Behrang Noohi. 22 July Behrang Noohi (QMUL) 1 / 18

Transcription

1 Behrang Noohi School of Mathematical Sciences Queen Mary University of London 22 July 2014 Behrang Noohi (QMUL) 1 / 18

2 Introduction Secure Communication How can one send a secret message? Steganography vs. Cryptography; Kerckhoff Principle Cryptography, before Diffie-Hellman (1976): substitution, transposition, polyalphabetic substitution (eg, Vigenere, Enigma),...; Problem: communicating the key More Public-key cryptography; Diffie-Hellman idea More More Behrang Noohi (QMUL) 2 / 18

3 Steganography Steganography: the art of hiding the message Behrang Noohi (QMUL) 3 / 18

4 Steganography The Ambassadors, by Hans Holbein the Younger (1533). The National Gallery. Modern methods Invisible ink, hidden words, microdots, DNA, watermarks... Return Behrang Noohi (QMUL) 4 / 18

5 Steganography The Ambassadors, by Hans Holbein the Younger (1533). The National Gallery. Modern methods Invisible ink, hidden words, microdots, DNA, watermarks... Return Behrang Noohi (QMUL) 4 / 18

6 Cryptography The model Alice and Bob share a secret key, unknown to Eve "Eavesdropper" Alice encrypts the plaintext message with the key, forming a ciphertext. Bob decrypts the ciphertext with the key, obtaining the original plaintext. Eve also receives the ciphertext, but cannot understand it. Kerckhoffs Principle Eve sees the communication AND knows the system. Only the key is secret. Encryption/decryption methods Substitution, transposition, codebook, stream ciphers,... Behrang Noohi (QMUL) 5 / 18

7 Cryptography The model Alice and Bob share a secret key, unknown to Eve "Eavesdropper" Alice encrypts the plaintext message with the key, forming a ciphertext. Bob decrypts the ciphertext with the key, obtaining the original plaintext. Eve also receives the ciphertext, but cannot understand it. Kerckhoffs Principle Eve sees the communication AND knows the system. Only the key is secret. Encryption/decryption methods Substitution, transposition, codebook, stream ciphers,... Behrang Noohi (QMUL) 5 / 18

8 Cryptography The model Alice and Bob share a secret key, unknown to Eve "Eavesdropper" Alice encrypts the plaintext message with the key, forming a ciphertext. Bob decrypts the ciphertext with the key, obtaining the original plaintext. Eve also receives the ciphertext, but cannot understand it. Kerckhoffs Principle Eve sees the communication AND knows the system. Only the key is secret. Encryption/decryption methods Substitution, transposition, codebook, stream ciphers,... Behrang Noohi (QMUL) 5 / 18

9 Substitution ciphers Monoalphabetic substitution Each letter is consistently replaced by another. Example Reversed alphabet: A Z, B Y, C X,... HELLO SVOOL. Permutations The key is a permutation of the alphabet: a bijective map σ : {A,..., Z} {A,..., Z}. Encryption: apply σ to each letter. Decryption: apply the inverse permutation σ 1 to each letter (σ(x) = y σ 1 (y) = x). Behrang Noohi (QMUL) 6 / 18

10 Substitution ciphers Monoalphabetic substitution Each letter is consistently replaced by another. Example Reversed alphabet: A Z, B Y, C X,... HELLO SVOOL. Permutations The key is a permutation of the alphabet: a bijective map σ : {A,..., Z} {A,..., Z}. Encryption: apply σ to each letter. Decryption: apply the inverse permutation σ 1 to each letter (σ(x) = y σ 1 (y) = x). Behrang Noohi (QMUL) 6 / 18

11 Substitution ciphers Monoalphabetic substitution Each letter is consistently replaced by another. Example Reversed alphabet: A Z, B Y, C X,... HELLO SVOOL. Permutations The key is a permutation of the alphabet: a bijective map σ : {A,..., Z} {A,..., Z}. Encryption: apply σ to each letter. Decryption: apply the inverse permutation σ 1 to each letter (σ(x) = y σ 1 (y) = x). Behrang Noohi (QMUL) 6 / 18

