Numbers on the Internet GCNU 1025 Numbers Save the Day

Numbers on the Internet

GCNU 1025Numbers Save the Day

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)

http://hk.apple.nextmedia.com/news/art/20050114/4587172

http://hk.apple.nextmedia.com/news/art/20050114/4587172

http://whatismyipaddress.com/

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 billion different addresses

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

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 15-30 seconds

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)



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)

Caesar cipher • Example: Classwork

• 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

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)

Vigenere cipher

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

Vigenere cipher

• Plaintext: THISISBORING• Key: YES = (24, 4, 18)

Encrypted message (cipher): RLAQMKZSJGRY

http://sharkysoft.com/misc/vigenere/

Substitution cipher

• Systematically replacing each letter by another letter• Examples: Caesar cipher and Vigenere cipher• Simple substitution cipher:

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


• Example: Kieron Bryan’s Murder attempt 2012• Encrypted letter sent to sister (during investigation) • Code broken by police in 3 days



• 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

http://www.manchestereveningnews.co.uk/news/greater-manchester-news/police-cracked-the-code-to-uncover-gunmans-682871

http://www.manchestereveningnews.co.uk/news/greater-manchester-news/police-cracked-the-code-to-uncover-gunmans-682871




• Frequency analysis• Example: frequencies for a plaintext passage


• Frequency analysis• Compare the standard frequencies with the frequencies obtained from the

encrypted message (cipher)• More useful for long messages• Patterns also considered


• 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”

Classwork: frequency analysis

• Identifying shifting number of a Caesar cipher by frequency analysis instead of trial-and-error

http://www.characterfrequencyanalyzer.com/english/index.php


• Frequency analysis• Example: frequencies for a Vigenere cipher

• Key: BRADPITT • Cipher:


• Frequency analysis• Example: frequencies for a Vigenere cipher

• Key: BRADPITT • Frequencies for cipher: are the numbers useful?


• 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




• Cryptogram• Common game in newspapers and magazines • Example:



• Step 1: frequency analysis (P likely to be cipher of E)



• Step 1: frequency analysis (P likely to be cipher of E)



• Step 2: look for particular patterns (such as “_EE_”)



• Step 3: look for short common words



http://www.youtube.com/watch?v=Pg1B8aB-yBk

http://www.cryptograms.org/



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?

Modular arithmetic

• Example: clock• 13 “=” 1, 15 “=” 3, 23 “=” 11, etc.• Key number: 12


Modular arithmetic

• Example: letter shifting• 26 0 (mod 26), 45 19 (mod 26), 71 19 (mod 26), etc.• Congruence modulo 26:


Modular arithmetic

• Example: clock• 13 1 (mod 12), 15 3 (mod 12), 23 11 (mod 12), etc.• Congruence modulo 12:


Modular arithmetic

• Example: (mod 26)? YES• Reason: 89 – 37 = 52 is divisible by 26

• Example: (mod 25)? NO• Reason: 89 – 38 = 51 is not divisible by 25

Modular arithmetic

• Example: congruence modulo 3• … (mod 3)• … (mod 3)• … (mod 3)

Modular arithmetic

• Application in letter shifting• Example: Caesar cipher with shifting number 4

• (mod 26)

Modular arithmetic

• Application in letter shifting• Example: Caesar cipher with shifting number 17



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: http://www.math.hkbu.edu.hk/~ajzhang/GCNU1025/Past_Midterm.pdf

http://www.math.hkbu.edu.hk/~ajzhang/GCNU1025/Past_Midterm.pdf

Modular arithmetic

• Application in letter shifting• Why does the cycle begin with 0?

• What is 64? • Answer: 12

• What is the remainder of ?• Answer: 12

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• A encrypted as Y

• (mod 26)

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

• (mod 26)

Modular arithmetic

• Properties:

• Examples: • Shifting number of 128 same as shifting number of 24

• (mod 26)• Shifting number of 88 same as shifting number of 10

• (mod 26)

Modular arithmetic

• Properties:

• Examples: • (mod 12)• (mod 12)• (mod 12)• (mod 26)



Modular arithmetic

Modular arithmetic

• Divisions and inverses in ordinary arithmetic• Division can be expressed as multiplication

• Example: • 3 and 1/3 are a pair• Property:

• Example: • 5 and 1/5 are a pair• Property:

• 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

Modular arithmetic

• Definition of inverse

• Example: is 3 an inverse of 9 modulo 26? YES

• Is 27 congruent to 1 modulo 26? YES• Example: is 5 an inverse of 7 modulo 26? NO

• (mod 26)? NO

Modular arithmetic


• Example: is 1 an inverse of 4 modulo 6? NO

• (mod 6)? NO• Example: is 2 an inverse of 4 modulo 6? NO

• (mod 6)? NO

Modular arithmetic




• (mod 6)? NO

Modular arithmetic




• (mod 6)? NO

Modular arithmetic

• Existence of inverse

• Example: 4 has no inverse modulo 6• Reason: 4 and 6 share 2 as a common factor

Modular arithmetic


• Example: 5 has an inverse modulo 6• Reason: 5 and 6 share no common factor other than 1

Modular arithmetic


• Example: 9 has an inverse modulo 26• Reason: 9 and 26 share no common factor other than 1

Modular arithmetic


• Example: 8 has no inverse modulo 26• Reason: 8 and 26 share 2 as a common factor



Modular arithmetic

• Uniqueness (modulo ) 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 (mod 6)• 11 is also an inverse of 5 since (mod 6)• 17 is also an inverse of 5 since (mod 6)• 5, 11 and 17 are all congruent modulo 6 • One inverse only in the sense of modulo 6

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: is not congruent to 1 (mod 6)• 2 is not an inverse of 5: is not congruent to 1 (mod 6)• 3 is not an inverse of 5: is not congruent to 1 (mod 6)• 4 is not an inverse of 5: is not congruent to 1 (mod 6)• 5 is an inverse of 5: (mod 6)

Modular arithmetic

• How to find an inverse?• Euclidean algorithm

• Example: inverse of 13 modulo 74• Target: find an expression so that (mod 74) and hence is an inverse of 13 modulo 74• Step 1: divide 74 by 13

Modular arithmetic


• Example: inverse of 13 modulo 74• Target: find an expression so that (mod 74) and hence 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

Modular arithmetic


• Example: inverse of 13 modulo 74• Target: find an expression so that (mod 74) and hence is an inverse of 13 modulo 74• Final step: inverse of 13 modulo 74 is -17 (or 57)



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


• 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


• 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


• 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

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 10062757?

• Tools: modular arithmetic and Euclidean algorithm

RSA algorithm


• 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)

RSA algorithm


• 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)

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”

http://ptrow.com/perl/calculator.pl

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”

http://ptrow.com/perl/calculator.pl

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

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!

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

Announcement

• Assignment No.3: next week • Coverage: Chapter 3


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Numbers on the Internet GCNU 1025 Numbers Save the Day