# Secure Network Communication Part II II Public Key Cryptography. Public Key Cryptography

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1 Kommunikationssysteme (KSy) - Block 8 Secure Network Communication Part II II Public Key Cryptography Dr. Andreas Steffen A. Steffen, , KSy_RSA.ppt 1 Secure Key Distribution Problem Bad scalability of symmetric key cryptosystems Public key distribution system as a solution Public Key Cryptography Inventors Basic Principles One-way functions with trap doors Hard problems Mathematical Operations in Finite Fields Addition Negative element Multiplication Inverse element RSA Public Key Cryptosystem Hard problem of factoring large numbers Key generation algorithm Public and private keys Encryption and decryption Efficient Exponentiation of large numbers Contest How to find large prime numbers Diffie-Hellman Key-Exchange Algorithm Generating a common secret key

2 The Secure Key Distribution Problem Alice K AB, K AC, K AD, K AE, K AF K AF, K BF, K CF, K DF, K EF Fred Bob K AB, K BC, K BD, K BE, K BF K AE, K BE, K CE, K AC, K BC, K CD, K DE, K EF Edna secure distribution of n 2 keys Dave Carol K CE, K CF K AD, K BD, K CD, K DE, K DF A. Steffen, , KSy_RSA.ppt 2 Key Distribution Problem in Dense Networks In densely-meshed networks where many parties communicate with each other, the required number of secret keys necessary when using symmetric encryption algorithms increases quadratically with the number of participants since in a fully-meshed network to each of the n communication partners (n-1) keys must be securely delivered. Take as an example a broadband communications network with 100 fully-meshed nodes were each session key is changed every hour, resulting in a requirement to safely distribute about keys each day. As can easily be seen, secret key distribution scales very badly with an increasing number of participants. Therefore for a long time people had been looking for alternative ways of establishing secure connections. A very efficient solution was finally found in 1976 with the novel concept of a Public Key Cryptosystem.

3 Public Key Distribution System Public Directory Fred Alice K A Bob Alice: K A Bob : K B Carol: K C Dave : K D Edna : K E Fred : K F K E K B K E K C public distribution of n keys Edna Dave K D Carol A. Steffen, , KSy_RSA.ppt 3 Public Key Distribution System In a Public Key Cryptosystem each user or host possesses a single key pair consisting of a private key which is kept secret by the no and a matching public key which is published in a public directory (usually an LDAP or WWW server). If a user Alice wants to send an encrypted message to user Bob then Alice encrypts her message with Bob s public key KB fetched from the public directory and sends it to Bob. Since Bob is the only one in possession of the matching private key, he alone can decrypt the encrypted message sent to him. Since only the public key of the recipient is required, with n users only n distinct keys are required. Under the assumption that each user generates her own public/ private key pair locally, no secure channels are required for the distribution of the public keys, since the don t contain any secret and must be put into the public domain anyway.

4 Public Key Cryptography The Inventors Whitfield Diffie and Martin Hellman 1976 Ralph Merkle 1978 Alice Trap Door Bob C = f K B (P) Encryption with one-way function One-way functions are often based on wellknown hard problems P = f -1 KB (C,T B) Computation of inverse function extremely expensive P = f -1 KB (C) K B Joe A. Steffen, , KSy_RSA.ppt 4 Inventors of Public Key Cryptography The concept of a Public Key Cryptosystem was invented at around the same time by Whitfield Diffie, Martin Hellman and Ralph Merkle. Whereas the first two researchers published their invention in 1976 and got all the fame, Ralph Merkle had the misfortune that the printing of his paper got delayed by more than a year so that it got published not until Today it is generally recognized that all three scientists are the fathers of public key cryptography. Recently it became known that already in 1970, James Ellis, at the time working for the British government as a member of the Communications-Electronics Security Group (CESG), formulated the idea of a Public Key Cryptosystem. Several practical algorithms including one variant very similar to RSA and another one identical to the Diffie-Hellman key exchange were discovered within the CESG. Unfortunately the British researchers were not allowed to publish their results due to state security reasons. Basic Principles of Public Key Cryptography All public key cryptosystems are based on the notion of a one-way function, which, depending on the public key, converts plaintext into ciphertext using a relatively small amount of computing power but whose inverse function is extremely expensive to compute, so that an attacker is not able to derive the original plaintext from the transmitted ciphertext within a reasonable time frame. Another notion used in public key cryptosystems is that of a trap door which each one-way function possesses and which can only be activated by the legitimate user holding the private key. Using the trapdoor, decryption of the ciphertext becomes easy. Many public key cryptosystems are based on known hard problems like the factoring of large numbers into their prime factors (RSA) or taking discrete logarithms over a finite field (Diffie-Hellman).

