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ProShares’ Bitcoin ETF becomes first of its kind to garner green signal from SEC

ProSharesBitcoin exchange-traded funds [ETF] have tried long and hard to make a debut in the US. This could finally be changing as ProShares’s application managed to garner the approval of the Securities and the Exchange Commission [SEC]. Just yesterday, Bloomberg shared a report noting that the USA would be getting its very first Bitcoin ETF […]

Complexity if there are more than one collision

Let $h: {0,1}^*$ → ${0,1}^l$ be a hash function. We define a k-collision as a set of k distinct messages in which $h(m_1)=h(m_2)=…=h(m_k)$ There is an attack running in time $O(2^l)$ which evaluates $h$ on $2^l + 1$ distinct inputs, by the pigeonhole principle, two of the outputs must be equal. A better way is::Listen

Let $h: {0,1}^*$${0,1}^l$ be a hash function. We define a k-collision as a set of k distinct messages in which $h(m_1)=h(m_2)=…=h(m_k)$

There is an attack running in time $O(2^l)$ which evaluates $h$ on $2^l + 1$ distinct inputs, by the pigeonhole principle, two of the outputs must be equal.
A better way is using birthday attack and $O(2^{l/2})$. How can we use this to find a complexity for multi-collision of $h$ that is a random oracle model? Another question that we can think of is what happens if k is some power of 2 in a Merkle-Damgård construction?
The Merkle-Damgård transform based on the textbook is as follows:

Let $(Gen, h)$ be a fixed-length hash function for inputs of length $2 n$ and with output length $n$. Construct hash function $(Gen, H)$ as follows:

  • $Gen$: remains unchanged.
  • $H$ : on input a key $s$ and a string $x in{0,1}^{*}$ of length $L<2^{n}$, do the following:
  1. Set $B:=leftlceilfrac{L}{n}rightrceil$ (i.e., the number of blocks in $x$ ). Pad $x$ with zeros so its length is a multiple of $n$. Parse the padded result as the sequence of $n$-bit blocks $x_{1}, ldots, x_{B}$. Set $x_{B+1}:=L$, where $L$ is encoded as an $n$-bit string.
  2. Set $z_{0}:=0^{n}$. (This is also called the $I V$.)
  3. For $i=1, ldots, B+1$, compute $z_{i}:=h^{s}left(z_{i-1} | x_{i}right)$.
  4. Output $z_{B+1}$.

ProShares’ Bitcoin ETF becomes first of its kind to garner green signal from SEC

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