Public Key Cryptography
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1 T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A Public Key Cryptography EECE EECE_ public_crypto.key - October 2, 2014
2 What is it? Two keys Sender uses recipient s public key to encrypt Receiver uses his private key to decrypt Based on trap door, one way function Easy to compute in one direction Hard to compute in other direction Trap door used to create keys Example: Given p and q, product N=pq is easy to compute, but given N, it is hard to find p and q 2 EECE_ public_crypto.key - October 2, 2014
3 How is it used? Encryption Suppose we encrypt M with Bob s public key Only Bob s private key can decrypt to find M Digital Signature Sign by encrypting with private key Anyone can verify signature by decrypting with public key But only private key holder could have signed Like a handwritten signature 3 EECE_ public_crypto.key - October 2, 2014
4 Topic Outline The Random Oracle model for Public Key Cryptosystems Public key encryption and trapdoor oneway permutations Digital signatures Looking under the hood Knapsack RSA Uses of Public Crypto The order of sign and encrypt 4 EECE_ public_crypto.key - October 2, 2014
5 Public Key Encryption and Trap-door One-Way Permutation as Random Oracle Public Key Encryption Scheme: Key pair (KR, KR -1 ) generation function from random string R KR KR -1 is infeasible C = {M} KR M = {C} KR -1!! Queries Responses H(K 1 ) K 1 H(K 2 ) K2 In: fixed size short string (plaintext) M, Key KR Out: fixed size short string (ciphertext) C 5 EECE_ public_crypto.key - October 2, 2014
6 Digital Signature as Random Oracle Public Key Signature Scheme: Key pair (σr, VR) generation function VR σr is infeasible S = Sig σr (M) {True, False} = Ver VR (S) Signing Queries Responses Verifying H(K 1 ) K 1 H(K 2 ) K2 Input Any string M + σr S + VR Output S = hash(m) cipher block True or False 6 EECE_ public_crypto.key - October 2, 2014
7 T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A Looking Under the Hood 7 EECE_ public_crypto.key - October 2, 2014
8 T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A Knapsack Cryptosystem 8 EECE_ public_crypto.key - October 2, 2014
9 Knapsack Problem Given a set of n weights W 0,W 1,...,W n-1 possible to find a i {0,1} so that and a sum S, is it S = a 0 W 0 +a 1 W a n-1 W n-1 (technically, this is subset sum problem) Example Weights (62,93,26,52,166,48,91,141) Problem: Find subset that sums to S=302 Answer: =302 The (general) knapsack is NP-complete 9 EECE_ public_crypto.key - October 2, 2014
10 Knapsack Problem General knapsack (GK) is hard to solve But super-increasing knapsack (SIK) is easy SIK: each weight greater than the sum of all previous weights SIK Example Weights (2,3,7,14,30,57,120,251) Problem: Find subset that sums to S=186 Work from largest to smallest weight Answer: = EECE_ public_crypto.key - October 2, 2014
11 Knapsack Cryptosystem 1. Generate super-increasing knapsack (SIK) 2. Convert SIK into general knapsack (GK) 3. Public Key: GK 4. Private Key: SIK plus conversion factors Easy to encrypt with GK With private key, easy to decrypt (convert ciphertext to SIK) Without private key, must solve GK (???) 11 EECE_ public_crypto.key - October 2, 2014
12 Knapsack Cryptosystem Let (2,3,7,14,30,57,120,251) be the SIK Choose m = 41 and n = 491 with m, n relatively prime and n greater than sum of elements of SIK General knapsack (2 41) mod 491 = mod 491 = mod 491 = mod 491 = mod 491 = mod 491 = mod 491 = mod 491 = 471 General knapsack: (82,123,287,83,248,373,10,471) 12 EECE_ public_crypto.key - October 2, 2014
13 Knapsack Example Private key: (2,3,7,14,30,57,120,251), n = 491, m -1 =12 m 1 mod n = 41 1 mod 491 = 12 (x 1 x) mod n = 1 mod n Public key: (82,123,287,83,248,373,10,471) Throw away: m = 41 Example: Encrypt 150 = = 548 = C To decrypt, (C m 1 ) mod n = (548 12) mod 491 = 193 mod 491 Solve (easy) SIK with S = 193 Obtain plaintext = EECE_ public_crypto.key - October 2, 2014
14 Knapsack Weakness Trapdoor: Convert SIK into general knapsack using modular arithmetic One-way: General knapsack easy to encrypt, hard to solve; SIK easy to solve This knapsack cryptosystem is insecure Broken by Shamir in 1983 with Apple II computer The attack uses lattice reduction General knapsack is not general enough! This special knapsack is easy to solve! 14 EECE_ public_crypto.key - October 2, 2014
15 T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A RSA Cocks (GCHQ), independently, by Rivest, Shamir and Adleman (MIT)! 15 EECE_ public_crypto.key - October 2, 2014
16 Rivest, Shamir, and Adleman EECE_ public_crypto.key - October 2, 2014