12 Cryptanalysis: breaking a cipher The security of a cipher How easy/hard is it to break? (Understand message / obtain key). Brute force Any cipher can be broken by trying all possible keys. How long will it take? Number of substitution cipher keys 26! = = = age of universe in nanoseconds! Computational feasibility Security is relative to our powers of computation. Behrang Noohi (QMUL) 7 / 18

13 Cryptanalysis: breaking a cipher The security of a cipher How easy/hard is it to break? (Understand message / obtain key). Brute force Any cipher can be broken by trying all possible keys. How long will it take? Number of substitution cipher keys 26! = = = age of universe in nanoseconds! Computational feasibility Security is relative to our powers of computation. Behrang Noohi (QMUL) 7 / 18

14 Cryptanalysis: breaking a cipher The security of a cipher How easy/hard is it to break? (Understand message / obtain key). Brute force Any cipher can be broken by trying all possible keys. How long will it take? Number of substitution cipher keys 26! = = = age of universe in nanoseconds! Computational feasibility Security is relative to our powers of computation. Behrang Noohi (QMUL) 7 / 18

15 Cryptanalysis: breaking a cipher The security of a cipher How easy/hard is it to break? (Understand message / obtain key). Brute force Any cipher can be broken by trying all possible keys. How long will it take? Number of substitution cipher keys 26! = = = age of universe in nanoseconds! Computational feasibility Security is relative to our powers of computation. Behrang Noohi (QMUL) 7 / 18

16 Statistical analysis Letter frequencies Some letters are more common than others. The most common letters in English writing are E, T, A, O, I/N, H/S/R,... Frequency analysis Count letter frequencies in the ciphertext; replace the most common ones by E, T, A, etc.; try to guess the others. ZH GRQW JHW SHDU WDUWV IURP SHDFK WUHHV Ze GRQt Jet SeaU tautv IURP SeaFK tueev Ze GRQt Jet Sear tarts IrRP SeaFK trees Ze GoQt Jet Sear tarts IroP SeaFK trees we dont get pear tarts from peach trees Behrang Noohi (QMUL) 8 / 18

17 Statistical analysis Letter frequencies Some letters are more common than others. The most common letters in English writing are E, T, A, O, I/N, H/S/R,... Frequency analysis Count letter frequencies in the ciphertext; replace the most common ones by E, T, A, etc.; try to guess the others. ZH GRQW JHW SHDU WDUWV IURP SHDFK WUHHV Ze GRQt Jet SeaU tautv IURP SeaFK tueev Ze GRQt Jet Sear tarts IrRP SeaFK trees Ze GoQt Jet Sear tarts IroP SeaFK trees we dont get pear tarts from peach trees Behrang Noohi (QMUL) 8 / 18

18 Statistical analysis Letter frequencies Some letters are more common than others. The most common letters in English writing are E, T, A, O, I/N, H/S/R,... Frequency analysis Count letter frequencies in the ciphertext; replace the most common ones by E, T, A, etc.; try to guess the others. ZH GRQW JHW SHDU WDUWV IURP SHDFK WUHHV Ze GRQt Jet SeaU tautv IURP SeaFK tueev Ze GRQt Jet Sear tarts IrRP SeaFK trees Ze GoQt Jet Sear tarts IroP SeaFK trees we dont get pear tarts from peach trees Behrang Noohi (QMUL) 8 / 18

19 Statistical analysis Letter frequencies Some letters are more common than others. The most common letters in English writing are E, T, A, O, I/N, H/S/R,... Frequency analysis Count letter frequencies in the ciphertext; replace the most common ones by E, T, A, etc.; try to guess the others. ZH GRQW JHW SHDU WDUWV IURP SHDFK WUHHV Ze GRQt Jet SeaU tautv IURP SeaFK tueev Ze GRQt Jet Sear tarts IrRP SeaFK trees Ze GoQt Jet Sear tarts IroP SeaFK trees we dont get pear tarts from peach trees Behrang Noohi (QMUL) 8 / 18