5 RSA Public Key Cryptosystem The Inventors R - Ron Rivest S - Adi Shamir A - Leonard Adleman The One-Way Function The exponentiation function y = f(x) = x e mod n can be computed with reasonable effort. Its inverse x = f -1 (y) is extremely difficult to compute. The Hard Problem Securing the Trap Door The RSA public key algorithm is based on the well-known hard problem of factoring large numbers into its prime factors that has been studied over many centuries. A. Steffen, , KSy_RSA.ppt 5

6 The RSA-155 Challenge The Effort 512 bit number (155 decimal digits) factored on August 22, 1999 after 7 months of cracking 300 workstations and Pentium PCs, 1 Cray supercomputer =? * A. Steffen, , KSy_RSA.ppt 6

7 Secure Network Communication Part II Mathematical Operations in in Finite Fields A. Steffen, , KSy_RSA.ppt 7

8 Mathematical Operations in Finite Fields Addition c = (a + b) mod n = ( a mod n + b mod n) mod n Example with modulus n = b a (7 + 9) mod 5 = 16 mod 5 = 1 (7 + 9) mod 5 = (7 mod mod 5) mod 5 = ( ) mod 5 = 6 mod 5 = 1 c A. Steffen, , KSy_RSA.ppt 8

9 Mathematical Operations in Finite Fields Negative Element y = -x mod n or y + x mod n = 0 Example with modulus n = 5 x y A. Steffen, , KSy_RSA.ppt 9

10 Mathematical Operations in Finite Fields Multiplication c = (a b) mod n = ( a mod n b mod n) mod n Example with modulus n = b a (7 9) mod 5 = 63 mod 5 = 3 (7 9) mod 5 = (7 mod 5 9 mod 5) mod 5 = ( 2 4 ) mod 5 = 8 mod 5 = 3 c A. Steffen, , KSy_RSA.ppt 10

11 Mathematical Operations in Finite Fields Inverse Element y = x -1 mod n or y x mod n = 1 Example with modulus n = 5 x y A. Steffen, , KSy_RSA.ppt 11

12 Secure Network Communication Part II RSA Public Key Cryptosystem A. Steffen, , KSy_RSA.ppt 12

13 RSA Public Key Cryptosystem Key Generation Algorithm Step 1: Choose two random large prime numbers p and q For maximum security, choose p and q of about equal length, e.g bits each. Step 2: Compute the product n = p q Step 3: Choose a random integer e < (p-1)(q-1) The numbers e and (p-1)(q-1) must be relatively prime, i.e. they should not share common prime factors. Step 4: Compute the unique inverse d = e -1 mod (p-1)(q-1) The equation d e mod (p-1)(q-1) = 1 can be solved using the Euclidian algorithm. A. Steffen, , KSy_RSA.ppt 13

14 RSA Public Key Cryptosystem Key Generation Example p = 3, q = 11: n = p q = 33 (p-1) (q-1) = 2 10 = = 20 the public exponent e must be relatively prime to (p-1) (q-1), i.e. it cannot contain any factors of 2 and 5 e d e d e d mod all possible choices for the exponents e and d A. Steffen, , KSy_RSA.ppt 14

15 RSA Public Key Cryptosystem Public and Private Keys Public Key: modulus n and public exponent e Publish n and e in a public directory, so that anybody wanting to send you a confidential message can retrieve it. Private Key: modulus n and private exponent d The private exponent d is your secret. It should be protected either by storing it in a tamper-proof smart card or when stored on a disk by encrypting it with a symmetric cipher secured by a secret passphrase of your choice. The large primes p and q that were used for key generation are not needed any more and should be erased. A. Steffen, , KSy_RSA.ppt 15

16 RSA Public Key Cryptosystem Encryption and Decryption Encryption of a plaintext block x: The sender uses the public key of the recipient to encrypt x < n. Decryption of a ciphertext block y: The recipient uses her private key to recover the plaintext block x y = x e mod n x = y d mod n Without proof: y d =(x =(x e e )) d =x =x e d e d =x =x m (p-1) (q-1) m (p-1) (q-1) + 11 =x =x 11 = x (mod n )) Encryption and Decryption are symmetric operations The order of the exponentiation with the public exponent e and the private exponent d can be exchanged. A. Steffen, , KSy_RSA.ppt 16

