Salient Features of India constitution especially power and functions
Mat 140 Cryptology Day 4
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
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?
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
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