20 Statistical analysis Letter frequencies Some letters are more common than others. The most common letters in English writing are E, T, A, O, I/N, H/S/R,... Frequency analysis Count letter frequencies in the ciphertext; replace the most common ones by E, T, A, etc.; try to guess the others. ZH GRQW JHW SHDU WDUWV IURP SHDFK WUHHV Ze GRQt Jet SeaU tautv IURP SeaFK tueev Ze GRQt Jet Sear tarts IrRP SeaFK trees Ze GoQt Jet Sear tarts IroP SeaFK trees we dont get pear tarts from peach trees Behrang Noohi (QMUL) 8 / 18

21 Statistical analysis Letter frequencies Some letters are more common than others. The most common letters in English writing are E, T, A, O, I/N, H/S/R,... Frequency analysis Count letter frequencies in the ciphertext; replace the most common ones by E, T, A, etc.; try to guess the others. ZH GRQW JHW SHDU WDUWV IURP SHDFK WUHHV Ze GRQt Jet SeaU tautv IURP SeaFK tueev Ze GRQt Jet Sear tarts IrRP SeaFK trees Ze GoQt Jet Sear tarts IroP SeaFK trees we dont get pear tarts from peach trees Behrang Noohi (QMUL) 8 / 18

22 Statistical analysis Letter frequencies Some letters are more common than others. The most common letters in English writing are E, T, A, O, I/N, H/S/R,... Frequency analysis Count letter frequencies in the ciphertext; replace the most common ones by E, T, A, etc.; try to guess the others. ZH GRQW JHW SHDU WDUWV IURP SHDFK WUHHV Ze GRQt Jet SeaU tautv IURP SeaFK tueev Ze GRQt Jet Sear tarts IrRP SeaFK trees Ze GoQt Jet Sear tarts IroP SeaFK trees we dont get pear tarts from peach trees Behrang Noohi (QMUL) 8 / 18

23 Modular arithmetic Caesar cipher Previous example used shift by 3: A D, B E, C F,..., Z C. A numerical interpretation Identify A,..., Z with 0,..., 25. Encode e(x) = x + 3 mod 26. Decode d(x) = x 3 mod 26. General shifts Suppose we use an m-letter alphabet, identified with 0,..., m 1. Encode e n (x) = x + n mod m. Decode d n (x) = x n mod m. Behrang Noohi (QMUL) 9 / 18

24 Modular arithmetic Caesar cipher Previous example used shift by 3: A D, B E, C F,..., Z C. A numerical interpretation Identify A,..., Z with 0,..., 25. Encode e(x) = x + 3 mod 26. Decode d(x) = x 3 mod 26. General shifts Suppose we use an m-letter alphabet, identified with 0,..., m 1. Encode e n (x) = x + n mod m. Decode d n (x) = x n mod m. Behrang Noohi (QMUL) 9 / 18

25 Modular arithmetic Caesar cipher Previous example used shift by 3: A D, B E, C F,..., Z C. A numerical interpretation Identify A,..., Z with 0,..., 25. Encode e(x) = x + 3 mod 26. Decode d(x) = x 3 mod 26. General shifts Suppose we use an m-letter alphabet, identified with 0,..., m 1. Encode e n (x) = x + n mod m. Decode d n (x) = x n mod m. Behrang Noohi (QMUL) 9 / 18

