Download - How to Destroy the World with Number Theory
![Page 1: How to Destroy the World with Number Theory](https://reader034.vdocument.in/reader034/viewer/2022052305/56816763550346895ddc3fe0/html5/thumbnails/1.jpg)
How to Destroy the World with Number Theory
Daniel DreibelbisUniversity of North Florida
![Page 2: How to Destroy the World with Number Theory](https://reader034.vdocument.in/reader034/viewer/2022052305/56816763550346895ddc3fe0/html5/thumbnails/2.jpg)
Basic CryptographyAlice wants to send a message to Bob.“Dr. Hamid eats kittens for breakfast.”Eve is listening to any communication
between Alice and Bob.Goal: Encrypt the message in a way that Alice
and Bob know, but Eve does not.
![Page 3: How to Destroy the World with Number Theory](https://reader034.vdocument.in/reader034/viewer/2022052305/56816763550346895ddc3fe0/html5/thumbnails/3.jpg)
Secret Decoder RingSimple substitution cipher.Each letter is replaced by a letter k letters
down the alphabet.
![Page 4: How to Destroy the World with Number Theory](https://reader034.vdocument.in/reader034/viewer/2022052305/56816763550346895ddc3fe0/html5/thumbnails/4.jpg)
Secret Decoder Ring.Let’s do k = 3.“Dr. Hamid eats kittens for breakfast.”
becomes “Gu. Kdplg hdwv nlwwhqv iru euhdnidvw.”
Bob decodes by removing k from each letter.The number k is called the key. Our SDR has
26 different keys.
![Page 5: How to Destroy the World with Number Theory](https://reader034.vdocument.in/reader034/viewer/2022052305/56816763550346895ddc3fe0/html5/thumbnails/5.jpg)
Real Life SDROur SDR has 26 different keys.In Real Life, we use an encryption method
called AES (Advanced Encryption System).AES has 2128 different keys2128 =
340,282,366,920,938,463,463,374,607,431,768,211,456That’s 340 undecillion. That’s a whole bunch
of keys.A brute force key search is infeasible.A typical key looks like sixteen characters, something like:
A4gf5*nTb[Q@21’7
![Page 6: How to Destroy the World with Number Theory](https://reader034.vdocument.in/reader034/viewer/2022052305/56816763550346895ddc3fe0/html5/thumbnails/6.jpg)
Key Exchange ProblemEve hears everything that Alice says to Bob
and Bob says to Alice.If Alice and Bob try to agree on a key k, Eve
will hear this also, and she will know the key.KEP: How can Alice and Bob agree on a key
without Eve knowing its value?
![Page 7: How to Destroy the World with Number Theory](https://reader034.vdocument.in/reader034/viewer/2022052305/56816763550346895ddc3fe0/html5/thumbnails/7.jpg)
Modular ArithmeticWe define a mod n to be the remainder when a is
divided by n.100 mod 17 = 15Computers are really good at mods.Example: a = 10100+5, b=1099+7, n=10101+3Mathematica worked out ab mod n in about 0.00034 seconds.Answer is: 29748478515601709481956621265827578332435037952560102437932180281116087872397108919020459135080599359
![Page 8: How to Destroy the World with Number Theory](https://reader034.vdocument.in/reader034/viewer/2022052305/56816763550346895ddc3fe0/html5/thumbnails/8.jpg)
Diffie-HellmanAlice and Bob agree on two numbers: n and c.Alice comes up with a number a, and she doesn’t tell
anyone what it is.Bob comes up with a number b, and he doesn’t tell
anyone what it is.Alice computes the number f = ca mod n. She sends f to
Bob.Bob computes the number g = cb mod n. He sends g to
Alice. Alice computes the number k = ga mod n. This is her key.Bob computes the number k = fb mod n. This is his key.They now have the same key.
![Page 9: How to Destroy the World with Number Theory](https://reader034.vdocument.in/reader034/viewer/2022052305/56816763550346895ddc3fe0/html5/thumbnails/9.jpg)
ExampleAlice and Bob choose n = 26 and c = 11.Alice picks a = 10, Bob picks b = 14. They both
keep their number private.Alice computes f = 1110 mod 26 = 23. She sends
this number to Bob.Bob computes g = 1114 mod 26 = 17. He sends
this number to Alice.Alice computes k = 1710 mod 26 = 9. She will use
this as her key to encode her message in SDR.Bob computes k = 2314 mod 26 = 9. He will use
this as his key to decode the message in SDR.
![Page 10: How to Destroy the World with Number Theory](https://reader034.vdocument.in/reader034/viewer/2022052305/56816763550346895ddc3fe0/html5/thumbnails/10.jpg)
Why does it work?Alice computes k = ga mod n = (cb)a mod n.Bob computes k = fb mod n = (ca)b mod n.They are the same!
![Page 11: How to Destroy the World with Number Theory](https://reader034.vdocument.in/reader034/viewer/2022052305/56816763550346895ddc3fe0/html5/thumbnails/11.jpg)
What does Eve know?Eve knows n, c, f, g. She needs to find out a or b, and then she can break the code.To find out a, she needs to solve:
ca mod n = fThis is called the discrete log problem. No one knows
how to solve it other than guessing values of a. For our problem, it looks like:
11a mod 26 = 23If we use large numbers (which we do in Real Life),
then guessing a will take thousands of years, even with the help of computers.
![Page 12: How to Destroy the World with Number Theory](https://reader034.vdocument.in/reader034/viewer/2022052305/56816763550346895ddc3fe0/html5/thumbnails/12.jpg)
Real Lifec = 316912650057057350374175801351a = 1267650600228229401496703205653b = 5070602400912917605986812821771n = 170141184728119831959916705212587311361f = 161287865144798146040576922608605193658g = 61813267884160838151925223195196755176k = 116582602641953240322793154442983171347Computers can work out the mods very, very quickly,
even with these big numbers.Discrete log problem: 316912650057057350374175801351a mod 170141184728119831959916705212587311361 =
161287865144798146040576922608605193658
![Page 13: How to Destroy the World with Number Theory](https://reader034.vdocument.in/reader034/viewer/2022052305/56816763550346895ddc3fe0/html5/thumbnails/13.jpg)
Crypto’s Dirty SecretEvery form of public key cryptography or key
exchange relies on our inability to solve a certain math problem quickly (factoring, DLP, ECDLP, SVP, etc).
It is still possible that these “hard math problems” have quick solutions. All we know is that no one has found a quick solution yet (or at least has admitted to this publicly).
Research Problem: Find a quick solution to the DLP (thus making Diffie-Hellman useless) OR prove that no quick solution exists (thus making every other form of crypto useless).
![Page 14: How to Destroy the World with Number Theory](https://reader034.vdocument.in/reader034/viewer/2022052305/56816763550346895ddc3fe0/html5/thumbnails/14.jpg)
Wkh Hqg!Thanks!www.unf.edu/~ddreibel