1. Numbers on the Internet
GCNU 1025
Numbers Save the Day

2. Internet security Anonymity on internet
Example: internet crime
IP (internet protocol) address: identification of computer
Information of network/host ID
Assigned by internet service providers (ISP)
Static/Dynamic (details not required)

3. Internet security IPv4: Internet Protocol version 4 4 parts (octets):
8 bits for each part (32 bits in total)
Each part expressed as base 10 numbers for convenience
At most ≈4 billion different addresses

4. Internet security IPv6: Internet Protocol version 6 8 parts:
16 bits for each part (128 bits in total)
Each part expressed as base 16 numbers
At most different addresses

5. Internet security Example: online banking
Double security: password and security device
Number-generating security device
A 6-digit code generated from device ID and time (by a secret method) for each log-in session (access granted only if entered code agrees with one generated by bank server)
Generated code effective only within seconds

6. Simple cryptography
Encryption: from plaintexts to ciphers (with a specific rule)
Decryption: from ciphers to plaintexts (with a specific rule)
Cryptography: study of encryption and decryption
Encryption and decryption can be symmetric (same key for both processes) or asymmetric (different keys)

7. Numbers on the Internet
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Numbers Save the Day

8. Caesar cipher
Cipher obtained by simple letter shifting with a fixed (forward) shifting number (A-Z treated as a cycle)
Example:
Plaintext: IT’S NOT A JOURNEY
Cipher: JU'T OPU B KPVSOFZ
Shifting number: 1 (encryption: letters shifted forward by 1 place)

9. Caesar cipher Example: Classwork How to decrypt?
Plaintext: NUMBER IS FUNNY
Shifting number: 4
Cipher: RYQFIV MW JYRRC
How to decrypt?
Key: shifting number
Caesar cipher: easy to break by simple trial-and-error
Key feature: single shifting number

10. Vigenere cipher
Vigenere cipher: higher security level with multiple shifting numbers
Example:
Shifting numbers (key used in both encryption and decryption): (12, 0, 19, 7)

11. Vigenere cipher
Shifting numbers represented by a word according to the rule
Example: (12, 0, 19, 7) represented by “MATH”

12. Vigenere cipher Plaintext: THISISBORING Key: YES = (24, 4, 18)
Encrypted message (cipher): RLAQMKZSJGRY

13. Substitution cipher
Systematically replacing each letter by another letter
Examples: Caesar cipher and Vigenere cipher
Simple substitution cipher:

14. Ways of breaking codes Look for single-letter words (A or I)
Look for special features such as apostrophes (’)
Look for particular patterns
Tackle shorter words first
Frequency analysis

15. Ways of breaking codes Example: Kieron Bryan’s Murder attempt 2012
Encrypted letter sent to sister (during investigation)
Code broken by police in 3 days

16. Ways of breaking codes

17. Ways of breaking codes Example: Kieron Bryan’s Murder attempt 2012
Particular pattern: 33, 9, 5, 10, 3, 5 (PLEASE)
Feb 2012 News: Police cracked the code to uncover gunman's bribery bid

18. Numbers on the Internet
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Numbers Save the Day

19. Ways of breaking codes Frequency analysis
Example: frequencies for a plaintext passage

20. Ways of breaking codes Frequency analysis
Compare the standard frequencies with the frequencies obtained from the encrypted message (cipher)
More useful for long messages
Patterns also considered

21. Ways of breaking codes Frequency analysis
Example: Kieron Bryan’s Murder attempt 2012
“E” is generally the most frequently used letter
“5” appears most frequently in cipher
Reasonable guess: “5” is the cipher for “E”

22. Classwork: frequency analysis
Identifying shifting number of a Caesar cipher by frequency analysis instead of trial-and-error

23. Ways of breaking codes Frequency analysis
Example: frequencies for a Vigenere cipher
Key: BRADPITT
Cipher:

24. Ways of breaking codes Frequency analysis
Example: frequencies for a Vigenere cipher
Key: BRADPITT
Frequencies for cipher: are the numbers useful?

