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Chapter 3 Encryption Algorithms & Systems (Part C). Outline. NP-completeness & Encryption Symmetric (secret key) vs Asymmetric (public key) Encryptions Popular Encryption Algorithms Merkle-Hellman Knapsacks RSA Encryption El Gamal Algorithms DES Hashing Algorithms Key Escrow & Clipper.
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Outline • NP-completeness & Encryption • Symmetric (secret key) vs Asymmetric (public key) Encryptions • Popular Encryption Algorithms • Merkle-Hellman Knapsacks • RSA Encryption • El Gamal Algorithms • DES • Hashing Algorithms • Key Escrow & Clipper csci5233 computer security & integrity (Chap. 3)
RSA Encryption • 1978: Rivest, Shamir, Adelman • Public key encryption • Remains secure to date • Encryption key (e) and decryption key (d) are interchangeable. • The two keys, e and d, are carefully chosen such that C = Pe mod n (encryption) and P = Cd mod n (decryption). csci5233 computer security & integrity (Chap. 3)
Euler Totient Function • (n): the number of positive integers less than n and are relatively prime to n. • If n is prime: (n) = n – 1 • When n = p * q, where both p and q are primes and p q: (n) = (p) * (q) = (p – 1) * (q – 1) csci5233 computer security & integrity (Chap. 3)
RSA Encryption • Public key = (e, n) • Private key = (d, n) • Step 1: Choose n, p, & q n = p * q, where both p and q are primes and p q Example: n = 143 = p * q = 11 * 13 csci5233 computer security & integrity (Chap. 3)
RSA Encryption • Step 2: Choose e. e is relatively prime to (n). That is, e is relatively prime to (p-1)*(q-1). Example: e = 17, which is relatively prime to 10*12. • Step 3: Compute d. d is the inverse of e mod (p-1)*(q-1). Use the algorithm on page 81 to compute inverses. Note: A Java implementation of the algorithm is available at the class page. Example: d = e-1 mod (p-1)*(q-1) = 17-1 mod 120= 113 csci5233 computer security & integrity (Chap. 3)
RSA Encryption • An example (pp.94-95): P = 7 Let n = 143, p = 11, q = 13, and e = 11. Note: e is relprime to (p-1)*(q-1). Then d = 11 Note: d is the inverse of e mod (p-1)*(q-1). Encryption: C = Pe mod n = 711 mod 143 = 106 Decryption: P = Cd mod n = 5011 mod 143 = 7 csci5233 computer security & integrity (Chap. 3)
RSA Encryption • Another example: P = 7 Let n = 143, p = 11, q = 13, and e = 17. Note: e is relprime to (p-1)*(q-1). Then d = 113 Note: d is the inverse of e mod (p-1)*(q-1). Encryption: C = Pe mod n = 717 mod 143 = 50 Decryption: P = Cd mod n = 50113 mod 143 = 7 csci5233 computer security & integrity (Chap. 3)
RSA Encryption • Still another example: P = 55 Let n = 285, p = 19, q = 17, and e = 37. Note: e is relprime to (p-1)*(q-1), 288. d = 109 Note: d is the inverse of e mod (p-1)*(q-1). Encryption: C = Pe mod n = 5537 mod 288 = 55 Decryption: P = Cd mod n = 55109 mod 288 = 55 csci5233 computer security & integrity (Chap. 3)
RSA Encryption • The cryptographer’s job: • Find three primes, p, q, and e, where p * q = n and e is relatively prime to (p-1)*(q-1). • Compute d based on e and n. The challenge: p, q, and e must be large enough primes. See discussions on p.95. csci5233 computer security & integrity (Chap. 3)
RSA Encryption • The cryptanalyst’s job: P = Cd mod n • Available: (e, n). • Find two primes p and q, such that p * q = n and e is relatively prime to (p-1)*(q-1). • Compute d: d = inverse (e, (p-1)*(q-1)) Q: Where’s the secrecy? Q: Given n and a prime e, how hard is it to find two distinct primes, p and q, such that p*q = n and (p-1)*(q-1) is relprime to e? csci5233 computer security & integrity (Chap. 3)
El Gamal Algorithm • A public key algorithm • 1984 • Important in the U.S. DSS (Digital Signature Standard) • Digital Signatures The sender computes the digital signature using his own private key. DS = E (Keypriv, P) The receiver verifies the signature using the sender’s public key. P = D (Keypub, DS) csci5233 computer security & integrity (Chap. 3)
El Gamal Algorithm • To generate a key pair: • Choose a prime p and two integers, a and x, such that a < p and x < p. • The prime p should be chosen so that (p-1) has a large prime factor q. • Calculate the public key: y = ax mod p. • Private key: x • Public key: y csci5233 computer security & integrity (Chap. 3)
El Gamal Algorithm • (The sender) To sign a message m: • Choose a new random integer k, 0 < k < p-1 and k is relprime to (p-1). • Compute r = ak mod p. • Compute s = k-1 ( m – xr ) mod (p-1) • The message signature: r and s. • Verification: A recipient use the public key (y) to compute ( y r r s ) mod p and determine if it is equivalent to am mod p. csci5233 computer security & integrity (Chap. 3)
Hash Algorithm • A hash algorithm is a check function that protects data against modifications. • C.f., checksum in network transmission • Hash functions produce a reduced form of a body of data (called a digest or check value) such that most changes to the data will also change the reduced form. • A cryptographic hash function uses a cryptographic function as part of the hash function. • 1992: Secure Hash Algorithm (SHA) csci5233 computer security & integrity (Chap. 3)
Secure Hash Algorithm (SHA) • 1992: NIST • Input data < 264 bits • 160-bit digest • Strength: diffusion, the avalanche effect • See Fig. 3-9, p.99 • C.f., MD4, MD5 Both MD5 and SHA are variants of the MD4 by Rivest. Strength: MD4 < MD5 < SHA csci5233 computer security & integrity (Chap. 3)
Summary • Public key encryption algorithms: Merkle-Hellman, RSA, El Gamal • SHA • Next: DES, Key Escrow csci5233 computer security & integrity (Chap. 3)