17 basics Let p and q be two large (e.g., 200 digits) prime numbers use probabilistic primality tests to find p & q quickly Let n = p q be the modulus Factoring n is supposed to be hard (i.e., billions of years) e relatively prime to (p-1)(q-1) -- encryption exponent d = e -1 mod (p-1)(q-1) -- decryption exponent Throw Away: p, q Public key: (n, e) Private key: d Notation: public is in cyan, secret is in red 17 EECE_ public_crypto.key - October 2, 2014
18 encrypting & decrypting To encrypt message M compute C = M e mod n -- fast with modular exponentiation To decrypt C compute M = C d mod n Recall that e and n are public If attacker can factor n, he can use e to easily find d since ed = 1 mod (p 1)(q 1) Factoring the modulus breaks RSA It is not known whether factoring is the only way to break RSA 18 EECE_ public_crypto.key - October 2, 2014
19 T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A RSA in the works 19 EECE_ public_crypto.key - October 2, 2014
20 simple RSA example: initialization Select large primes p = 43, q = 59 Then n = p q = 2537 and (p 1)(q 1) = 2436 Choose e = 13 (relatively prime to 2436) Find d such that ed = 1 mod (p 1)(q 1), we find that d = 937 works note: d exists because gcd(e, (p-1)(q-1)) = 1 Public key: (N, e) = (2537, 13) Private key: d = EECE_ public_crypto.key - October 2, 2014
21 simple RSA example: encryption plain text: M = STOP = (18 19, 14 15) ciphertext: C = M e mod n = ( mod 2537, mod 2537) = = UDVE fast modular exponentiation 21 EECE_ public_crypto.key - October 2, 2014
22 simple RSA example: decryption ed = 1 mod (p 1)(q 1) k s.t. ed = k(p-1)(q-1)+1 C d (M e ) d = M de = M 1+ k(p-1)(q-1) (mod n) M p-1 1 mod p and M q-1 1 mod q by Fermat s Little Theorem: If p is prime and a is an integer not divisible by p, then a p-1 1 mod p. Furthermore a p a mod p C d M ((M p-1 ) k(q-1) ) mod p M 1 mod p M mod p C d M ((M q-1 ) k(p-1) ) mod p M 1 mod p M mod q Because gcd(p,q) =1, C d M mod p q by Chinese Remainder Theorem 22 EECE_ public_crypto.key - October 2, 2014
23 simple RSA example: decryption Decrypt message M C d mod p q mod 2537 = 0704 = HE mod 2537 = 1115 = LP HELP 23 EECE_ public_crypto.key - October 2, 2014
24 T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A Uses for Public Key Crypto 24 EECE_ public_crypto.key - October 2, 2014
25 Uses for Public Key Crypto Confidentiality Transmitting data over insecure channel Secure storage on insecure media Authentication Digital signature provides integrity and nonrepudiation No non-repudiation with symmetric keys 25 EECE_ public_crypto.key - October 2, 2014
26 Non-non-repudiation Alice orders 100 shares of stock from Bob Alice computes MAC using symmetric key Stock drops, Alice claims she did not order Can Bob prove that Alice placed the order? No! Since Bob also knows symmetric key, he could have forged message Problem: Bob knows Alice placed the order, but he can t prove it 26 EECE_ public_crypto.key - October 2, 2014
27 Non-repudiation Alice orders 100 shares of stock from Bob Alice signs order with her private key Stock drops, Alice claims she did not order Can Bob prove that Alice placed the order? Yes! Only someone with Alice s private key could have signed the order This assumes Alice s private key is not stolen (revocation problem) 27 EECE_ public_crypto.key - October 2, 2014
28 T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A Sign and Encrypt vs Encrypt and Sign 28 EECE_ public_crypto.key - October 2, 2014
29 Public Key Notation Sign message M with Alice s private key: [M] Alice Encrypt message M with Alice s public key: {M} Alice Then {[M] Alice } Alice = M [{M} Alice ] Alice = M 29 EECE_ public_crypto.key - October 2, 2014
30 Confidentiality and Non-repudiation Suppose that we want confidentiality and nonrepudiation Can public key crypto achieve both? Alice sends message to Bob Sign and encrypt {[M] Alice } Bob Encrypt and sign [{M} Bob ] Alice Can the order possibly matter? (see Stamp) 30 EECE_ public_crypto.key - October 2, 2014
31 Sign and Encrypt M = I love you {[M] Alice } Bob {[M] Alice } Charlie Alice Bob Q: What is the problem? A: Charlie misunderstands crypto! Charlie 31 EECE_ public_crypto.key - October 2, 2014
![[{M} Bob ] Alice [{M} Bob ] Charlie Alice](/docs-images/60/45019010/images/32-1.png "Charlie Note that Charlie cannot decrypt M")
32 Encrypt and Sign M = My theory, which is mine, is this:. [{M} Bob ] Alice [{M} Bob ] Charlie Alice Charlie Note that Charlie cannot decrypt M Q: What is the problem? A: Bob misunderstands crypto! Bob EECE_ public_crypto.key - October 2, 2014
33 Summary The Random Oracle model for Public Key Cryptosystems Public key encryption and trapdoor one-way permutations Digital signatures Looking under the hood Knapsack RSA Uses of Public Crypto The order of sign and encrypt 33 EECE_ public_crypto.key - October 2, 2014
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