# Public-Key Cryptography. Oregon State University

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1 Public-Key Cryptography Çetin Kaya Koç Oregon State University 1

2 Sender M Receiver Adversary Objective: Secure communication over an insecure channel 2

3 Solution: Secret-key cryptography Exchange the key over a secure channel key: e or d Sender C Receiver C := f(e, M) M := g(d, C) Adversary 3

4 Secret-Key Cryptography Functions f(e, ) and g(d, ) are inverses of one another Encryption and decryptions processes are symmetric: Either Or f = g and e d C := f(e, M) and M := f(d, C) d is easily deduced from e e is easily deduced from d f g and e = d C := f(e, M) and M := g(e, C) g is easily deduced from f f is easily deduced from g 4

5 Example: Hill Algebra Encode 26 letters A,B,C,..,Z as 0,1,2,..,25 Select d d matrix Z of integer elements and find its inverse Z 1 modulo 26, e.g., Z = Verify: and Z 1 = = = Encryption: C := ZM mod 26 Encryption Key: e = Z Decryption: M := Z 1 C mod 26 Decryption Key: d = Z 1 Functions: f = g : matrix-vector product 5

6 Plaintext : HELP M 1 = H E = 7 4 ; M 2 = L P = Encryption: C 1 := ZM = = 7 8 = H I Encryption: C 2 := ZM = = 0 19 = A T Cryptotext: HIAT 6

7 Cryptotext: HIAT C 1 = H I = 7 8 ; C 2 = A T = 0 19 Decryption: M 1 := Z 1 C = = 7 4 = H E Decryption: M 2 := Z 1 C = = = L P Plaintext: HELP 7

8 Problems with secret-key cryptography: requires establishment of a secure channel for key exchange two parties cannot start communication if they never met Alternative: Public-Key Cryptography requires establishment of a public-key directory in which everyone publishes their encryption keys two parties can start communication even if they never met provides ability to sign digital data 8

9 Key Exchange Parties S and R agree on a large prime number p (This can be accomplished in public) S selects a GF (p) and computes a 1 such that aa 1 mod p 1 S keeps these integers secret R selects b GF (P ) and computes b 1 such that bb 1 mod p 1 R keeps these integers secret Suppose S wants to pass the key x to R 9

10 S computes x a GF (p) and sends it to R R computes (x a ) b = x ab GF (p) and sends it to S S computes (x ab ) a 1 = x b GF (p) and sends it to R R Computes (x b ) b 1 = x GF (p) Adversary knows the field GF (p) and sees x a, x ab, x b Computing α from x α in the field GF (p) is discrete analogue of taking logarithms Discrete logarithm problem is known to be very hard 10

11 public-key directory User S R Key e S e R Sender C Receiver C := f(e R, M) M := f(d R, C) Adversary 11

12 Public-Key Cryptography Functions f(e R, ) and f(d R, ) are inverse of one another Encryption and decryptions processes are asymmetric: e R d R C := f(e R, M) and M := f(d R, C) e R is public; known to everyone d R is private; known only to User R e R is easily deduced from d R d R is NOT easily deduced from e R 12

13 A public-key cryptography system is based on a function f(x) such that Given x, computing y = f(x) is EASY Given y = f(x), computing x is HARD x easy hard f(x) We call f(x) a one-way function In order to decide what is hard: Sometimes we use the theory of complexity Often the test of time determines 13

14 Example: Discrete Logarithm Given x, a, and p, computing y x a mod p is EASY However, given y, x, and p, computing a is HARD Example: Factoring Given x and y, computing n = xy is EASY However, given n, computing the factors x and y is HARD Example: Discrete Square-root Given x and n, computing a x 2 mod n is EASY However, given a and n, computing x is HARD 14

15 Discrete Logarithm Example For x = 6, a = 9, p = 11, we compute y x a x((x 2 ) 2 ) 2 mod p with 4 multiplications: y = 6((6 2 ) 2 ) 2 = 6((36) 2 ) 2 = 6((3) 2 ) 2 = 6(9) 2 = 6(81) = 6(4) = 24 = 2 However, finding an a such that is hard 6 a 2 mod 11 We need to try all possibilities (from 1 to p 1) to obtain such a 15

16 One additional structure about the function y = f(x) is needed to design a public-key cryptosystem Given y and some special information about f(x), computing x is EASY Given y without this special information, computing x is HARD We call f(x) a one-way trapdoor function Special information is trapdoor information 16

17 Example: Trapdoor Knapsack Knapsack Problem: Let I = {0, 1,..., n 1}. Given the integer vector A = {a 0, a 1,..., a n 1 } and another integer X, is there a J I such that i J a i = X Easy Knapsack Problem: If the numbers a i have the superincreasing property, i.e., j 1 i=0 a i < a j then the knapsack problem is easy Hard Knapsack Problem: Without the superincreasing property the knapsack problem (in general) is hard 17