25. Ways of breaking codes Frequency analysis Caesar cipher
Preserves frequencies
Easy to break with trial-and-error with the help of frequency analysis
The most frequent cipher letter is likely to represent E or A
Vigenere cipher
Does not preserve frequencies
Not easy to break even with frequency analysis

26. Numbers on the Internet
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Numbers Save the Day

27. Ways of breaking codes Cryptogram
Common game in newspapers and magazines
Example:

28. Ways of breaking codes Cryptogram
Common game in newspapers and magazines
Example:
Step 1: frequency analysis (P likely to be cipher of E)

29. Ways of breaking codes Cryptogram
Common game in newspapers and magazines
Example:
Step 1: frequency analysis (P likely to be cipher of E)

30. Ways of breaking codes Cryptogram
Common game in newspapers and magazines
Example:
Step 2: look for particular patterns (such as “_EE_”)

31. Ways of breaking codes Cryptogram
Common game in newspapers and magazines
Example:
Step 3: look for short common words

32. Ways of breaking codes Cryptogram
Common game in newspapers and magazines
Example:

33. Numbers on the Internet
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Numbers Save the Day

34. Modular arithmetic Example: letter shifting
26 “=” 0, 27 “=” 1, 28 “=” 2, 45 “=” 19, 71 “=” 19, etc.
Key number: 26
Do the numbers differ by a certain number of complete cycles?
Is the difference a multiple of 26?

35. Modular arithmetic Example: clock 13 “=” 1, 15 “=” 3, 23 “=” 11, etc.
Key number: 12
Do the numbers differ by a certain number of complete cycles?
Is the difference a multiple of 12?

36. Modular arithmetic Example: letter shifting
26  0 (mod 26), 45  19 (mod 26), 71  19 (mod 26), etc.
Congruence modulo 26:
Do the numbers differ by a certain number of complete cycles?
Is the difference a multiple of 26?

37. Modular arithmetic Example: clock
13  1 (mod 12), 15  3 (mod 12), 23  11 (mod 12), etc.
Congruence modulo 12:
Do the numbers differ by a certain number of complete cycles?
Is the difference a multiple of 12?

38. Modular arithmetic Example: 37  89 (mod 26)? YES
Reason: 89 – 37 = 52 is divisible by 26
Example: 38  89 (mod 25)? NO
Reason: 89 – 38 = 51 is not divisible by 25

39. Modular arithmetic Example: congruence modulo 3

40. Modular arithmetic Application in letter shifting
Example: Caesar cipher with shifting number 4
24+4=28≡2 (mod 26)

41. Modular arithmetic Application in letter shifting
Example: Caesar cipher with shifting number 17

42. Numbers on the Internet
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Numbers Save the Day

43. Announcement Mid-term test on Nov 7 (Friday) 20% of final score 1-hour
Coverage: up to Chapter 2
Closed book
DO NOT use calculators in MOBILE PHONES!
DO NOT use electronic devices except calculators!
Past midterm paper:

44. Modular arithmetic Application in letter shifting
Why does the cycle begin with 0?
What is 64?
Answer: 12
What is the remainder of 64÷26?

45. Modular arithmetic Application in letter shifting
Example: Caesar cipher (encryption) with shifting number 128
Operation #1: adding 128
A encrypted as Y
Operation #2: adding 24
128≡24 (mod 26)

46. Modular arithmetic Application in letter shifting
Example: Caesar cipher (decryption) with shifting number 88
Operation #1: subtracting by 88
A decrypted as Q
Operation #2: subtracting by 10
A encrypted as Q
88≡10 (mod 26)

47. Modular arithmetic Properties: Examples:
Shifting number of 128 same as shifting number of 24
0+128≡26+24 (mod 26)
Shifting number of 88 same as shifting number of 10
1−88≡27−10 (mod 26)

48. Modular arithmetic Properties: Examples: 2×13≡2×25 (mod 12)

49. Numbers on the Internet
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50. Modular arithmetic

51. Modular arithmetic Divisions and inverses in ordinary arithmetic
Division can be expressed as multiplication
Example: 6÷3=6× 1 3
3 and 1/3 are a pair
Property: 3× 1 3 =1
Example: 8÷5=8× 1 5
5 and 1/5 are a pair
Property: 5× 1 5 =1
Inverse of a number: a number with which the product is 1
Example: inverse of 3 is 1/3
Example: inverse of 1/5 is 5

