Create Presentation
Download Presentation

Download Presentation
## CSC 482/582: Computer Security

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**CSC 482/582: Computer Security**Public Key Cryptography CSC 482/582: Computer Security**Topics**• Number Theory Review • Public Key Cryptography • One-Way Trapdoor Functions • Diffie-Hellman Key Exchange • RSA Cipher • Modern Steganography CSC 482/582: Computer Security**Number Theory Review**• Prime Numbers • Fundamental Theorem of Arithmetic • Greatest Common Divisors • Relatively Prime Numbers • Modular Inverses • Euler’s Totient Function CSC 482/582: Computer Security**Fundamental Thm of Arithmetic**Definition: An integer n > 1 is prime if its only factors are 1 and n. Definition: An integer n > 1 that is not prime is composite. Theorem: An integer n > 1 can be written as a product of prime numbers. n = p1a p2b p3c p4d … CSC 482/582: Computer Security**Greatest Common Divisor**Definition: The greatest common divisor of integers a and b > 0, gcd(a,b) is the largest number that divides both a and b. Definition: Two integers a and b > 0 are relatively prime if gcd(a,b) = 1. Theorem: If n is prime, then every integer from 1 to n-1 is relatively prime to n. CSC 482/582: Computer Security**Modular Arithmetic**The operation x mod n, where x and n are integers, is taking the remainder of x divided by n, which is one of n possible values: {0, 1, 2, …, (n - 1)} Congruence is equality of two numbers modulo n a = b mod n iff a = b + kn Equivalent to performing arithmetic in Zn, which Zn= {0, 1, 2, …, (n - 1)} CSC 482/582: Computer Security**Laws of Modular Arithmetic**(a + b) mod N = (a mod N + b mod N) mod N (a - b) mod N = (a mod N - b mod N) mod N ab mod N = (a mod N)(b mod N) mod N a(b+c) mod N = ((ab mod N) + (ac mod N)) mod N CSC 482/582: Computer Security**Modular Inverses**Multiplicative Inverse: The inverse of a number x is a number x-1 such that xx-1 = 1. Modular Inverse: An integer x-1 such that 1 = (x x-1) mod n For example, 4 × 3 mod 11 = 12 mod 11 = 1 1 and n-1 are their own modular inverses. However, there is not always a solution. 2 has no inverse mod 16 If n is prime, all elements but 0 have an inverse. CSC 482/582: Computer Security**Modular Exponentiation**Identities ab ac = a(b+c) mod n (ab)c= abc mod n Can take modulo operator inside a2 mod n = (a mod n)(a mod n) mod n a2 mod n = (a mod n) 2 mod n CSC 482/582: Computer Security**Euler Totient Function**Euler’s totient function (n) • Number of positive integers less than nthat are relatively prime to n. • Or equivalently, the number of integers k in the range 1 ≤ k ≤ n for which gcd(n,k)=1. Example:(10) = 4 • 1, 3, 7, 9 are relatively prime to 10 Theorem: If gcd(a,b)=1, (ab) = (a) (b) Note: If n is prime, (n) = n – 1 Result: If a, b prime, (ab) = (a-1)(b-1) CSC 482/582: Computer Security**Euler’s Totient Theorem**Theorem: If n > 1 and gcd(a,n) = 1, then a(n) mod n = 1. Corollary: If n > 1 and gcd(a,n) = 1, then a(n)-1 mod n is the modular inverse of a mod n. Ex: 74 ≡ 1 (mod 10), which we use to compute 7222 mod 10: 7222 ≡ 74x55 + 2 ≡ (74)55x72 ≡ 155x72 ≡ 49 ≡ 9 (mod 10). CSC 482/582: Computer Security**Public Key Cryptography**Two keys Private key known only to owner. Public key available to anyone. Applications Confidentiality: • Sender enciphers using recipient’s public key, • Receiver deciphers using their private key. Integrity/authentication: • Sender enciphers using own private key, • Recipient deciphers using sender’s public key. CSC 482/582: Computer Security**Requirements**Based on mathematical problems such that • It must be computationally easy to encipher or decipher a message given the appropriate key. • It must be computationally infeasible to derive the private key from the public key. • It must be computationally infeasible to determine the private key from a chosen plaintext attack. Computationally infeasible means exponential time complexity, so that encryption with an n+1 bit key is 2X harder to crack than an n-bit key based encryption. • It is faster to solve a computationally infeasible problem than to do a brute force exhaustive key search. • Therefore keys for public key encryption must be longer than keys for symmetric encryption algorithms. CSC 482/582: Computer Security**Diffie-Hellman Key Exchange**Compute a common, shared key • Called a symmetric key exchange protocol Based on discrete logarithm problem • One way function: f(k) = gk mod p. • Invert function by computing logarithm to find k. • Solutions known for small p. • Computationally infeasible for large p. Shared Constants • prime modulus p, • integer base g ≠ {0, 1, p–1} CSC 482/582: Computer Security**Algorithm**• Alice chooses a random private key k • Computes public key A= gk mod pand sends to Bob. • Bob chooses a random private key k’ • Computes public key B = gk’mod p and sends to Alice. • Alice computes secret key • K1 = Bk mod p • Bob computes secret key • K2 = Ak’ mod p • Encrypt messages w/ symmetric cipher using K1=K2 as the secret key. CSC 482/582: Computer Security**How do we know K1=K2?