26 Caesar s revenge: polyalphabetic ciphers Examples: one-time pad, Vigenere, Enigma Machine,... The one-time pad Keep changing amount we shift by! Let s use binary alphabet {0, 1}. The secret key is a random binary string, say k = Encryption, decryption both m m + k (bitwise addition mod 2): e.g. e( ) = = , d( ) = = Pro: Unbreakable! If k is random then so is m + k: it contains no information about m. (Shannon s Theorem) Con: Inefficient! k is as long as m: it begs the question of how Alice and Bob managed to agree on k. More efficient: short k and long m, break m into blocks b 1, b 2,, encode as b 1 + k, b 2 + k,.... (But this is breakable.) Behrang Noohi (QMUL) 10 / 18

27 Caesar s revenge: polyalphabetic ciphers Examples: one-time pad, Vigenere, Enigma Machine,... The one-time pad Keep changing amount we shift by! Let s use binary alphabet {0, 1}. The secret key is a random binary string, say k = Encryption, decryption both m m + k (bitwise addition mod 2): e.g. e( ) = = , d( ) = = Pro: Unbreakable! If k is random then so is m + k: it contains no information about m. (Shannon s Theorem) Con: Inefficient! k is as long as m: it begs the question of how Alice and Bob managed to agree on k. More efficient: short k and long m, break m into blocks b 1, b 2,, encode as b 1 + k, b 2 + k,.... (But this is breakable.) Behrang Noohi (QMUL) 10 / 18

28 Caesar s revenge: polyalphabetic ciphers Examples: one-time pad, Vigenere, Enigma Machine,... The one-time pad Keep changing amount we shift by! Let s use binary alphabet {0, 1}. The secret key is a random binary string, say k = Encryption, decryption both m m + k (bitwise addition mod 2): e.g. e( ) = = , d( ) = = Pro: Unbreakable! If k is random then so is m + k: it contains no information about m. (Shannon s Theorem) Con: Inefficient! k is as long as m: it begs the question of how Alice and Bob managed to agree on k. More efficient: short k and long m, break m into blocks b 1, b 2,, encode as b 1 + k, b 2 + k,.... (But this is breakable.) Behrang Noohi (QMUL) 10 / 18

29 Caesar s revenge: polyalphabetic ciphers Examples: one-time pad, Vigenere, Enigma Machine,... The one-time pad Keep changing amount we shift by! Let s use binary alphabet {0, 1}. The secret key is a random binary string, say k = Encryption, decryption both m m + k (bitwise addition mod 2): e.g. e( ) = = , d( ) = = Pro: Unbreakable! If k is random then so is m + k: it contains no information about m. (Shannon s Theorem) Con: Inefficient! k is as long as m: it begs the question of how Alice and Bob managed to agree on k. More efficient: short k and long m, break m into blocks b 1, b 2,, encode as b 1 + k, b 2 + k,.... (But this is breakable.) Behrang Noohi (QMUL) 10 / 18

30 The Vigenere cipher We don t communicate in binary! Cipher easier to remember if we use A..Z. The secret key is a word; each letter represents the shift from A to that letter; e.g. CAESAR +2,+0,+4,+18,+0,+17. Example: The rain in Spain falls mainly on the plain. Confusion is created since at different times (i) the same letter may be encoded differently, and (ii) different letters may be encoded identically! Behrang Noohi (QMUL) 11 / 18

31 The Vigenere cipher We don t communicate in binary! Cipher easier to remember if we use A..Z. The secret key is a word; each letter represents the shift from A to that letter; e.g. CAESAR +2,+0,+4,+18,+0,+17. Example: The rain in Spain falls mainly on the plain. Confusion is created since at different times (i) the same letter may be encoded differently, and (ii) different letters may be encoded identically! Behrang Noohi (QMUL) 11 / 18

32 The Vigenere cipher We don t communicate in binary! Cipher easier to remember if we use A..Z. The secret key is a word; each letter represents the shift from A to that letter; e.g. CAESAR +2,+0,+4,+18,+0,+17. Example: The rain in Spain falls mainly on the plain. Confusion is created since at different times (i) the same letter may be encoded differently, and (ii) different letters may be encoded identically! Behrang Noohi (QMUL) 11 / 18

