CMSC 414 Computer and Network Security Lecture 6

CMSC 414Computer and Network SecurityLecture 6 Jonathan Katz

Administrative announcements • Midterm I • March 6 • GRACE accounts set up • Read the essays and papers linked from the course webpage • Will discuss next Tuesday and/or Thursday

Public-key cryptography

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

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

Private- vs. public-key I • Disadvantages of private-key cryptography • Need to securely share keys • What if this is not possible? • Need to know in advance the parties with whom you will communicate • Can be difficult to distribute/manage keys in a large organization • O(n2) keys needed for person-to-person communication in an n-party network • All these keys need to be stored securely • Inapplicable in open systems (think: e-commerce)

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

Private- vs. public-key III • Public-key cryptography is not a cure-all • Still requires secure 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 (for public-key encryption) • Can be too inefficient for certain applications

Cryptographic primitives

Public-key encryption

Functional definition • 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

Security? • Just as in the case of private-key encryption, but the attacker gets to see the public key pk • That is: • 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) • Security against ciphertext-only attacks implies security against chosen-plaintext attacks

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

Security • If the DDH assumption holds, the El Gamal encryption scheme is secure against chosen-plaintext attacks

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 N

RSA key generation • Generate random 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) • We have an asymmetry! • Given c=xe mod N, receiver can compute x=cd mod N • No apparent way for anyone else to recover x

Hardness of the RSA problem? • The RSA problem: • Compute x given N, e, and xe 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

“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?

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

Padded RSA • 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 idea has been standardized as RSA PKCS #1 v1.5

Hybrid encryption • Public-key encryption is “slow” • Encrypting “block-by-block” would be inefficient for long messages • Hybrid encryption gives the functionality of public-key encryption at the (asymptotic) efficiency of private-key encryption!

Enc’ Enc Hybrid encryption message “encrypted message” ciphertext k “encapsulated key” pk Enc = public-key encryption scheme Enc’ = private-key encryption scheme

Security • If public-key component and private-key component are secure against chosen-plaintext attacks, then hybrid encryption is secure against chosen-plaintext attacks

Malleability/chosen-ciphertext security • All the public-key encryption schemes we have seen so far are malleable • Given a ciphertext c that encrypts an (unknown) message m, may be possible to generate a ciphertext c’ that encrypts a related message m’ • In many scenarios, this is problematic • E.g., auction example; password example • Note: the problem is not integrity (there is no integrity in public-key encryption, anyway), but malleability

CMSC 414 Computer and Network Security Lecture 6