Number Theory in Cryptography
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1 Number Theory in Cryptography Introduction September 20, 2006 Universidad de los Andes 1
2 Guessing Numbers 2
3 Guessing Numbers (person x) (last 6 digits of phone number of x) 3
4 Guessing Numbers (person x) (last 6 digits of phone number of x) A Hash Function is a function f from A to B such that It is easy to compute f(x) for any x A. For any y B, it is hard to find an x A with f(x) = y. It is hard to find x, x A with x x and f(x) = f(x ). 4
5 Caesar Cipher VIXYVR XS VSQI 5
6 Caesar Cipher VIXYVR XS VSQI A B C D E F G H I J K L M N O P Q R S T U V W X Y Z W X Y Z A B C D E F G H I J K L M N O P Q R S T U V 6
7 Caesar Cipher VIXYVR XS VSQI A B C D E F G H I J K L M N O P Q R S T U V W X Y Z W X Y Z A B C D E F G H I J K L M N O P Q R S T U V RETURN TO ROME 7
8 Caesar Cipher VIXYVR XS VSQI A B C D E F G H I J K L M N O P Q R S T U V W X Y Z W X Y Z A B C D E F G H I J K L M N O P Q R S T U V RETURN TO ROME Breaking the code: just try all 26 shifts. 8
9 Substitution Cipher MQWE WE B YXM QBLHGL A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Q A Z X S W E D C V F R T G B N H Y U J M K I O L P 9
10 Substitution Cipher MQWE WE B YXM QBLHGL A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Q A Z X S W E D C V F R T G B N H Y U J M K I O L P THIS IS A LOT HARDER 10
11 Substitution Cipher MQWE WE B YXM QBLHGL A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Q A Z X S W E D C V F R T G B N H Y U J M K I O L P THIS IS A LOT HARDER Breaking the code: Can not try 26! = permutations... 11
12 Solution: Letter Frequencies English Spanish A B C D E F 29 7 G H 53 7 I J 1 4 K 4 0 L M English Spanish N O P Q 1 9 R S T U V 9 9 W 15 0 X 2 2 Y 20 9 Z 1 5 out of 1000 letters 12
13 Viginere Cipher HVD PZAHSQ JMLEIDRXPSG ZVZ UCH OVZZSFUIY 13
14 Viginere Cipher HVD PZAHSQ JMLEIDRXPSG ZVZ UCH OVZZSFUIY Shift the letters of the encrypted message according to the value of the letters of the secret keyword LLAVES. (a= 1, b= 2,...). A B C D E F G H I J K L M N O P Q R S T U V W X Y Z H V D P Z A H S Q J M L E I D R X P S G Z V Z U C H O V Z Z S F U I Y L L A V E S L L A V E S L L A V E S L L A V E S L L A V E S L L A V E T H E L E T T E R F R E Q U E N C I E S A R E N O T P R E S E R V E D 14
15 Viginere Cipher HVD PZAHSQ JMLEIDRXPSG ZVZ UCH OVZZSFUIY Shift the letters of the encrypted message according to the value of the letters of the secret keyword LLAVES. (a= 1, b= 2,...). A B C D E F G H I J K L M N O P Q R S T U V W X Y Z H V D P Z A H S Q J M L E I D R X P S G Z V Z U C H O V Z Z S F U I Y L L A V E S L L A V E S L L A V E S L L A V E S L L A V E S L L A V E T H E L E T T E R F R E Q U E N C I E S A R E N O T P R E S E R V E D E N E S E N E S Repeated bigrams stay repeated bigrams if their distance is a multiple of the length of the key. 15
16 Security All these ciphers are breakable once the enemy knows the type of encryption. 16
17 Enigma A German WW-II encryption machine, broken by the allies 17
18 ROTORS Period of 26 3 substitutions A REFLECTOR B C D E F 18
19 REFLECTOR ROTORS A B C D E F Period of 26 3 substitutions Weaknesses: Permutations are involutions Letter x does not map to x Rotors can be stolen Book of initial settings too 19
20 REFLECTOR ROTORS A B C D E F Period of 26 3 substitutions Weaknesses: Permutations are involutions Letter x does not map to x Rotors can be stolen Book of initial settings too User errors: repeated initial 3 letters nonrandom initial 3 letters test message with only T s 20
21 REFLECTOR ROTORS A B C D E F Period of 26 3 substitutions Weaknesses: Permutations are involutions Letter x does not map to x Rotors can be stolen Book of initial settings too User errors: repeated initial 3 letters nonrandom initial 3 letters test message with only T s British could decipher until 1932, then extra keyboard permutation. 21
22 REFLECTOR ROTORS A B C D E F Period of 26 3 substitutions Weaknesses: Permutations are involutions Letter x does not map to x Rotors can be stolen Book of initial settings too User errors: repeated initial 3 letters nonrandom initial 3 letters test message with only T s British could decipher until 1932, then extra keyboard permutation. Polish until 1939, then extra rotors, no repeated 3 letters. 22
23 REFLECTOR ROTORS A B C D E F Period of 26 3 substitutions Weaknesses: Permutations are involutions Letter x does not map to x Rotors can be stolen Book of initial settings too User errors: repeated initial 3 letters nonrandom initial 3 letters test message with only T s British could decipher until 1932, then extra keyboard permutation. Polish until 1939, then extra rotors, no repeated 3 letters. At the end of the war all messages could be deciphered in 2 days. The Germans were still confident about ENIGMA. 23
24 Lesson learned A crypto system should be safe even if the enemy knows your encryption algorithm the enemy knows lots of plain texts together with their encryptions (no chosen plain text attacks) 24
