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I found this paper that explains how to attack a reduced round tiger hash, but I'm having a hard time understanding some of the mathematical notation. Could someone explain the section "Message Modification by Meet-in-the-Middle" in plain English?

Here's a screenshot of that section:

screenshot of "3.2 Message Modification by Meet-in-the-Middle"

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I'm afraid we probably can't explain this to you. We want to help, but there are limits to what kinds of help can be expected from this kind of site. It looks like you first need to gain more fundamental knowledge about cryptanalysis before you will be in a good position to understand that research paper. Mathematics is the language of cryptanalysis; asking us to explain it without mathematics is not reasonable. If you have a specific question about a specific aspect of that article, I suggest you edit the question accordingly; otherwise, this is simply too broad as it currently stands. –  D.W. 23 hours ago
Have you read the meet-in-the-middle article on Wikipedia? –  Gilles 17 hours ago

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