An Introduction to Cryptography
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1 An Introduction to Cryptography Spotlight on Science J. Robert Buchanan Department of Mathematics 25 March 2009 J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
2 Introduction 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
3 Introduction Cryptology Throughout History (1 of 3) Subject as old as recorded history. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
4 Introduction Cryptology Throughout History (1 of 3) Subject as old as recorded history. Julius Caesar used substitution ciphers for military and personal communication. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
5 History (2 of 3) Introduction Enigma was patented by Arthur Scherbius in 1918 and became the foundation of German and Japanese codes during WWII. (Photo credit: JRB) J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
6 History (3 of 3) Introduction Since WWII the foundation of cryptology has been mathematics, particularly number theory. (Photo credit: JRB) J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
7 Introduction 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
8 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
9 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
10 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!). J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
11 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
12 Outline Public Key Cryptography RSA 1 Introduction 2 Public Key Cryptography RSA Euclidean Algorithm and Friends Euler Phi Function How it all works 3 For More Information J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
13 RSA 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
14 RSA A Little Number Theory I 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.
15 RSA A Little Number Theory II Definition If gcd(a, b) = 1 we say that a and b are relatively prime.
16 RSA Examples Example 4 = gcd(16, 60) 13 = gcd(13, 65)? = gcd(72, 81) J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
17 Outline Public Key Cryptography Euclidean Algorithm and Friends 1 Introduction 2 Public Key Cryptography RSA Euclidean Algorithm and Friends Euler Phi Function How it all works 3 For More Information J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
18 Euclidean Algorithm and Friends 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
19 Example Public Key Cryptography Euclidean Algorithm and Friends Example Find gcd(105, 56) using the Euclidean Algorithm. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
20 Example Public Key Cryptography Euclidean Algorithm and Friends Example Find gcd(105, 56) using the Euclidean Algorithm. a b gcd(105, 56) = 7 J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
21 Euclidean Algorithm and Friends 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
22 Euclidean Algorithm and Friends 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
23 Euclidean Algorithm and Friends 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) Example (mod 26) since = 2. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
24 Euclidean Algorithm and Friends 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) Example (mod 26) since = 2. Corollary If a and b are relatively prime then b has a multiplicative inverse modulo a. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
25 Extended Euclidean Algorithm Euclidean Algorithm and Friends 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??? J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
26 Extended Euclidean Algorithm Euclidean Algorithm and Friends 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
27 Example Public Key Cryptography Euclidean Algorithm and Friends x y d ( 1)(105) + (2)(56) = 7 A Java applet implementing the Extended Euclidean Algorithm is also available. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
28 Euclidean Algorithm and Friends Finding a Multiplicative Inverse Verify that 23 and 29 are relatively prime and find the multiplicative inverse of 29 modulo 23. x y d (4)(29) + ( 5)(23) (mod 23) 1 (4)(29) (mod 23) So 4 is the multiplicative inverse of 29 modulo 23. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
29 Outline Public Key Cryptography Euler Phi Function 1 Introduction 2 Public Key Cryptography RSA Euclidean Algorithm and Friends Euler Phi Function How it all works 3 For More Information J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
30 Euler Phi Function 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
31 Euler Phi Function (1 of 2) Euler Phi Function Do you notice a pattern? n J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
32 Euler Phi Function (2 of 2) Euler Phi Function n φ(n) n φ(n) J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
33 Euler Phi Function Euler Phi Function (2 of 2) n φ(n) n φ(n) Lemma If p is prime, then φ(p) = p 1. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
34 Euler Phi Function 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). J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
35 Outline Public Key Cryptography How it all works 1 Introduction 2 Public Key Cryptography RSA Euclidean Algorithm and Friends Euler Phi Function How it all works 3 For More Information J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
36 How it all works 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 = J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
37 How it all works 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 = J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
38 How it all works 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) = J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
39 How it all works 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) J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
40 How it all works 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) J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
41 How it all works 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, ). J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
42 How it all works 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). Remark: only the Key Utility knows p and q. 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, ). J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
43 Exchanging Messages How it all works 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 = J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
44 Exchanging Messages How it all works 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). J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
45 Exchanging Messages How it all works 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 the message hi to Bob, she looks up Bob s public key (e b, n b ) and must do two things. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
46 How it all works Encrypting Using RSA 1 Alice converts hi to a number, m. Letter h i ASCII Code m = J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
47 How it all works Encrypting Using RSA 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
48 How it all works Encrypting Using RSA 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 Alice breaks the m into chunks smaller than n b. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
49 How it all works 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 = J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
50 How it all works 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 J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
51 Is RSA Secure? Public Key Cryptography How it all works The security of RSA depends on the difficulty in factoring large numbers like n. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
52 How it all works 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
53 How it all works 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
54 How it all works 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
55 How it all works Will We Run Out of Primes? Theorem There are infinitely many prime numbers. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
56 How it all works 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
57 How it all works 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
58 How it all works 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 ) J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
59 How it all works 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
60 How it all works 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
61 For More Information 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. J. Robert Buchanan (Millersville Univ.) An Introduction to Cryptography 25 March / 33
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