Public Key Cryptography
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1 Public Key Cryptography Spotlight on Science J. Robert Buchanan Department of Mathematics 2011
2 What is Cryptography? cryptography: study of methods for sending messages in a form that only be understood by the recipient. cryptanalysis: study of methods for defeating cryptography. cryptology: study of both cryptography and cryptanalysis.
3 Terminology Plaintext: the un-encrypted, clearly readable form of a message. Ciphertext: the encrypted form of the plaintext, readable only with the key to the cipher.
4 Public Key Cryptography In some cryptographic systems, the key used to encipher the message must be kept hidden from the public and should be known only by the sender and recipient of the message. There are other cryptographic systems in which each party has a private key (which they keep secret) and a public key which they publish so that others may send them encrypted messages which only they can decrypt. The public key functions as the recipient s cryptographic telephone number.
5 Overview of Public Key Cryptography Suppose Alice wants to send Bob a message which only Bob will be able to read. 1 Alice looks up Bob s public key and enciphers the plaintext using his public key.
6 Overview of Public Key Cryptography Suppose Alice wants to send Bob a message which only Bob will be able to read. 1 Alice looks up Bob s public key and enciphers the plaintext using his public key. 2 Alice sends Bob the ciphertext (or even publishes it in the newspaper!).
7 Overview of Public Key Cryptography Suppose Alice wants to send Bob a message which only Bob will be able to read. 1 Alice looks up Bob s public key and enciphers the plaintext using his public key. 2 Alice sends Bob the ciphertext (or even publishes it in the newspaper!). 3 Bob decrypts the ciphertext using his private key.
8 RSA Public Key Cryptography One of the most popular public key cryptographic systems is known as RSA. It was developed by Adi Shamir Ronald Rivest Leonard Adelman.
9 A Little Number Theory Definition A prime number is a whole number p greater than 1 whose only divisors are p and 1. An integer which is not prime is called a composite number.
10 A Little Number Theory Definition A prime number is a whole number p greater than 1 whose only divisors are p and 1. An integer which is not prime is called a composite number. Definition If a and b are positive integers, the greatest common divisor of a and b is number denoted gcd(a, b) and is the positive integer d satisfying both of the following properties: d divides a and d divides b, if c is an integer and c divides a and c divides b, then c divides d.
11 A Little Number Theory Definition A prime number is a whole number p greater than 1 whose only divisors are p and 1. An integer which is not prime is called a composite number. Definition If a and b are positive integers, the greatest common divisor of a and b is number denoted gcd(a, b) and is the positive integer d satisfying both of the following properties: d divides a and d divides b, if c is an integer and c divides a and c divides b, then c divides d. Definition If gcd(a, b) = 1 we say that a and b are relatively prime.
12 Examples 4 = gcd(16, 60) 13 = gcd(13, 65)? = gcd(72, 81)
13 Euclidean Algorithm To find the gcd(a, b): 1 If b divides a evenly, gcd(a, b) = b. 2 If not, replace a by b and replace b by the remainder of a b and go back to step 1. A Java applet implementing the Euclidean Algorithm is also available.
14 Example Find gcd(105, 56) using the Euclidean Algorithm.
15 Example Find gcd(105, 56) using the Euclidean Algorithm. a b gcd(105, 56) = 7
16 Linear Combinations Definition If a and b are integers, then an expression of the from ax + by where x and y are also integers is called a linear combination of a and b.
17 Linear Combinations Definition If a and b are integers, then an expression of the from ax + by where x and y are also integers is called a linear combination of a and b. Theorem If a and b are integers, then gcd(a, b) is the smallest positive integer which can be written as a linear combination of a and b.
18 Modular Arithmetic Definition Two integers n and m are congruent modulo k (or equivalent modulo k) written symbolically as when k divides n m evenly. n m (mod k) (mod 26) since = 2.
19 Modular Arithmetic Definition Two integers n and m are congruent modulo k (or equivalent modulo k) written symbolically as when k divides n m evenly. n m (mod k) (mod 26) since = 2. Corollary If a and b are relatively prime then b has a multiplicative inverse modulo a.
20 Extended Euclidean Algorithm To find x and y such that use the tabular method. gcd(a, b) = ax + by Start the table like this: x y d 1 0 a 0 1 b???