33 The Vigenere cipher We don t communicate in binary! Cipher easier to remember if we use A..Z. The secret key is a word; each letter represents the shift from A to that letter; e.g. CAESAR +2,+0,+4,+18,+0,+17. Example: The rain in Spain falls mainly on the plain. Confusion is created since at different times (i) the same letter may be encoded differently, and (ii) different letters may be encoded identically! Behrang Noohi (QMUL) 11 / 18

34 Breaking the Vigenere cipher Much harder than a substitution, but it has weaknesses... Suppose we know the key length, say it is 6. Just look at the letters in positions 6,12,18,... they are encoded by the same shift: can use frequency analysis! Repeat for other remainders mod 6. How to get the key length? Could guess. Or use more sophisticated statistics... Kasiski method: Look for repeated consecutive pairs (digrams) or triples (trigrams). The key length probably divides the distance between them. Behrang Noohi (QMUL) 12 / 18

35 Breaking the Vigenere cipher Much harder than a substitution, but it has weaknesses... Suppose we know the key length, say it is 6. Just look at the letters in positions 6,12,18,... they are encoded by the same shift: can use frequency analysis! Repeat for other remainders mod 6. How to get the key length? Could guess. Or use more sophisticated statistics... Kasiski method: Look for repeated consecutive pairs (digrams) or triples (trigrams). The key length probably divides the distance between them. Behrang Noohi (QMUL) 12 / 18

36 Breaking the Vigenere cipher Much harder than a substitution, but it has weaknesses... Suppose we know the key length, say it is 6. Just look at the letters in positions 6,12,18,... they are encoded by the same shift: can use frequency analysis! Repeat for other remainders mod 6. How to get the key length? Could guess. Or use more sophisticated statistics... Kasiski method: Look for repeated consecutive pairs (digrams) or triples (trigrams). The key length probably divides the distance between them. Behrang Noohi (QMUL) 12 / 18

37 Breaking the Vigenere cipher Much harder than a substitution, but it has weaknesses... Suppose we know the key length, say it is 6. Just look at the letters in positions 6,12,18,... they are encoded by the same shift: can use frequency analysis! Repeat for other remainders mod 6. How to get the key length? Could guess. Or use more sophisticated statistics... Kasiski method: Look for repeated consecutive pairs (digrams) or triples (trigrams). The key length probably divides the distance between them. Behrang Noohi (QMUL) 12 / 18

38 Enigma Machine A polyalphabetic substitution cipher used by Germans in WWII. (Broken by Polish and British cryptologists.) Permutation of alphabet implemented by a set of rotors (and a plugboard) The permutation changes with each keystroke (ie, rotors turn). German procedural flaws, operator mistakes, laziness, failure to systematically introduce changes in encipherment procedures, and Allied capture of key tables and hardware that, during the war, enabled Allied cryptologists to succeed. Behrang Noohi (QMUL) 13 / 18

39 Enigma Machine A polyalphabetic substitution cipher used by Germans in WWII. (Broken by Polish and British cryptologists.) Permutation of alphabet implemented by a set of rotors (and a plugboard) The permutation changes with each keystroke (ie, rotors turn). German procedural flaws, operator mistakes, laziness, failure to systematically introduce changes in encipherment procedures, and Allied capture of key tables and hardware that, during the war, enabled Allied cryptologists to succeed. Behrang Noohi (QMUL) 13 / 18

40 Enigma Machine A polyalphabetic substitution cipher used by Germans in WWII. (Broken by Polish and British cryptologists.) Permutation of alphabet implemented by a set of rotors (and a plugboard) The permutation changes with each keystroke (ie, rotors turn). German procedural flaws, operator mistakes, laziness, failure to systematically introduce changes in encipherment procedures, and Allied capture of key tables and hardware that, during the war, enabled Allied cryptologists to succeed. Behrang Noohi (QMUL) 13 / 18

