Russian Peasant Multiplication Explained and Generalised

The recent(ish) NumberPhile video about Russian Peasant Multiplication seems to have struck a chord, being submitted here three times[0][1][2] (that I've counted).

It seems to be new to a few people, even though it's old (obviously), based on binary (not so obviously), and ubiquitous in learning about taking powers (critical in cryptography).

So I thought I'd submit this post that I wrote ages ago. The material isn't time critical, it's always true, it's always relevant, so I didn't put a date on the title.

Comments and feedback welcome and invited.

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[0] https://news.ycombinator.com/item?id=22241502

[1] https://news.ycombinator.com/item?id=22256230

[2] https://news.ycombinator.com/item?id=22265913