52. Modular arithmetic Definition of inverse
Example: is 3 an inverse of 9 modulo 26? YES
3×9=27
Is 27 congruent to 1 modulo 26? YES
Example: is 5 an inverse of 7 modulo 26? NO
5×7=35
35≡1 (mod 26)? NO

53. Modular arithmetic Definition of inverse
Example: is 1 an inverse of 4 modulo 6? NO
1×4=4
4≡1 (mod 6)? NO
Example: is 2 an inverse of 4 modulo 6? NO
2×4=8
8≡1 (mod 6)? NO

54. Modular arithmetic Definition of inverse
Example: is 3 an inverse of 4 modulo 6? NO
3×4=12
12≡1 (mod 6)? NO
Example: is 4 an inverse of 4 modulo 6? NO
4×4=16
16≡1 (mod 6)? NO

55. Modular arithmetic Definition of inverse
Example: is 5 an inverse of 4 modulo 6? NO
5×4=20
20≡1 (mod 6)? NO
Example: is 15 an inverse of 4 modulo 6? NO
3×4=12
12≡1 (mod 6)? NO

56. Modular arithmetic Existence of inverse
Example: 4 has no inverse modulo 6
Reason: 4 and 6 share 2 as a common factor

57. Modular arithmetic Existence of inverse
Example: 5 has an inverse modulo 6
Reason: 5 and 6 share no common factor other than 1

58. Modular arithmetic Existence of inverse
Example: 9 has an inverse modulo 26
Reason: 9 and 26 share no common factor other than 1

59. Modular arithmetic Existence of inverse
Example: 8 has no inverse modulo 26
Reason: 8 and 26 share 2 as a common factor

60. Numbers on the Internet
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61. Modular arithmetic Uniqueness (modulo 𝑛) of inverse
Example: 5 has an inverse modulo 6
Reason: 5 and 6 share no common factor other than 1
5 is an inverse of 5 since 5×5=25≡1 (mod 6)
11 is also an inverse of 5 since 11×5=55≡1 (mod 6)
17 is also an inverse of 5 since 17×5=85≡1 (mod 6)
5, 11 and 17 are all congruent modulo 6
One inverse only in the sense of modulo 6

62. Modular arithmetic How to find an inverse? Trial-and-error
Example: 5 has an inverse modulo 6
Reason: 5 and 6 share no common factor other than 1
Candidates: 1, 2, 3, 4 and 5
1 is not an inverse of 5: 1×5=5 is not congruent to 1 (mod 6)
2 is not an inverse of 5: 2×5=10 is not congruent to 1 (mod 6)
3 is not an inverse of 5: 3×5=15 is not congruent to 1 (mod 6)
4 is not an inverse of 5: 4×5=20 is not congruent to 1 (mod 6)
5 is an inverse of 5: 5×5=25≡1 (mod 6)

63. Modular arithmetic How to find an inverse? Euclidean algorithm
Example: inverse of 13 modulo 74
Target: find an expression 13𝑢+74𝑣=1 so that 13𝑢=1−74𝑣 ≡1 (mod 74) and hence 𝑢 is an inverse of 13 modulo 74
Step 1: divide 74 by 13

64. Modular arithmetic How to find an inverse? Euclidean algorithm
Example: inverse of 13 modulo 74
Target: find an expression 13𝑢+74𝑣=1 so that 13𝑢=1−74𝑣 ≡1 (mod 74) and hence 𝑢 is an inverse of 13 modulo 74
Step 1: divide 74 by 13
Subsequent steps: divide the divisor of the previous division by the remainder of the previous division until the remainder is 1

65. Modular arithmetic How to find an inverse? Euclidean algorithm
Example: inverse of 13 modulo 74
Target: find an expression 13𝑢+74𝑣=1 so that 13𝑢=1−74𝑣 ≡1 (mod 74) and hence 𝑢 is an inverse of 13 modulo 74
Final step: inverse of 13 modulo 74 is -17 (or 57)

