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ECE454/CS594 Computer and Network Security Dr. Jinyuan (Stella) Sun Dept. of Electrical Engineering and Computer Science University of Tennessee Fall 2011 1 Public Key Cryptography • • • • Modular Arithmetic RSA Diffie-Hellman Elliptic Curve Cryptography 2 Public Key Cryptography Aka: asymmetric cryptography, invented in 1970s Use two keys: a public key known to everyone, a private key kept secret to the owner Encryption/decryption: encryption can be done by everyone using the recipient’s public key, decryption can be done only by the recipient with his/her private key Digital signature: signing is done with signer’s private key, and verification is done with signer’s public key Key exchange: establish a shared session key with PKC, SKC is used afterwards 3 Modular Arithmetic Fundamental to PKC Modulo n or mod n: non-negative integers < some integer n, sometimes “mod n” is omitted Modular addition Modular multiplication Modular exponentiation 4 Modular Addition Example: mod 10 5+5=0 3+9=? 2+2=? 9+9=? Additive inverse: an additive inverse of x is the number we need to add to x to get 0, e.g., what’s the additive inverse of 4 mod 10? 5 Modular Multiplication Example: 3 7 = 1 mod 10 • Multiplicative inverse: if xy = 1 mod n, then x and y are each other’s multiplicative inverse mod n • Relatively prime: no common factors other than 1 • Existence of multiplicative inverse: x has multiplicative inverse mod n iff x is relatively prime to n • Euclid’s algorithm: provides efficient method to find multiplicative inverses mod n • 6 Modular Multiplication (Cont’d) (n): totient function • number of integers < n and relatively prime to n • (n) = n – 1 if n is prime • (pq) = pq – (p + q – 1) = (p – 1)(q – 1), if p and q are prime 7 Euclid’s Algorithm—gcd Finds the greatest common divisor (gcd) of two integers: subtract y from x and still have the same gcd • subtract as many ys as possible and replace x with the remainder • switch x and y when x becomes smaller than y • each step looks like: <x, y> <y, remainder(x/y)> • eventually x or y will be zero and the other is the gcd • Example: gcd (408, 595) = 17 595/408 = 1 remainder 187 408/187 = 2 remainder 34 187/34 = 5 remainder 17 34/17 = 2 remainder 0 Each step we can write the remainder r = ux + vy (e.g., 17 = -16x408 + 11x595) 8 Euclid’s Algorithm— Multiplicative Inverse Efficiently finds a number m’s multiplicative inverse mod n • We are looking for u that makes um = 1 mod n • In other words, we are looking for u and v s.t. um + vn = 1 • Calculate r = gcd (m, n) = um + vn and find r = 1 (only exists when m and n are relatively prime, i.e., gcd (m, n) = 1) Q: Will there be more than one multiplicative inverse for a given number? 9 Modular Exponentiation Example: 46 = 4096 = 6 mod 10 • xy mod n = x(y mod (n)) mod n • (n) = 4 • If y = 1 mod (n), then xy mod n = x mod n 10 RSA Named after Rivest, Shamir, and Adleman • Public key / private key, use one to encrypt and the other to decrypt • Key length: variable, most commonly 512 bits • Plaintext block: smaller than the key length • Ciphertext block: same as key length • Advantage: Easy key management • Disadvantage: much slower than secret key algorithms • 11 RSA Algorithm Choose two large primes, p and q, ~256 bits each • n = pq, (n) = (p – 1)(q – 1) • Choose e that is relatively prime to (n) • By Euclid’s algorithm, find d that is the multiplicative inverse of e mod (n), i.e., ed = 1 mod (n) • Let <e, n> be the public key, d the private key • 12 Encryption and Decryption • Encryption with public key <e, n>: c = me mod n • Decryption with private key d: m = cd mod n c d mod n = (m e mod n) d mod n = (m e ) d mod n = m mod n =m 13 Signature and Verification • Sign with private key d: Sig = md mod n • Verify with public key <e, n>: m = Sige mod n Sig e mod n = (m d mod n) e mod n = (m d ) e mod n = m mod n =m 14 Why is RSA Secure? • Given n, it is hard to factor it to get p and q • RSA misuse: Alice uses Bob’s public key to encrypt a message sent to Bob. If Frank knows the message is one of many possible messages, he can use the same public key to compute and compare the ciphertexts to find the message (Solution?) 15 Efficiency of RSA Operations • Exponentiation of large numbers of several hundred digits • Find big primes, p and q • Find e and d 16 Exponentiating With Big Numbers • Page 154 – 155 17 Finding Big Primes p and q • The probability of a randomly chosen number n to be prime is 1 / ln n, which is about one in 230 for n of a hundred digit • Test whether a random number n is a prime - Fermat’s Theorem: if n is a prime and 0 < a < n, then a n-1 = 1 mod n - For a non-prime n of a hundred bits, the chance of a n-1 = 1 mod n is about 1 in 1013 (run more times) - Unfortunately, there are Carmichael numbers (very rare) that show a n-1 = 1 mod n for all a’s - Use Miller-Rabin algorithm 18 An Efficient Method of Finding Primes 1. 2. 