How to Destroy the World with Number Theory Daniel Dreibelbis University of North Florida Basic Cryptography Alice wants to send a message to Bob Dr Hamid eats kittens for breakfast Eve is listening to any communication between Alice and Bob ID: 761582
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How to Destroy the World with Number Theory Daniel Dreibelbis University of North Florida
Basic Cryptography Alice 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.
Secret Decoder Ring Simple substitution cipher. Each letter is replaced by a letter k letters down the alphabet.
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.
Real Life SDR Our SDR has 26 different keys. In Real Life, we use an encryption method called AES (Advanced Encryption System). AES has 2 128 different keys 2128 = 340,282,366,920,938,463,463,374,607,431,768,211,456 That’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
Key Exchange Problem Eve 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?
Modular Arithmetic We define a mod n to be the remainder when a is divided by n.100 mod 17 = 15 Computers 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
Diffie-Hellman Alice 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 = g a mod n . This is her key. Bob computes the number k = fb mod n. This is his key.They now have the same key.
Example Alice 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 = 11 10 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.
Why does it work? Alice computes k = g a mod n = (cb)a mod n.Bob computes k = fb mod n = (ca)b mod n.They are the same!
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: 11 a mod 26 = 23 If we use large numbers (which we do in Real Life), then guessing a will take thousands of years, even with the help of computers.
Real Life c = 316912650057057350374175801351 a = 1267650600228229401496703205653 b = 5070602400912917605986812821771n = 170141184728119831959916705212587311361 f = 161287865144798146040576922608605193658g = 61813267884160838151925223195196755176k = 116582602641953240322793154442983171347Computers can work out the mods very, very quickly, even with these big numbers.Discrete log problem: 316912650057057350374175801351a mod 170141184728119831959916705212587311361 = 161287865144798146040576922608605193658
Crypto’s Dirty Secret Every 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).
Wkh H q g ! Thanks! www.unf.edu/~ddreibel