21 Extended Euclidean Algorithm To find x and y such that gcd(a, b) = ax + by use the tabular method. Start the table like this: x y d 1 0 a 0 1 b??? To fill in the next row: 1 Find q and r such that a b = q + r b. 2 Multiply next to last row by q and subtract from third to last row. 3 Keep appending rows to the table until r = 0.
22 Example x y d ( 1)(105) + (2)(56) = 7 A Java applet implementing the Extended Euclidean Algorithm is also available.
23 More Number Theory Definition If N is the set of all positive integers then for each positive integer n define φ(n) to be the number of positive integers less than or equal to n which are relatively prime to n. Such a function φ is called the Euler phi function.
24 Euler Phi Function (1 of 2) Do you notice a pattern? n
25 Euler Phi Function (2 of 2) n φ(n) n φ(n)
26 Euler Phi Function (2 of 2) n φ(n) n φ(n) Lemma If p is prime, then φ(p) = p 1.
27 Euler Phi Function (2 of 2) n φ(n) n φ(n) Lemma If p is prime, then φ(p) = p 1. Lemma If p and q are distinct primes, then φ(pq) = φ(p)φ(q) = (p 1)(q 1).
28 Getting Public and Private Keys A public agency (call it the Key Utility ) 1 selects two large primes p and q, 1 let p = 1009 and q = 2003.
29 Getting Public and Private Keys A public agency (call it the Key Utility ) 1 selects two large primes p and q, 2 forms the product n = pq, 1 let p = 1009 and q = n =
30 Getting Public and Private Keys A public agency (call it the Key Utility ) 1 selects two large primes p and q, 2 forms the product n = pq, 3 computes φ(n), 1 let p = 1009 and q = n = φ( ) = (1009 1)(2003 1) =
31 Getting Public and Private Keys A public agency (call it the Key Utility ) 1 selects two large primes p and q, 2 forms the product n = pq, 3 computes φ(n), 4 selects integer e such that gcd(e, φ(n)) = 1, 1 let p = 1009 and q = n = φ( ) = (1009 1)(2003 1) = e = 1003 (for example)
32 Getting Public and Private Keys A public agency (call it the Key Utility ) 1 selects two large primes p and q, 2 forms the product n = pq, 3 computes φ(n), 4 selects integer e such that gcd(e, φ(n)) = 1, 5 finds the multiplicative inverse d of e (mod φ(n)), 1 let p = 1009 and q = n = φ( ) = (1009 1)(2003 1) = e = 1003 (for example) 5 d = (for this e)
33 Getting Public and Private Keys A public agency (call it the Key Utility ) 1 selects two large primes p and q, 2 forms the product n = pq, 3 computes φ(n), 4 selects integer e such that gcd(e, φ(n)) = 1, 5 finds the multiplicative inverse d of e (mod φ(n)), 6 private key is d, public key is (e, n). 1 let p = 1009 and q = n = φ( ) = (1009 1)(2003 1) = e = 1003 (for example) 5 d = (for this e) 6 private key is , public key is (1003, ).
34 Getting Public and Private Keys A public agency (call it the Key Utility ) 1 selects two large primes p and q, 2 forms the product n = pq, 3 computes φ(n), 4 selects integer e such that gcd(e, φ(n)) = 1, 5 finds the multiplicative inverse d of e (mod φ(n)), 6 private key is d, public key is (e, n). 1 let p = 1009 and q = n = Remark: only the Key Utility knows p and q. 3 φ( ) = (1009 1)(2003 1) = e = 1003 (for example) 5 d = (for this e) 6 private key is , public key is (1003, ).
35 Public Keys are Like Telephone Numbers Suppose Alice and Bob have the following keys. Key Holder Public Key Private Key Alice (e a, n a ) = (1003, ) d a = Bob (e b, n b ) = (10007, ) d b =
36 Public Keys are Like Telephone Numbers Suppose Alice and Bob have the following keys. Key Holder Public Key Private Key Alice (e a, n a ) = (1003, ) d a = Bob (e b, n b ) = (10007, ) d b = Private key d a is known only to Alice (and the Key Utility), while private key d b is known only to Bob (and the Key Utility).