3. Pick an odd random number n in the proper range. Test n's divisibility by small primes and go back to 1 if you find a factor. Repeat the following until n is proven not prime (in which case go back to step 1) or as many times as you feel necessary to show that n is probably prime (using Miller-Rabin algorithm): • Pick an a at random and compute ac mod n (where c is the odd number for which n-1 = 2bc). • Each time squaring of ac mod n is performed, check if the result is 1; if so, check if the number that was squared is ±1; if not, n is not prime (Chinese remainder theorem: 1 has many square roots if n is not prime). • Next, if the result of ac mod n is ±1, n passes the primality test for this a. • Otherwise, at most b-1 times, replace the result by its square and check if it is ±1. If it is 1, n is not prime (because the previous result is a square root of 1 different from ±1). If it is -1, n passes the primality test for this a. • If you've done the squaring b-1 times, n is not prime (because a(n-1)/2 is not ±1). Even if n is a Carmichael number, at least ¾ a’s will show it 19 Finding e and d • e: can be randomly chosen, relatively prime to (n) • d: calculated by Euclid’s algorithm, s.t. ed =1 mod (n) • If e is chosen to be small such as 3, the encryption and signature verification will be faster, while the decryption and digital signature remain the same • d should not be small (Why?) 20 Popular Values of e • 3 and 65537 (216 + 1) • Advantage: computationally efficient - 3: 2 multiplies - 65537: 17 multiplies 21 Problems of e=3 • Problem 1: c = m e mod n, if e is 3 and m is less than n1/3, then m 3 < n and thus c = m 3 mod n = m 3 m = c1/3 Solution: pad m to be larger than n1/3 • Problem 2: If a message is encrypted for three recipients using their public keys, <3, n1> <3, n2> <3, n3> to get three ciphertexts, c1 = m3 mod n1, c2 = m3 mod n2, c3 = m3 mod n3 , an attacker can compute c = m3 mod n1n2n3 by Chinese Remainder Theorem. Since m is smaller than n1, n2, and n3, c = m3 m = c1/3 Solution: pad m with different numbers for c1, c2, c3 • Problem 3: Need to choose p and q s.t. 3 is relatively prime to (p-1)(q-1). It is easier to choose eligible p and q for 65537. 22 Attacks on RSA • Brute-force attacks: trying all possible private keys • Mathematical attacks: trying to factor the product of two primes • Timing attacks: depend on the running time of the decryption algorithm (one type of side channel attacks) • Chosen ciphertext attacks: exploit properties of the RSA algorithm 23 Countermeasures • Brute-force attacks: use a large key space • Mathematical attacks: use large enough n (1024-2048 bits), select p and q with constraints • Timing attacks: constant exponentiation time, random delay, blinding the ciphertext • Chosen ciphertext attacks: randomly pad the plaintext before encryption, e.g., optimal asymmetric encryption padding (OAEP) 24 PKCS—Public Key Cryptography Standard: Encryption • Standard for the encoding of information that will be signed or encrypted through RSA • A suite of standards PKCS #1—15 • PKCS #1 for formatting a message to be encrypted: • The encoding addresses several RSA threats: - guessable message - sending same encrypted message to >=3 recipients (e=3) - Encrypting messages<1/3 length of n (e=3) 25 PKCS—Public Key Cryptography Standard: Signature • PKCS #1 for formatting a message to be signed: • The encoding addresses several RSA threats: - padding avoids smooth numbers w.h.p. - avoids cube root problem - including digest type avoids an obscure threat: MD4(m’)=MD5(m) 26 Diffie-Hellman • The first public key cryptosystem • But does neither encryption nor signatures • Used for key exchange: Alice and Bob negotiate a shared secret key over a public communication channel 27 DiffieHellman Key Exchange 28 Why Is Diffie-Hellman Secure? • It is difficult to compute discrete logarithm: knowing g and gx, it is difficult to compute x 29 Man-in-the-Middle Attack Alice A, gA Bob B, gB gA gB KAB=gAB Frank F, gF Alice A, gA gA Bob B, gB gF gB gF KAF=gAF KFB=gFB 30 Countermeasures Publish public numbers: • Alice keeps x private, but publishes X = g x mod p through a reliable, trusted service such as PKI • Bob keeps y private, but publishes Y = g y mod p • Alice retrieves Y from the trusted service • Bob retrieves X from the trusted service • No place for Frank to get in the middle. The key between Alice and Bob is in fact pre-determined. 31 Countermeasures (Cont’d) Authenticated Diffie-Hellman: • Encrypt the Diffie-Hellman exchange with the pre-shared secret • Encrypt the Diffie-Hellman public number with the other side’s public key • Sign the Diffie-Hellman public number with your private key • Following the Diffie-Hellman exchange, transmit a hash of the agreed key and the pre-shared secret • Following the Diffie-Hellman exchange, transmit a hash of the pre-shared secret and your public number 32 Encryption with Diffie-Hellman • Alice uses Diffie-Hellman to generate a shared secret key, gAB, with Bob • Encryption: Alice uses any secret key encryption scheme with the above secret key • The secret key need not be shared with Bob before encryption: Bob can retrieve the encrypted message and the secret key simultaneously given that he has published his <pB, gB, TB= gB> 33 ElGamal Signatures • Each party has a long-term public/private key pair • Public key is <g, p, T= gS mod p> and private key is S • For each message m signed, generate a new public/private key pair: <g, p, Tm= gSm mod p>, Sm (a random number) • Signature: X=Sm+dmS mod (p-1), where dm=MD(m|Tm) • Transmitted to the recipient: <m, Tm, X> • Verification: check if gX=Tm Tdm mod p • If signature is valid, verification will pass because: • Other requirements for a secure signature: verification will fail if message/signature is modified, knowing signature will not divulge S, not knowing S will not be able to sign 34 Elliptic Curve Cryptography • Known subexponential algorithms for breaking RSA and Diffie-Hellman (a brute-force attack requires exponential amount of computation), so required key size is large • No known subexponential algorithm for breaking ECC • ECC offers the same security with much smaller key size Comparable key sizes in terms of computational effort for cryptanalysis 35 Assignments Read [Kaufman] Chapter 6 36