Cryptography and Mathematics

1. Cryptography and Mathematics Sarah Brown March 2, 2005

2. Cryptography: The enciphering and deciphering of messages in secret code or cipher. We will focus on how data is securely transmitted over the internet. Source: Merriam-Webster Online Dictionary m-w.com

3. Alice and Bob: Two parties communicating over the internet.?

4. Security Requirements Authentication Privacy/confidentiality Integrity Non-repudiation Source: http://www.garykessler.net/library/crypto.html

5. Authentication Proving one's identity. Alice must know that Mallory, a malicious person, cannot pretend to be Bob. ?

6. Privacy/confidentiality Ensuring Eve, an eavesdropper, cannot read the message. ? ?

7. Integrity Bob needs assurance that the message has not been altered by malicious Mallory. ? ?

8. Non-repudiation Bob must prove that Alice really sent the message. ?

9. Definitions key: the set of instructions governing the encipherment and decipherment of messages plaintext: the initial unencrypted data ciphertext: the encrypted plaintext Source: Merriam-Webster Online Dictionary m-w.com Source: http://www.garykessler.net/library/crypto.html

10. Three Encryption Techniques Secret Key Cryptography Public Key Cryptography Hash Functions Source: http://www.garykessler.net/library/crypto.html

11. Secret Key Cryptography Alice and Bob have same key Both can encrypt and decrypt Most common scheme: Data Encryption Standard (DES) Problem: How does Alice get the key to Bob without anyone intercepting it? Source: http://www.garykessler.net/library/crypto.html

12. Public Key Cryptography Public key—how to encrypt Secret key—how to decrypt Alice wants to send confidential message to Bob. Bob’s public key tells her how to scramble the message. Bob uses his secret key to unscramble it. Source: http://www.garykessler.net/library/crypto.html

13. Inverse Functions! Easy to scramble message, hard to unscramble It’s easy to find f(x) but very hard to find f-1(x)

14. Examples Multiplication vs. factorization: 45462979 is the product of two prime numbers. Factor it. Multiply 5741 and 7919. Exponentiation vs. logarithms: Find x and y satisfying logx 59049 = y. Evaluate 310.

15. Nonrepudiation Alice uses her secret key to scramble a message. Bob uses Alice’s public key to unscramble it. Why? Bob knows Alice must have sent it. Source: http://www.garykessler.net/library/crypto.html

16. RSA Most common public key algorithm Developed by Ronald Rivest, Adi Shamir, and Leonard Adleman Easy to multiply two prime numbers. Given the product of two huge prime numbers, it is difficult to factor. Source: http://www.garykessler.net/library/crypto.html

17. Basic Steps Let p and q be prime. Calculate n = pq. Choose a number e that is relatively prime to (p-1)(q-1). Choose d so that (ed-1)/[(p-1)(q-1)] is an integer. n,e are public information d is private Source: http://www.garykessler.net/library/crypto.html

18. Encrypting/Decrypting the Message To encrypt a message M, create the ciphertext C = Me mod n The receiver decrypts the ciphertext using M = Cd mod n Source: http://www.garykessler.net/library/crypto.html

19. Simplified Example Let p=3 and q=5. Then n = pq = 15. Choose e to be relatively prime to (p-1)(q-1) = (2)(4) = 8. Select e=11 The value (11d-1)/[(2)(4)] = (11d-1)/8 must be an integer. Calculate one possible value, d=3. Let's say we wish to send the string SECRET, which has the decimal representation of the ASCII values of the characters 83 69 67 82 69 84. The sender encrypts each digit one at a time using the public key value (e,n)=(11,15). Thus, each ciphertext character Ci = Mi11 mod 15. The input digit string 0x836967826984 will be transmitted as 0x2c696d286924. The receiver decrypts each digit using the private key value (d,n)=(3,15). Thus, each plaintext character Mi = Ci3 mod 15. The input digit string 0x2c696d286924 will be converted to 0x836967826984 and, presumably, reassembled as the plaintext string SECRET. Source: http://www.garykessler.net/library/crypto.html

20. Money for Factoring RSA-640 Prize: $20,000 Status: Not Factored Decimal Digits: 193 3107418240490043721350750035888567930037346022842727545720161948823206440518081504556346829671723286782437916272838033415471073108501919548529007337724822783525742386454014691736602477652346609 Source: RSA Laboratories http://www.rsasecurity.com/rsalabs/

21. RSA-2048 Prize: $200,000 Status: Not Factored Decimal Digits: 617 25195908475657893494027183240048398571429282126204032027777137836043662020707595556264018525880784406918290641249515082189298559149176184502808489120072844992687392807287776735971418347270261896375014971824691165077613379859095700097330459748808428401797429100642458691817195118746121515172654632282216869987549182422433637259085141865462043576798423387184774447920739934236584823824281198163815010674810451660377306056201619676256133844143603833904414952634432190114657544454178424020924616515723350778707749817125772467962926386356373289912154831438167899885040445364023527381951378636564391212010397122822120720357 Source: RSA Laboratories http://www.rsasecurity.com/rsalabs/

22. Hash Functions Guarantee data integrity. Cannot decrypt the message! Source: http://www.garykessler.net/library/crypto.html

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24. Source: http://www.garykessler.net/library/crypto.html

25. Pros and Cons Secret key: -getting the key to Bob without someone intercepting it +1000 times faster than public key Public key: -slow +don’t worry about someone finding the key Hash function -+can’t get the message back Source: http://www.garykessler.net/library/crypto.html

26. Hybrid Source: http://www.garykessler.net/library/crypto.html

27. Career Information Potential employers: NSA (National Security Agency), IBM, RSA Security, national labs such as Sandia, NRL (Naval Research Lab), MIT’s Lincoln Labs. Math classes to take: linear algebra, number theory, combinatorics Computer classes: general programming, algorithms Thanks to Jason Holt of BYU for most of this info.

28. Last Word It’s easier to hack than decrypt.

29. Main Source http://www.garykessler.net/library/crypto.html Recommended Reading: The Code Book by Simon Singh