41 Enigma Machine A polyalphabetic substitution cipher used by Germans in WWII. (Broken by Polish and British cryptologists.) Permutation of alphabet implemented by a set of rotors (and a plugboard) The permutation changes with each keystroke (ie, rotors turn). German procedural flaws, operator mistakes, laziness, failure to systematically introduce changes in encipherment procedures, and Allied capture of key tables and hardware that, during the war, enabled Allied cryptologists to succeed. Behrang Noohi (QMUL) 13 / 18

42 Enigma Machine A polyalphabetic substitution cipher used by Germans in WWII. (Broken by Polish and British cryptologists.) Permutation of alphabet implemented by a set of rotors (and a plugboard) The permutation changes with each keystroke (ie, rotors turn). German procedural flaws, operator mistakes, laziness, failure to systematically introduce changes in encipherment procedures, and Allied capture of key tables and hardware that, during the war, enabled Allied cryptologists to succeed. Behrang Noohi (QMUL) 13 / 18

43 Key exchange A one-time pad attempt: Alice Eve Bob p m = p + k 1 m 1 A m 2 m2 = m + k 1 B m = m + k 3 2 A m 3 m = m + k = p 4 3 B Return Problem! m 1 + m 2 + m 3 = p. Behrang Noohi (QMUL) 14 / 18

44 Public key cryptography Diffie-Hellman idea: method where key is public knowledge?! How could this possibly work? One-way function e: computing e(x) easy; computing d(y) = e 1 (y) hard. RSA cryptosystem: power map e(x) = x l mod n; l, n public. Inverse problem given y, find x with x l = y mod n believed hard. Trapdoor function: n = pq with p, q large primes, secret key k with kl = 1 mod (p 1)(q 1), d(y) = y k mod n. Behrang Noohi (QMUL) 15 / 18

45 Public key cryptography Diffie-Hellman idea: method where key is public knowledge?! How could this possibly work? One-way function e: computing e(x) easy; computing d(y) = e 1 (y) hard. RSA cryptosystem: power map e(x) = x l mod n; l, n public. Inverse problem given y, find x with x l = y mod n believed hard. Trapdoor function: n = pq with p, q large primes, secret key k with kl = 1 mod (p 1)(q 1), d(y) = y k mod n. Behrang Noohi (QMUL) 15 / 18

46 Public key cryptography Diffie-Hellman idea: method where key is public knowledge?! How could this possibly work? One-way function e: computing e(x) easy; computing d(y) = e 1 (y) hard. RSA cryptosystem: power map e(x) = x l mod n; l, n public. Inverse problem given y, find x with x l = y mod n believed hard. Trapdoor function: n = pq with p, q large primes, secret key k with kl = 1 mod (p 1)(q 1), d(y) = y k mod n. Behrang Noohi (QMUL) 15 / 18

47 Public key cryptography Diffie-Hellman idea: method where key is public knowledge?! How could this possibly work? One-way function e: computing e(x) easy; computing d(y) = e 1 (y) hard. RSA cryptosystem: power map e(x) = x l mod n; l, n public. Inverse problem given y, find x with x l = y mod n believed hard. Trapdoor function: n = pq with p, q large primes, secret key k with kl = 1 mod (p 1)(q 1), d(y) = y k mod n. Behrang Noohi (QMUL) 15 / 18

48 Public key cryptography Diffie-Hellman idea: method where key is public knowledge?! How could this possibly work? One-way function e: computing e(x) easy; computing d(y) = e 1 (y) hard. RSA cryptosystem: power map e(x) = x l mod n; l, n public. Inverse problem given y, find x with x l = y mod n believed hard. Trapdoor function: n = pq with p, q large primes, secret key k with kl = 1 mod (p 1)(q 1), d(y) = y k mod n. Behrang Noohi (QMUL) 15 / 18

