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CMSC 414 Computer and Network Security Lecture 7

CMSC 414 Computer and Network Security Lecture 7. Jonathan Katz. Public-key cryptography. The public-key setting. A party (Alice) generates a public key along with a matching secret key (aka private key )

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CMSC 414 Computer and Network Security Lecture 7

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  1. CMSC 414Computer and Network SecurityLecture 7 Jonathan Katz

  2. Public-key cryptography

  3. The public-key setting • A party (Alice) generates a public key along with a matching secret key (aka private key) • The public key is widely distributed, and is assumed to be known to anyone (Bob) who wants to communicate with Alice • We will discuss later how this can be ensured • Alice’s public key is also known to the attacker! • Alice’s secret key remains secret • Bob may or may not have a public key of his own

  4. pk c = Encpk(m) pk c = Encpk(m) The public-key setting

  5. Private- vs. public-key I • Disadvantages of private-key cryptography • Need to securely share keys • What if this is not possible? • Difficult to distribute/manage keys in a large organization • Private-key crypto requires private+authenticated channel to share keys; public-key crypto requires authenticated channel • Consider communication in an n-party network: each party must store n-1 secret keys • Inapplicable in open systems (i.e., e-commerce) • Need to know in advance the parties with whom you will communicate • Not suited for “many-to-one” communication

  6. Private- vs. public-key II • Why study private-key setting at all? • Private-key is orders of magnitude more efficient • Private-key still has domains of applicability • Military settings, disk encryption, … • Bidirectional, “one-to-one” authenticated communication • Public-key crypto is “harder” to get right • Need stronger assumptions, easier to attack • Can combine private-key primitives with public-key techniques to get the best of both (for encryption) • So still need to understand the private-key setting! • Can distribute keys using trusted entities (KDCs)

  7. Private- vs. public-key III • Public-key cryptography is not a cure-all • Still requires authenticated distribution of public keys • May (sometimes) be just as hard as sharing a key • Technically speaking, requires only an authenticated channel instead of an authenticated + private channel • Not clear with whom you are communicating (unless the sender has a public key) • Can be too inefficient for certain applications

  8. Cryptographic primitives

  9. Public-key encryption • Key-generation algorithm: randomized algorithm that outputs (pk, sk) • Encryption algorithm: • Takes a public key and a message (plaintext), and outputs a ciphertext; c  Epk(m) • Decryption algorithm: • Takes a private key and a ciphertext, and outputs a message (or perhaps an error); m = Dsk(c) • Correctness: for all (pk, sk), Dsk(Epk(m)) = m

  10. Security? • Just as in the case of private-key encryption, but the attacker gets to see the public key pk • For all m0, m1, no adversary running in time T, given pk and an encryption of m0 or m1, can determine the encrypted message with probability better than 1/2 +  • Public-key encryption must be randomized (even to achieve security against ciphertext-only attacks) • In the public-key setting, security against ciphertext-only attacks implies security against chosen-plaintext attacks • And security for encryption of multiple messages

  11. p, g, hA = gx p, g hA = gx mod p c = (KBA. m) mod p hB = gy mod p hB, c El Gamal encryption • We have already (essentially) seen one encryption scheme: Receiver Sender KAB = (hB)x KBA = (hA)y

  12. El Gamal encryption • (Some aspects of the actual scheme are simplified) • Key generation • Choose a large prime p, and an element g Zp* • Choose random x  {0, …, p-2}, set h=gx • The public key is (p, g, h), and the private key is x • Encryption • View the message m as an element of Zp* • Choose random r  {0, …, p-2} • The ciphertext is (gr, hr m) • To decrypt ciphertext (c1, c2) output c2/c1x

  13. Security? • Security of El Gamal encryption is equivalent to the decisional Diffie-Hellman assumption • Best known algorithm for decisional Diffie-Hellman in Zp* runs in time ≈ exp(log p1/3) • So if p is a 1024-bit prime, best current attack on El Gamal encryption requires time ≈ 260 • In other groups, the Diffie-Hellman problem is currently ‘harder’ • E.g., for elliptic curve groups, best current algorithms require time exp(log |G|/2) • Can use ~120-bit group elements to get 260 security

  14. RSA background • N=pq, p and q distinct, odd primes • (N) = (p-1)(q-1) • Easy to compute (N) given the factorization of N • Hard to compute (N) without the factorization of N • Fact: for all x  ZN*, it holds that x(N) = 1 mod N • Proof: take CMSC 456! • If ed=1 mod (N), then for all x it holds that (xe)d = x mod NI.e., given d, we can compute eth roots

  15. We have an asymmetry! • Let e be relatively prime to (N) • Needed so that ed=1 mod (N) has a solution • Given e and the factors of N, can compute d and hence compute eth roots • Without the factorization of N, no apparent way to compute eth roots • Let’s use this to encrypt…

  16. Hardness of computing eth roots? • The RSA problem: • Given N, e, and c, compute c1/e mod N • If factoring is easy, then the RSA problem is easy • We know of no other way to solve the RSA problem besides factoring N • But we do not know how to prove that the RSA problem is as hard as factoring • The upshot: we believe factoring is hard, and we believe the RSA problem is hard

  17. How hard is factoring? • Best current algorithms for factoring N=pq a product of two equal-length primes, run in time ≈ exp(log N1/3) • So need |N| ≈ 1024 for reasonable security • Currently |N| ≈ 2048 recommended for good security margins

  18. RSA key generation • Generate random primes p, q of sufficient length • Compute N=pq and (N) = (p-1)(q-1) • Compute e and d such that ed = 1 mod (N) • e must be relatively prime to (N) • Typical choice: e = 3; other choices possible • Public key = (N, e); private key = (N, d)

  19. “Textbook RSA” encryption • Public key (N, e); private key (N, d) • To encrypt a message m  ZN*, compute c = me mod N • To decrypt a ciphertext c, compute m = cd mod N • Correctness clearly holds… • …what about security?

  20. Textbook RSA is insecure! • It is deterministic! • Furthermore, it can be shown that the ciphertext leaks specific information about the plaintext

  21. Padded RSA • Introduce randomization… • Public key (N, e); private key (N, d) • Say |N| = 1024 bits • To encrypt m  {0,1}895, • Choose random r  {0,1}128 • Compute c = (r | m)e mod N • Decryption done in the natural way… • Essentially this is standardized as PKCS #1 v1.5 (since superseded)

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