**K1 = Bk mod p = (gk’ mod p)k mod p = gk’kmod p = gkk’mod p = (gkmod p)k’mod p = Ak’ mod p = K2 CSC 482/582: Computer Security**Example**Assume p = 53 and g = 17 Alice chooses kAlice = 5 Then KAlice = 175 mod 53 = 40 Bob chooses kBob = 7 Then KBob = 177 mod 53 = 6 Shared key: KBobkAlice mod p = 65 mod 53 = 38 KAlicekBob mod p = 407 mod 53 = 38 CSC 482/582: Computer Security**Activity: DH with a Partner**Use an arbitrary precision calculator like python/ruby interpreter or https://apfloat.appspot.com/for activity. • Together agree on a prime modulus > 100 for p. • Together select g ≠ {0, 1, p–1}. • Individually, each person selects a private key. • Individually, each person computes public key. • Share public keys with your partner. • Individually, each person computes shared key. • Compare shared keys and verify they’re identical. CSC 482/582: Computer Security**Elliptic Curve Cryptography**ECC is based on the algebra of elliptic curves (like y2 = x3 + ax + b) over finite fields instead of being based on simple finite groups like modular arithmetic. • Analogs to discrete logarithm and other one-way functions suitable for public key crypto exist in EC. • Patents may prevent use of some ECC algorithms. Suite B is NIST-recommended crypto including ECC • Includes Elliptic Curve Diffie-Hellman (ECDH) • NSA recommended until August 2015, when it announced a rapid transition away from Suite B to a new set of quantum-resistant algorithms to be published. CSC 482/582: Computer Security**One-Way Trapdoor Functions**Trapdoor one-way Function: One-way function whose inverse is easy to calculate only if given a special piece of information. Example: Prime factoring • Easy to calculate product. • Difficult to calculate prime factors from product. • Easy to calculate one prime factor, given others. CSC 482/582: Computer Security**RSA**• Modular exponentiation cipher. • Treat message blocks as large numbers. • Relies on the difficulty of determining the number of integers relatively prime to a large integer n. CSC 482/582: Computer Security**Algorithm**Choose two large prime numbers p, q • Let n = pq; then (n) = (p–1)(q–1) • Choose e < n such that e relatively prime to (n). • Compute inverse of e, d • ed mod (n) = 1 • d = e-1 mod (n) Public key: (e, n) Private key: d Encipher: c = me mod n Decipher: m = cd mod n CSC 482/582: Computer Security**Example: Confidentiality**Take p = 7, q = 11, so n = 77 and (n) = 60 Alice chooses e = 17, making d = 53 Bob wants to send Alice secret message HELLO, which we represent by the numbers 07 04 11 11 14 0717 mod 77 = 28 0417 mod 77 = 16 1117 mod 77 = 44 1117 mod 77 = 44 1417 mod 77 = 42 Bob sends 28 16 44 44 42 CSC 482/582: Computer Security**Example**Alice receives 28 16 44 44 42 Alice uses private key, d = 53, to decrypt message: 2853 mod 77 = 07 1653 mod 77 = 04 4453 mod 77 = 11 4453 mod 77 = 11 4253 mod 77 = 14 Alice translates message to letters to read HELLO No one else could read it, as only Alice knows her private key and that is needed for decryption. CSC 482/582: Computer Security**Ex: Integrity/Authentication**Take p = 7, q = 11, so n = 77 and (n) = 60 Alice chooses e = 17, making d = 53 Alice wants to send Bob message HELLO (07 04 11 11 14) so Bob knows it is what Alice sent (no changes in transit, and authenticated) 0753 mod 77 = 35 0453 mod 77 = 09 1153 mod 77 = 44 1153 mod 77 = 44 1453 mod 77 = 49 Alice sends 35 09 44 44 49 CSC 482/582: Computer Security**Example**Bob receives 35 09 44 44 49 Bob uses Alice’s public key, e = 17, n = 77, to decrypt message: 3517 mod 77 = 07 0917 mod 77 = 04 4417 mod 77 = 11 4417 mod 77 = 11 4917 mod 77 = 14 Bob translates message to letters to read HELLO Alice sent it as only she knows her private key If (enciphered) message’s blocks (letters) altered in transit, would not decrypt properly CSC 482/582: Computer Security**Example: Both**Alice wants to send Bob message HELLO both enciphered and authenticated (integrity-checked) Alice’s keys: public (17, 77); private: 53 Bob’s keys: public: (37, 77); private: 13 Alice enciphers HELLO (07 04 11 11 14): (0753 mod 77)37 mod 77 = 07 (0453 mod 77)37 mod 77 = 37 (1153 