18 Easy Example: A = {1, 2, 4, 8, 16}; Superincreasing 1 < 2; 1+2 < 4; < 8; < 16 Let X = 23. Solution is found by computing the binary expansion of X = 23 = (10111) 2, thus = 23 Hard Example: A = {3, 4, 5, 12, 13}; Nonsuperincreasing Let X = 19. We need to try all subsets of A to find out which one of these sums to = = = = = = = = = = = = = = = = 30 18

19 How to design a trapdoor knapsack Take an easy knapsack and disguise it Example: A = {1, 2, 4, 8, 16} Select a prime p larger than the sum 31, for example p = 37 Select t and compute t 1 mod p, for example, t = 17 and t 1 = 24 Produce a new knapsack vector B from A such that b i a i t mod p This gives B = {17, 34, 31, 25, 13} This knapsack problem is nonsuperincreasing However, with the special trapdoor information t = 17, t 1 = 24, and p = 37, we can convert this problem to the easy version 19

20 Example: Given B and X = 72, is there a subset of B summing to X? Solve the easy knapsack with X X Xt 1 mod 37 = 26 A = {1, 2, 4, 8, 16} and X = = 26 This gives the solution for the hard knapsack: B = {17, 34, 31, 25, 13} and X = = 72 20

21 Knapsack Public-Key Cryptosystem User R: Selects an easy vector A with A = n > 100 Selects a prime p larger than the sum n 1 i=0 a i Selects t and t 1 such that tt 1 1 mod p Obtains the hard knapsack B from A using b i a i t mod p Publishes B Keeps A, t, t 1, and p secret 21

22 User S: wants to send a message to R User S: Takes the message M and breaks into n bits Let m i be the ith bit of M (m i = 0 or m i = 1) Computes C as and sends C to R C := n 1 i=0 m i b i User R: Receives C and computes C Ct 1 mod p and solves the easy knapsack Uses this solution to obtain M 22

23 Example User R: Publishes B = {17, 34, 31, 25, 13} Keeps A = {1, 2, 4, 8, 16}, t = 17, t 1 = 24, and p = 37 secret User S: wants to send the message M = 12 to User R User S: Takes M = 12 = (01100) 2 Computes C := which gives C = 65 Send C = 65 to User R 23

24 User R: Receives C = 65 Computes C = 65t 1 = mod 37 Solves the easy knapsack problem: 6 = This gives the message as (01100) 2 = 12 24

25 Other important public-key cryptosystems: RSA (1978): factoring problem McEliece (1978): decoding problem for general linear codes ElGamal (1985): discrete log problem DSS (1992): discrete log problem 25

26 The RSA Algorithm The RSA algorithm was invented by Rivest, Shamir, and Adleman in 1977 p and q be two distinct large random primes The modulus n is the product of these two primes: n = pq Euler s totient function of n is given by φ(n) = (p 1)(q 1) Now, select a number 1 < e < φ(n) such that gcd(e, φ(n)) = 1 and compute d with d = e 1 (mod φ(n)) using extended Euclid s algorithm 26

27 Here, e is the public exponent and d is the private exponent Usually one selects a small public exponent, e.g., e = The modulus n and the public exponent e are published The value of d and the prime numbers p and q are kept secret Encryption is performed by computing C = M e (mod n) where M is the plaintext such that 0 M < n Decryption is performed by computing M = C d (mod n) 27

28 The correctness of the RSA algorithm follows from Euler s theorem: Let n and a be positive, relatively prime integers. Then a φ(n) = 1 (mod n) Since we have ed = 1 mod φ(n), i.e., ed = 1 + Kφ(n) for some integer K, we can write C d = (M e ) d (mod n) = M ed (mod n) = M 1+Kφ(n) (mod n) = M (M φ(n) ) K (mod n) = M 1 (mod n) provided that gcd(m, n) = 1 28

29 The exception gcd(m, n) > 1 can be dealt as follows: According to Carmichael s theorem M λ(n) = 1 (mod n) where λ(n) is Carmichael s function λ(n) = λ(pq) = (p 1)(q 1) gcd(p 1, q 1) λ(n) is always a proper divisor of φ(n) when n is the product of distinct odd primes; in this case λ(n) is smaller than φ(n) The relationship between e and d is given by M ed = M (mod n) if ed = 1 (mod λ(n)) If n is a product of distinct primes, the above holds for all M, thus covering the exception gcd(m, n) > 1 in Euler s theorem 29

30 As an example, we construct a simple RSA cryptosystem as follows: Pick p = 11 and q = 13, and compute n = p q = = 143 φ(n) = (p 1) (q 1) = = 120 We can also compute Carmichael s function of n as λ(pq) = (p 1)(q 1) gcd(p 1, q 1) Thus, = gcd(10, 12) = = 60 M ed = M (mod 143) if ed = 1 (mod 60) for all 0 M

