"And now that we have those computers, mod p arithmetic (where p is a huge prime) is the cornerstone of cryptographic security for internet commerce." Not so much. There have been so many problems with RSA that the most recent recommendations are to disable RSA completely, in favor of elliptic curves. "A number of the most popular websites and services online, including Facebook and *PayPal*, are vulnerable to an exploit which has resurfaced from 1998." "We believe RSA encryption modes are so risky that the only safe course of action is to disable them." http://www.zdnet.com/article/robot-exploit-from-1998-resurrected-leaves-top-... https://eprint.iacr.org/2017/1189.pdf At 08:42 AM 12/15/2017, James Propp wrote:
Hi,
I'm almost finished with an essay entitled "The Roots of Unity" and would love to get people's feedback. I'll be publishing it in less than 48 hours, and I don't expect many people to have a chance to read it (let alone offer comments) before I publish it. But I'll continue to tinker with it as comments come in, so if you find mistakes or other opportunities for improvement, please let me know, even if it's after the 17th.
I tried (and failed) to find a doable puzzle that makes use of the the multiplicative structure of the complex numbers. Can you think of a good one that'd be accessible to a bright high school student? I was going to include the question "Let p be the probability that the sum of the numbers you get when you roll six fair dice is a multiple of seven. Do you think p is equal to, greater than, or less than 1/7? What if we roll seven dice instead of six?" (There's a nice anaysis that uses 7th roots of unity.) But this strikes me as likely to be too tricky for nearly all of my readers. Can anyone suggest a substitute?
I hope you'll sign your comments, even negative ones; sometimes I have a follow-up question (like "Okay, would this way of phrasing it be less dreadful?"), which I can't ask you if you've commented anonymously. I want to make the essay better, and I have a fairly thick skin.
Please leave your feedback here: https://mathenchant.wordpress.com?p=2052&shareadraft=5a33f99f1b1af
Title: The Roots of Unity Beginning: Two primal pleasures for me from the years of my childhood (and maybe the years of infancy that I can't remember) are feelings that I'll call coziness and spaciousness. The first is the feeling I'd get at night pulling the blankets up over my head; the other is the feeling I'd get entering a landsca... Read more: https://mathenchant.wordpress.com?p=2052&shareadraft=5a33f99f1b1af Thanks, jamespropp