66. Numbers on the Internet
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67. Public-key & private-key cryptography
Private-key cryptography: same key used for encryption and decryption
Example: Caesar cipher and Vigenere cipher
Private key: secret between sender and receiver
Symmetric: same key in encryption and decryption
Potential risk: interception by third party during transfer of key

68. Public-key & private-key cryptography
Private-key cryptography: same key used for encryption and decryption
Example: Caesar cipher and Vigenere cipher
Private key: secret between sender and receiver
Symmetric: same key in encryption and decryption
Potential drawback: high number of keys needed in a network

69. Public-key & private-key cryptography
Public-key cryptography: different keys used for encryption and decryption
Analogy: padlock example
Open padlock made public
Sender uses open padlock (public key) to secure message
Receiver uses private key to unlock

70. Public-key & private-key cryptography
Public-key cryptography: different keys used for encryption and decryption
Public key: known to public for encryption
Private key: known to receiver only for decryption
No potential risk of interception during key transfer

71. RSA algorithm
Construction of a pair of public key and private key for public-key cryptography
Important ingredient for asymmetry: difficulty of factorization of large number into prime factors
Multiplying 2 big prime numbers to form large number: simple
Example: what is the product of 1009 and 9973?
Factorizing the product without knowing any of the primes: challenging
Example: how to factorize ?
Tools: modular arithmetic and Euclidean algorithm

72. RSA algorithm
Construction of a pair of public key and private key for public-key cryptography
Example: construction of a public key for others (for encryption) and a private key for yourself (for decryption)
Construction of public key
Choose two prime numbers (known to you only): p = 5 and q = 11
Product of the two primes (known to public): n = 55
Modulo (known to you only): m = (p – 1)(q – 1) = 4 x 10 = 40
Choose a number e (known to public) so that (e, m) = 1: e = 7
Public key: (n, e) = (55, 7)

73. RSA algorithm
Construction of a pair of public key and private key for public-key cryptography
Example: construction of a public key for others (for encryption) and a private key for yourself (for decryption)
Construction of public key
Public key: (n, e) = (55, 7)
Modulo (known to you only): m = (p – 1)(q – 1) = 4 x 10 = 40
Construction of private key
Inverse d of e modulo m (known to you only) via Euclidean algorithm: inverse of 7 modulo 40 is 23
Private key: (n, d) = (55, 23)

74. RSA algorithm
Example (cont’): how to use the public key for encryption?
Public key: (n, e) = (55, 7)
Message: “OK”
Step 1: convert message into numbers according to a rule
“OK” converted into “14 10”
Step 2: encryption by raising to the power of e modulo n
“14 10” encrypted as “9 10”

75. RSA algorithm
Example (cont’): how to use the private key for decryption?
Private key: (n, d) = (55, 23)
Received message: “9 10”
Step 1: decryption by raising to the power of d modulo n
“9 10” decrypted as “14 10”
Step 2: convert numbers back into message
“14 10” converted back into “OK”

76. RSA algorithm Padlock analogy revisited
Open padlock (public key: (55, 7)) made public
Sender uses open padlock (public key: (55, 7)) to secure message
Receiver uses private key (55, 23) to unlock

77. RSA algorithm
Example revisited: construction of a public key for others (for encryption) and a private key for yourself (for decryption)
Construction of public key
Choose two prime numbers (known to you only): p = 5 and q = 11
Product of the two primes (known to public): n = 55
Modulo (known to you only): m = (p – 1)(q – 1) = 4 x 10 = 40
Choose a number e (known to public) so that (e, m) = 1: e = 7
Public key: (n, e) = (55, 7)
Construction of private key
Inverse d of e modulo m (known to you only) via Euclidean algorithm: inverse of 7 modulo 40 is 23
Private key: (n, d) = (55, 23)
Security loophole: 55 is too easy to factorize!

78. RSA algorithm
Example revisited: construction of a public key for others (for encryption) and a private key for yourself (for decryption)
Public key: (n, e) = (55, 7)
Private key: (n, d) = (55, 23)
Security loophole: 55 is too easy to factorize!
Real-life example: very big n used for security reason

79. Announcement
Assignment No.3: next week
Coverage: Chapter 3

80. Numbers on the Internet
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