31 The public exponent e is selected such that 1 < e < φ(n) and gcd(e, φ(n)) = gcd(e, 120) = 1 For example, e = 17 would satisfy this constraint The private key exponent d is computed by d = e 1 (mod φ(n)) = 17 1 (mod 120) = 113 which is computed using the extended Euclid algorithm, or any other algorithm for computing the modular inverse 31

32 The user publishes the public exponent and the modulus pair: (e, n) = (13, 143), and keeps the following as private: d = 113, p = 11, q = 13 A typical encryption/decryption process is executed as follows: Plaintext: M = 50 Encryption: C := M e (mod n) C := (mod 143) C = 85 Ciphertext: C = 85 Decryption: M := M d (mod n) M := (mod 143) M = 50 32

33 Exchange of Private Messages The public-key directory contains pairs (e, n) for each user. The users wishing to send private and/or signed messages to one another refer to the directory to get these parameters. For example, the directory might be arranged as follows: User Public Keys Alice (e a, n a ) Bob (e b, n b ) Charlie (e c, n c ) where the pair n a and e a respectively are the modulus and the public exponent for Alice, and so on. As an example, we show how Alice sends her private message M to Bob. 33

34 In our simple protocol example Alice executes the following steps: 1. Alice locates Bob s name in the directory and obtains his public exponent and the modulus: (e b, n b ) 2. Alice computes C := M e b (mod n b ) 3. Alice sends C to Bob over the network 4. Bob receives C 5. Bob computes M = C d b (mod n b ) in order to obtain M 34

35 Exchange of Signed Private Messages The procedure for sending a signed private message is a little more involved. For example, suppose Alice wants to send her signed private message to Bob. 1. Alice locates Bob s name in the directory and obtains his public-key exponent and the modulus: (e b, n b ) 2. Alice compares her modulus n a to Bob s modulus. One of these will be true: n a < n b, or n b < n a The order of signing and encrypting are reversed for each of these cases 35

36 Alice first forms her message M such that 0 M < min(n a, n b ) Assuming n a < n b : 3. Alice signs: C 1 = M d a (mod n a ) 4. Alice encrypts: C 2 = C e b 1 (mod n b ) 5. Alice sends C 2 to Bob 6. Bob receives C 2 7. Bob decrypts: C 1 = C d b 2 (mod n b ) 8. Bob verifies: M = C e a 1 (mod n a ) 36

37 Assuming n b < n a : 3. Alice encrypts: C 1 = M e b (mod n b ) 4. Alice signs: C 2 = C d a 1 (mod n a ) 5. Alice sends C 2 to Bob 6. Bob receives C 2 7. Bob verifies: C 1 = C e a 2 (mod n a ) 8. Bob decrypts: M = C d b 1 (mod n b ) 37

38 Example User Private Keys Alice d a = 113, p a = 11, q a = 13 Bob d b = 263, p b = 17, q b = 19 User Public Keys Alice e a = 17, n a = 143 Bob e b = 23, n b = 323 Alice executes the following steps in order to send a signed and private message: 1. Alice locates Bob s name in the directory and obtains his public-key exponent and the modulus: (e b, n b ) = (23, 323) 2. Alice compares her modulus n a = 143 to Bob s modulus n b = 323. Since n a < n b, she executes the following steps: 38

39 3. Alice forms her message 0 M < 143, let M = Alice signs: C 1 = mod 143, and obtains C 1 = Alice encrypts: C 2 = mod 323, and obtains C 2 = Alice sends C 2 = 232 to Bob 7. Bob decrypts: C 1 = mod 323, obtaining C 1 = Bob verifies: M = mod 143, obtaining M = 32 39

40 Note that changing the order of signing and encrypting would be incorrect. If Alice encrypts before signing, then she would compute: M = 32 C 1 = (mod 323) = 280 C 2 = (mod 143) = 37 and send C 2 = 37 to Bob. After having received 37, Bob would compute C 1 = (mod 143) = 137 M = (mod 323) = 35 which is not equal to the message:

41 Practice In practice the protocols are more complicated; for example, the order of signing and encryption are not interchanged and signing apply to messages of arbitrary length. The signature is often computed by first computing a hash value of the long message and then signing this hash value. 41

42 Security of the RSA Algorithm If one can factor quickly, then one can break the RSA algorithm: Alice s public keys are published: (e a, n a ) Factor n a to get p a and q a Compute φ(n a ) = φ(p a q a ) = (p a 1)(q a 1) Compute d a = e 1 a (mod φ(n a )) Thus, we can now intercept and decrypt all messages sent to Alice Thus, factoring = breaking RSA However, breaking RSA? = factoring No proof exists that breaking RSA is equivalent factoring 42

43 Difficulty of Factoring The best algorithm (Number Field Sieve) requires roughly e 1.92(ln n)1/3 (ln ln n) 2/3 operations to factor the composite number n Digits MIPS-years

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