An overview of public-key cryptography and a wrap-up of this minicourse. For MAT 140: Introduction to the Mathematical Sciences at Franklin College.

1. Cryptology
Day 4: Going public
MAT 140: Introduction to the
Mathematical Sciences
24 September 2008
Robert Talbert, PhD
Associate Professor of Mathematics
and Computing Science
rtalbert@franklincollege.edu

2. Recap of Day 3
• Bits, bytes, and binary representation of text using ASCII
• The XOR operation
• The Simple XOR cipher
• Perfect security and one-time pads

4. Alice

5. Alice Bob

6. Alice Bob

7. Plaintext
Dear Bob - The
meeting will be at
the embassy.
Alice Bob

8. Encryption
function
Plaintext
Dear Bob - The
meeting will be at
the embassy.
Alice Bob

9. Encryption
function
Plaintext Ciphertext
Dear Bob - The Qrne Obo - Gur
meeting will be at zrrgvat jvyy or
the embassy. ng gur rzonffl.
Alice Bob

10. Encryption
function
Plaintext Ciphertext
Dear Bob - The Qrne Obo - Gur
meeting will be at zrrgvat jvyy or
the embassy. ng gur rzonffl.
Alice Bob

11. Decryption
Encryption
function
function
Plaintext Ciphertext
Dear Bob - The Qrne Obo - Gur
meeting will be at zrrgvat jvyy or
the embassy. ng gur rzonffl.
Alice Bob

12. Decryption
Encryption
Original
function
function
Plaintext Ciphertext plaintext
Dear Bob - The Qrne Obo - Gur
meeting will be at zrrgvat jvyy or Dear Bob - The
the embassy. ng gur rzonffl. meeting will be
at the embassy.
Alice Bob

13. Decryption
Encryption
Original
function
function
Plaintext Ciphertext plaintext
Dear Bob - The Qrne Obo - Gur
meeting will be at zrrgvat jvyy or Dear Bob - The
the embassy. ng gur rzonffl. meeting will be
at the embassy.
Alice Bob
How are these
communicated?

14. The key distribution problem

15. The key distribution problem

16. The key distribution problem

17. The key distribution problem

18. The key distribution problem
Insecure channel

19. The key distribution problem
Insecure channel

20. The key distribution problem
Insecure channel

21. The key distribution problem
Insecure channel

22. The key distribution problem
Insecure channel
CENSORED

23. The key distribution problem
Insecure channel
CENSORED
Anonymity

24. The key distribution problem
Insecure channel
CENSORED
Anonymity

25. The key distribution problem
Insecure channel
CENSORED
Anonymity

26. The key distribution problem
Insecure channel
CENSORED
Anonymity

27. The key distribution problem
Insecure channel
CENSORED
Anonymity

28. The key distribution problem
Insecure channel
CENSORED
Anonymity

29. The key distribution problem
Insecure channel
CENSORED
Anonymity
Many-to-many
communication

30. Activity: A “public-key” cipher

31. Why does Kid Crypto work?
Can an adversary find the private key from the
public key?
How does this solve the key distribution problem?

32. Alice Bob

33. Alice Bob

34. Alice Bob

35. Alice Bob
One-way trapdoor function

36. Alice Bob
Public Private
One-way trapdoor function (e,n) d

37. Plaintext
Dear Bob - The
meeting will be at
the embassy.
Alice Bob
Public Private
One-way trapdoor function (e,n) d

38. Plaintext
Dear Bob - The
meeting will be at
the embassy.
Alice Bob
Public Private
One-way trapdoor function (e,n) d

39. Encryption
(ex) mod n
function
Plaintext
Dear Bob - The
meeting will be at
the embassy.
Alice Bob
Public Private
One-way trapdoor function (e,n) d

40. Encryption
(ex) mod n
function
Plaintext Ciphertext
Dear Bob - The Qrne Obo - Gur
meeting will be at zrrgvat jvyy or
the embassy. ng gur rzonffl.
Alice Bob
Public Private
One-way trapdoor function (e,n) d

41. Encryption
(ex) mod n
function
Plaintext Ciphertext
Dear Bob - The Qrne Obo - Gur
meeting will be at zrrgvat jvyy or
the embassy. ng gur rzonffl.
Alice Bob
Public Private
One-way trapdoor function (e,n) d

42. Decryption
(dy) mod n
Encryption
(ex) mod n
function
function
Plaintext Ciphertext
Dear Bob - The Qrne Obo - Gur
meeting will be at zrrgvat jvyy or
the embassy. ng gur rzonffl.
Alice Bob
Public Private
One-way trapdoor function (e,n) d

43. Decryption
(dy) mod n
Encryption
(ex) mod n
Original
function
function
Plaintext Ciphertext plaintext
Dear Bob - The Qrne Obo - Gur
meeting will be at zrrgvat jvyy or Dear Bob - The
the embassy. ng gur rzonffl. meeting will be
at the embassy.
Alice Bob
Public Private
One-way trapdoor function (e,n) d

44. RSA
Choose two large prime numbers p, q
Calculate n = pq and m = (p-1)(q-1)
Choose e such that gcd(e, (p-1)(q-1)) = 1
Choose d such that ed = 1 mod (p-1)(q-1)
Alice Publicize e, n. Keep d private.
To encrypt x: calculate y = xe mod n
To decrypt y: calculate z = yd mod n.
Fermat’s Little Theorem:
Bob xed = x mod n.

45. This number is a product of two primes:
74037563479561712828046796097
42957314259318888923128908493
62326389727650340282662768919
96419625117843995894330502127
58537011896809828673317327310
89309005525051168770632990723
96380786710086096962537934650
563796359

46. Modulus (1024 bits):
9d 88 94 32 62 30 a3 3b
0f 5b 30 7c d5 69 75 21
25 99 0f e1 a8 c2 7d 5b
2b e9 24 72 48 fa 13 47
7c 53 8b 6a 56 be 66 5a
93 7c 3b bb 84 d4 90 4e
ab d5 aa f1 33 de 93 2a
1c 80 e3 f5 19 25 a8 69
d3 55 44 fd 7f 76 bb 28
f5 90 62 69 57 eb 82 ba
dd 73 40 c0 dd fb 7e 2b
ea 86 8e 83 eb e1 5f cc
f4 a5 a4 be 99 4f 90 ff
66 79 b1 44 d4 13 60 96
80 00 3d f8 49 b5 e7 82
54 b3 9d 17 56 cc 90 bb
Exponent (24 bits):
01 00 01

47. Cutting edge
Design/analysis of Elliptic curve Quantum
block ciphers cryptography cryptography

48. Cryptology at Franklin College
MAT 135 CMP 141
MAT 140 Computing I
MAT 142
Calculus II
MAT 240 MAT 300
MAT 233
Methods of Topics in
Linear Algebra
Problem Solving Mathematics
MAT 361
Modern Algebra Careers: NSA and other
government; network security
MAT 490 and CISSP license; more
Individualized
Study

49. For more information
http://delicious.com/robert.talbert/crypto
http://www.slideshare.net/rtalbert
http://en.wikipedia.org/wiki/Portal:Cryptography