49 Issues in modern cryptography Message Integrity: Can Eve crucially change the meaning of a message she cannot entirely read (e.g. the amount in a bank transaction)? Digital Signatures: Eve sees some signed messages, can she forge a signature? Communication protocols: Zero-knowledge proof, Multiparty secrets, Elections, Digital cash... Behrang Noohi (QMUL) 16 / 18

50 Issues in modern cryptography Message Integrity: Can Eve crucially change the meaning of a message she cannot entirely read (e.g. the amount in a bank transaction)? Digital Signatures: Eve sees some signed messages, can she forge a signature? Communication protocols: Zero-knowledge proof, Multiparty secrets, Elections, Digital cash... Behrang Noohi (QMUL) 16 / 18

51 Issues in modern cryptography Message Integrity: Can Eve crucially change the meaning of a message she cannot entirely read (e.g. the amount in a bank transaction)? Digital Signatures: Eve sees some signed messages, can she forge a signature? Communication protocols: Zero-knowledge proof, Multiparty secrets, Elections, Digital cash... Behrang Noohi (QMUL) 16 / 18

52 Conclusion Cryptography concerns secure communication. Unlike steganography (making the message obscure), the assumption (Kerckhoffs Principle) is Eve knows the system; only the key is secret. Ciphers are various methods of using the secret key to encrypt/decrypt a message, e.g. Substitution, Vigenere, Permutation,... Security is always relative to computational power, and in fear of an ingenious unforseen attack. Public Key Cryptography provides great flexibility, but its security is only empirical. Modern cryptography has evolved into a diverse field of theoretical and practical research. Behrang Noohi (QMUL) 17 / 18

53 Conclusion Cryptography concerns secure communication. Unlike steganography (making the message obscure), the assumption (Kerckhoffs Principle) is Eve knows the system; only the key is secret. Ciphers are various methods of using the secret key to encrypt/decrypt a message, e.g. Substitution, Vigenere, Permutation,... Security is always relative to computational power, and in fear of an ingenious unforseen attack. Public Key Cryptography provides great flexibility, but its security is only empirical. Modern cryptography has evolved into a diverse field of theoretical and practical research. Behrang Noohi (QMUL) 17 / 18

54 Conclusion Cryptography concerns secure communication. Unlike steganography (making the message obscure), the assumption (Kerckhoffs Principle) is Eve knows the system; only the key is secret. Ciphers are various methods of using the secret key to encrypt/decrypt a message, e.g. Substitution, Vigenere, Permutation,... Security is always relative to computational power, and in fear of an ingenious unforseen attack. Public Key Cryptography provides great flexibility, but its security is only empirical. Modern cryptography has evolved into a diverse field of theoretical and practical research. Behrang Noohi (QMUL) 17 / 18

55 Conclusion Cryptography concerns secure communication. Unlike steganography (making the message obscure), the assumption (Kerckhoffs Principle) is Eve knows the system; only the key is secret. Ciphers are various methods of using the secret key to encrypt/decrypt a message, e.g. Substitution, Vigenere, Permutation,... Security is always relative to computational power, and in fear of an ingenious unforseen attack. Public Key Cryptography provides great flexibility, but its security is only empirical. Modern cryptography has evolved into a diverse field of theoretical and practical research. Behrang Noohi (QMUL) 17 / 18

56 Conclusion Cryptography concerns secure communication. Unlike steganography (making the message obscure), the assumption (Kerckhoffs Principle) is Eve knows the system; only the key is secret. Ciphers are various methods of using the secret key to encrypt/decrypt a message, e.g. Substitution, Vigenere, Permutation,... Security is always relative to computational power, and in fear of an ingenious unforseen attack. Public Key Cryptography provides great flexibility, but its security is only empirical. Modern cryptography has evolved into a diverse field of theoretical and practical research. Behrang Noohi (QMUL) 17 / 18

57 Thank you! I thank Peter Keevash for letting me use a modified version of his slides. Behrang Noohi (QMUL) 18 / 18