where $\mathbb S$ is a large finite subset of $\{0,1\}^*$, such that $H$ "compress data" [however this is defined], and $H$ is [conjectured] collision-resistant [thus second-preimage-resistant] and first-preimage-resistant; e.g., SHA-512, for $k=512$. Let $«0»$ and $«1»$ be two public distinct elements of $\mathbb ...


4

A long message is a message that, when padded, is longer than the block size of the hash function. That means that the hash function has to process the message in parts and keep track of state somehow, which may allow for attacks. Such attacks would not apply to messages shorter than the block size, and may additionally require a large number of blocks to ...


4

Without the specific reference I can't be sure this is what you are talking about, but generally a "long message" attack is a way to defeat second preimage resistance with less complexity than expected. It uses a time-space tradeoff to find a second preimage with complexity $2^{n/2}$ for a $n$-bit hash function (normally you would expect $2^n$). In the ...


3

I prefer using definitions that explicitly specify who does what. Weak collision resistance: After Bob creates some message x1, it is "computationally infeasible" for an attacker Mallory to compute some other message x2 such that h(x1) == h(x2). Strong collision resistance: It is "computationally infeasible" for an attacker Mallory to find any two messages ...


2

First, I must warn you that any definition that uses "feasible" will not be a rigorous one. The only way I know to rigorously define collision and preimage resistances is using function families, i.e. keyed hash functions. That said, if you believe the negations are equivalent, the definitions you are using are themselves equivalent (you correctly negated ...



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