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Friday Technical Tutorials

Introduction

Welcome to the seventh edition of Friday Technical Tutorials.  In Chapter One…MACD The Basics, we walked our readers through the basic construction of Geralds Appel’s  Moving Average Convergence-Divergence Indicator or MACD. This technical analysis tool, developed in the late 1970’s, is a trend following momentum indicator and is featured in…


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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 $z_{B+1}$.

Friday Technical Tutorials

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