17 RSA Public Key Cryptosystem Encryption / Decryption Example Encryption with Public Key n = 33, e = 3 Binary Plaintext Groups of 5 Bits Decimal Plaintext y = x y = x 3 mod Decryption with Private Key n = 33, d = 7 Decimal Ciphertext x = y x = y 7 mod A. Steffen, , KSy_RSA.ppt 17

18 RSA Public Key Cryptosystem Efficient Exponentiation of Large Numbers Multiplication in finite fields (a b) mod n = [ (a mod n) (b mod n) ] mod n Straight exponentiation method with e-1 multiplications y = x e = x x... x mod n Efficient exponentiation with < 2 log 2 e multiplications based on the binary representation of the exponent e = b k 2 k + b k-1 2 k b i 2 i b b b 1 with b i = {0,1} and k = log 2 e e j e j e j c h bg b b b b b k k k 1 k y = x 2 x 2 x 2 x 2 x mod n with the iterative squaring: x i i = ex j mod n A. Steffen, , KSy_RSA.ppt 18

19 RSA Public Key Cryptosystem Exponentiation Example Encryption with Public Key n = 33, e = 13 e = e = y = x e =(x 8 ) 1 (x 4 ) 1 (x2 ) 0 (x) 1 mod n y = x 8 x4 x mod n x 2 = x x mod n, x 4 = x 2 x2 mod n, x 8 = x 4 x4 mod n Decryption with Private Key n = 33, d = 17 d = d = x = y d =(y 16 ) 1 (y 8 ) 0 (y 4 ) 0 (y2 ) 0 (y) 1 mod n x = y 16 y mod n y 2 = y y mod n, y 4 = y 2 y2 mod n, y 8 = y 4 y4 mod n, y 16 = y 8 y8 mod n A. Steffen, , KSy_RSA.ppt 19

20 RSA Public Key Cryptosystem Contest Choose a plaintext number 1 < x < 33 and keep it secret! Encrypt x with RSA using the public key n = 33, e = 13. Exchange the encrypted number y with your neighbour. Decrypt your neighbour s number using the private key n = 33, d = 17. Check with your collegue if the decrypted number equals the original plaintext number. A. Steffen, , KSy_RSA.ppt 20

21 RSA Public Key Cryptosystem Plaintext to Ciphertext Mapping n = 33, e = 13, d = 17 y = x e mod n x y x y x y x y x y A. Steffen, , KSy_RSA.ppt 21

22 RSA Public Key Cryptosystem How to find large random prime numbers There are primes 512 bits in length or less. There are only atoms in the universe. The chance that two people choose the same prime factors for key generation is therefore near to nil! To prove that a randomly chosen number is really prime you would have to factor it. Try small factors (3, 5, 7, 11,...) Probabilistic Primality Tests (e.g. Rabin-Miller) Result of Primality Test not prime is prime 100 % 0.1 % 0 % 99.9 % random number is a composite number prime number After passing 5 tests, assume a random number to be prime A. Steffen, , KSy_RSA.ppt 22

23 Secure Network Communication Part II Diffie-Hellman Key-Exchange Algorithm A. Steffen, , KSy_RSA.ppt 23

24 The Diffie-Hellman Key-Exchange Algorithm Generating a Common Secret Key Alice and Bob agree on a large prime modulus n, a primitive element g and the one-way function y = f(x) = g x mod n. The integers n and g are not secret and can be published. Alice chooses a large random integer a and sends Bob A = g a mod n Bob chooses a large random integer b and sends Alice B = g b mod n Alice computes s = B a mod n = g ba mod n Bob computes s = A b mod n = g ab mod n Alice and Bob share now the secret key s = g ab mod n Since computing the inverse x = f -1 (y) is extremely difficult, no one listening to the key-exchange can compute the secret key s from the values A, B, n and g. A. Steffen, , KSy_RSA.ppt 24

25 The Diffie-Hellman Key-Exchange Algorithm Practical Example y = g x mod n Prime modulus: n = 31 Primitive element: g = 3 x y x y A: a = 8 A = 3 8 mod 31 = 20 s = 16 8 mod 31 = 4 B: b = 6 B = 3 6 mod 31 = 16 s = 20 6 mod 31 = 4 A. Steffen, , KSy_RSA.ppt 25

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