25 Lesson learned A crypto system should be safe even if the enemy knows your encryption algorithm the enemy knows lots of plain texts together with their encryptions (no chosen plain text attacks) Solution Use a public algorithm with a secret key. 25
26 Data Encryption Standard (DES, 1974) Xor: (x y) y = x 26
27 Data Encryption Standard (DES, 1974) Xor: (x y) y = x message key encryption
28 Data Encryption Standard (DES, 1974) Xor: (x y) y = x message key encryption encryption key = message 28
29 Data Encryption Standard (DES, 1974) Xor: (x y) y = x message key encryption encryption key = message message encryption = key!danger! 29
30 Data Encryption Standard (DES, 1974) Pick a secret shared key of 64 bits. Divide the message in blocks of 64 bits. Encrypting one block consists of a combination of repeated with parts of the key, permutations, breaking up in subblocks, and small functions by table. 30
31 Data Encryption Standard (DES, 1974) Pick a secret shared key of 64 bits. Divide the message in blocks of 64 bits. Encrypting one block consists of a combination of repeated with parts of the key, permutations, breaking up in subblocks, and small functions by table. Disadvantage: Need to agree on a key before hand... System uses a secret shared key 31
32 Data Encryption Standard (DES, 1974) Pick a secret shared key of 64 bits. Divide the message in blocks of 64 bits. Encrypting one block consists of a combination of repeated with parts of the key, permutations, breaking up in subblocks, and small functions by table. Disadvantage: Need to agree on a key before hand... System uses a secret shared key Problem: How do you prove a cryptography system is secure? 32
33 Public Keys English English Lonapse many Lonapse Lonapse English unique 33
34 Public Keys English English Lonapse public key Lonapse Lonapse English private key 34
35 Public Keys M E L2E M L B M E E2L M L A encrypting, sending, and decrypting a message 35
36 Public Keys M E L2E M L B M E E2L M L A encrypting, sending, and decrypting a message English and Lonapse have same words! 36
37 Public Keys M E L2E M L B M E L2E M NL M E E2L M L E2L A?M E? M NL encrypting, sending, and decrypting a message signing, sending, and checking the signature of a message English and Lonapse have same words! 37
38 Public Keys (RSA) RSA (Rivest, Shamir, Adleman): An n >> 0, a public key e, and a private key d, such that x de x mod n for all x. 38
39 Public Keys (RSA) 0 < M < n x de x mod n M (M e ) d M e B M M d M M e A M? (M d ) e M d encrypting, sending, and decrypting a message M signing, sending, and checking the signature of a message 39
40 Public Keys (RSA) Security of this system is based on our inability to take e-th roots. A factorization of n allows one to compute d from e. It is believed that finding d is as hard as factorizing n. So breaking this system would be as hard as factorizing n. 40
41 Public Keys (RSA) Security of this system is based on our inability to take e-th roots. A factorization of n allows one to compute d from e. It is believed that finding d is as hard as factorizing n. So breaking this system would be as hard as factorizing n. Advantages: compact, use in smart cards both encryption and signing 41
42 Public Keys (RSA) Security of this system is based on our inability to take e-th roots. A factorization of n allows one to compute d from e. It is believed that finding d is as hard as factorizing n. So breaking this system would be as hard as factorizing n. Advantages: compact, use in smart cards both encryption and signing Disadvantages: Computationally intensive only small messages man-in-the-middle attack (weakness of public keys) 42
43 RSA only encripts small messages B M [M, H(M) d ] For signing, you can just sign a hash-function of the message instead. A H(M)? (H(M) d ) e [M, H(M) d ] signing, sending, and checking the signature of a message 43
44 RSA only encripts small messages For encryption, one can use public-key systems to agree on a shared secret key for a more efficient encryption algorithm (like triple-des). A certain way of doing this is called PGP (Pretty Good Privacy) 44
45 Public key systems and the man-in-the-middle attack B A 45
46 Public key systems and the man-in-the-middle attack B A 46
47 Public key systems and the man-in-the-middle attack B M A 47
48 Public key systems and the man-in-the-middle attack B M A 48
49 Public key systems and the man-in-the-middle attack B M A 49
50 Public key systems and the man-in-the-middle attack B M A Solution: A trusted third party (online companies that garantee you are you by checking your credit card info) 50
51 Important Factorizing integers 51
52 Important Factorizing integers Discrete logarithms (tomorrow) 52
53 Important Factorizing integers Discrete logarithms (tomorrow) Coffee (now) 53
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