mod 77)37 mod 77 = 44 (1153 mod 77)37 mod 77 = 44 (1453 mod 77)37 mod 77 = 14 Alice sends 07 37 44 44 14 CSC 482/582: Computer Security**Activity: RSA with a Partner**Use an arbitrary precision calculator and an online Euler’s totient calculator at the following URL: http://www.javascripter.net/math/calculators/eulertotientfunction.htm • Together agree on primes p,q < 100. • Together compute n and (n) = (p–1)(q–1) • Individually choose e < n, s.t.e relatively prime to (n). • Individually compute d= e-1 mod (n). • Share public keys e with each other. • Individually encrypt “HELLO” with partner’s public key. • Share ciphertext with partner. • Individually decrypt message with your private key. CSC 482/582: Computer Security**Security Services**Confidentiality • Only the owner of the private key knows it, so text enciphered with public key cannot be read by anyone except the owner of the private key. Authentication • Only the owner of the private key knows it, so text enciphered with private key must have been generated by the owner. CSC 482/582: Computer Security**More Security Services**Integrity • Enciphered letters cannot be changed undetectably without knowing private key. Non-Repudiation • Message enciphered with private key came from someone who knew it. CSC 482/582: Computer Security**Warnings**Use large blocks for encryption: • If 1 character per block, RSA can be broken using statistical attacks (just like classical cryptosystems.) • Attacker cannot alter letters, but can rearrange them and alter message meaning. • Ex: reverse ciphertext of message ON to get NO CSC 482/582: Computer Security**Steganography**Hiding messages in another text (the covertext) so that no one except intended recipient knows a message has been sent. Wax Tablets: In ancient times, messages written in wax poured on top of a stone/wood tablet. Messages hidden by engraving in wood then pouring wax over them. Invisible Ink: Write message using lemon juice on paper. Write covertext in regular ink after dries. Heat to view hidden message. Null Cipher: Hide message in ordinary text, using nth letter of each word, or every nth word of the message. CSC 482/582: Computer Security**Digital Steganography**• Choose a cover medium file. JPEG, MP3, etc. • Identify redundant bits in cover medium. Low order bits in image and audio files. • Select and replace data. Steganographic Image Picture of cat hidden in the image above. CSC 482/582: Computer Security**JSteg: JPEG Steganography**JPEG image format • For each color component, a discrete cosine transform (DCT) turns 8x8 pixel blocks into 64 DCT coefficients. Derek Upham’sJSteg algorithm • LSBs of DCT coefficients are redundancy. • Modification of a single DCT coef affects all 64 pixels. CSC 482/582: Computer Security**Steganalysis**Compare stegographic file with original. • 100% effective at identifying presence. • Original file is “secret key” of steganography. Statistical analysis • Inserting high entropy changes histogram of color frequencies in predictable ways. • Reduces frequency difference between adjacent colors. Countermeasures • Insert less information to reduce impact. • Choose DCT coefficients to modify at random. • Alternate +/- DCT coefficient value to encode bits. • Use parity of groups of DCT LSBs to encode a message. CSC 482/582: Computer Security**Key Points**• Two types of cryptosystems: • classical (symmetric) • public key (asymmetric) • Public Key Cryptography • Based on mathematical problems; solving the problem is exponentially hard but easier than brute force search of key space. • Confidentiality: encipher with public, decipher with private • Integrity: encipher with private, decipher with public • Algorithms: Diffie-Hellman, RSA, ECC. • Steganography • Hide messages in text or media files. • Analyze using statistical techniques. CSC 482/582: Computer Security**References**• Matt Bishop, Introduction to Computer Security, Addison-Wesley, 2005. • Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, Handbook of Applied Cryptography, http://www.cacr.math.uwaterloo.ca/hac/, CRC Press, 1996. • Bruce Schneier, Applied Cryptography, 2nd edition, Wiley, 1996. • NIST, FIPS-198a, “The Keyed-Hash Message Authentication Code (HMAC)”,http://csrc.nist.gov/publications/fips/fips198/fips-198a.pdf • NielsProvos and Peter Honeyman, “Hide and Seek: An Introduction to Steganography,” IEEE Security & Privacy, May/June 2003. • John Viega and Gary McGraw, Building Secure Software, Addison-Wesley, 2002. • http://csis.bits-pilani.ac.in/faculty/murali/netsec-09/seminar/refs/anuroopsrep.pdf CSC 482/582: Computer Security