37 Public Keys are Like Telephone Numbers Suppose Alice and Bob have the following keys. Key Holder Public Key Private Key Alice (e a, n a ) = (1003, ) d a = Bob (e b, n b ) = (10007, ) d b = Private key d a is known only to Alice (and the Key Utility), while private key d b is known only to Bob (and the Key Utility). If Alice wants to send a message to Bob, she looks up Bob s public key (e b, n b ).
38 Encrypting Using RSA Alice wants to send the message hi to Bob. 1 Alice converts hi to a number, m. Letter h i ASCII Code m =
39 Encrypting Using RSA Alice wants to send the message hi to Bob. 1 Alice converts hi to a number, m. Letter h i ASCII Code m = Alice creates the ciphertext c using the formula c m e b (mod n b ) c = (mod ) c = and sends it to Bob. Alice can try this Java applet if she needs help with modular exponentiation.
40 Encrypting Using RSA Alice wants to send the message hi to Bob. 1 Alice converts hi to a number, m. Letter h i ASCII Code m = Alice creates the ciphertext c using the formula c m e b (mod n b ) c = (mod ) c = and sends it to Bob. Alice can try this Java applet if she needs help with modular exponentiation. Remark: if m > n b break m into chunks smaller than n b.
41 Bob Decrypts the Ciphertext When Bob receives the ciphertext c he uses his private key d b and computes m c d b (mod n b ) m = (mod ) m =
42 Bob Decrypts the Ciphertext When Bob receives the ciphertext c he uses his private key d b and computes m c d b (mod n b ) m = (mod ) m = ASCII Code Letter h i
43 ASCII Character Codes a 97 f 102 k 107 p 112 u 117 b 98 g 103 l 108 q 113 v 118 c 99 h 104 m 109 r 114 w 119 d 100 i 105 n 110 s 115 x 120 e 101 j 106 o 111 t 116 y 121 z 121
44 Is RSA Secure? The security of RSA depends on the difficulty in factoring large numbers like n.
45 Is RSA Secure? The security of RSA depends on the difficulty in factoring large numbers like n. If n = pq where p and q are prime numbers with more than 500 digits each then n has more than 1000 digits.
46 Is RSA Secure? The security of RSA depends on the difficulty in factoring large numbers like n. If n = pq where p and q are prime numbers with more than 500 digits each then n has more than 1000 digits. It takes approximately years to factor a 500-digit composite number using today s computers.
47 Is RSA Secure? The security of RSA depends on the difficulty in factoring large numbers like n. If n = pq where p and q are prime numbers with more than 500 digits each then n has more than 1000 digits. It takes approximately years to factor a 500-digit composite number using today s computers. As computers get faster, we can keep increasing n.
48 Will We Run Out of Primes? Theorem There are infinitely many prime numbers.
49 Will We Run Out of Primes? Theorem There are infinitely many prime numbers. Proof. Suppose there are only finitely many primes p 1 < p 2 < < p k.
50 Will We Run Out of Primes? Theorem There are infinitely many prime numbers. Proof. Suppose there are only finitely many primes p 1 < p 2 < < p k. Form the number q = (p 1 )(p 2 ) (p k ) + 1 > p k. This number is either prime or composite.
51 Will We Run Out of Primes? Theorem There are infinitely many prime numbers. Proof. Suppose there are only finitely many primes p 1 < p 2 < < p k. Form the number q = (p 1 )(p 2 ) (p k ) + 1 > p k. This number is either prime or composite. 1 q (mod p i )
52 Will We Run Out of Primes? Theorem There are infinitely many prime numbers. Proof. Suppose there are only finitely many primes p 1 < p 2 < < p k. Form the number q = (p 1 )(p 2 ) (p k ) + 1 > p k. This number is either prime or composite. 1 q (mod p i ) No (known) prime evenly divides q.
53 Will We Run Out of Primes? Theorem There are infinitely many prime numbers. Proof. Suppose there are only finitely many primes p 1 < p 2 < < p k. Form the number q = (p 1 )(p 2 ) (p k ) + 1 > p k. This number is either prime or composite. 1 q (mod p i ) No (known) prime evenly divides q. Either q itself is prime or there is a prime larger than p k which divides q.
54 Where Can You Learn More? The modern language of cryptology is mathematics. Major in mathematics in college. Take a course in number theory. The National Security Agency employs and educates cryptologists. The National Cryptologic Museum has exhibits on ancient and modern cryptography.
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