Cryptography Lecture 19

Advanced encryption standard (AES) Public design competition run by NIST Began in Jan 1997 15 algorithms submitted Workshops in 1998, 1999 Narrowed to 5 finalists Workshop in early 2000; winner announced in late 2000 Factors besides security taken into account

AES 128-bit block length 128-, 192-, and 256-bit key lengths Basically an SPN structure! 1-byte S-box (same for all bytes) Mixing permutation replaced by invertible linear transformation If two inputs differ in b bytes, outputs differ in ≥ 5-b bytes No attacks better than brute-force known

Hash functions

Security goal Main goal is collision resistance Want optimal birthday security Also want preimage resistance, 2nd-preimage resistance Want optimal security here as well “Optimal” measured relative to a random function Why not design H to be a “random function”?

The random-oracle (RO) model Treat H as a public, random function Then H(x) is uniform for any x… …unless the attacker computes H(x) explicitly

Many applications One canonical example: key derivation

Key derivation Consider deriving a (shared) key from (shared) high-entropy information E.g., biometric data E.g., generating randomness Cryptographic keys must be uniform, but shared data is only high-entropy

Min-entropy Let X be a distribution The min-entropy of X (measured in bits) is H(X) = - log maxx { Pr[X=x] } I.e., if H(X) = n, then the probability of guessing x sampled from X is (at most) 2-n Min-entropy is more suitable for crypto than standard (Shannon) entropy

Key derivation Given shared information x (sampled from distribution X), derive shared key k=H(x) In what sense can we claim that k is a good (i.e., uniform) cryptographic key?

The random-oracle (RO) model Treat H as a public, random function Then H(x) is uniform for any x… …unless the attacker computes H(x)… …but the attacker cannot do that (with high probability) if X has high min-entropy!

The RO model Intuitively Assume the hash function “is random” Models attacks that are agnostic to the specific hash function being used Security in the real world as long as “no weaknesses found” in the hash function

The RO model Formally Choose a uniform hash function as part of the security experiment Attacker can only evaluate H via explicit queries to an oracle Simulate H for the attacker as part of the security proof/reduction

The RO model In practice Prove security in the RO model Instantiate the RO with a “good” hash function Hope for the best…

Pros and cons of the RO model There is no such thing as a public hash function that “is random” Not even clear what this means formally Known counterexamples There are (contrived) schemes secure in the RO model, but insecure when using any real-world hash function Sometimes over-abused (arguably)

Pros and cons of the RO model No known example of “natural” scheme secure in the RO model being attacked in the real world If an attack is found, just replace the hash Proof in the RO model better than no proof at all Evidence that the basic design principles are sound

Ideal-cipher model “Stronger” than the RO model! Model block cipher F: {0,1}n x {0,1}n  {0,1}n as a collection of public, independent, random permutations I.e., for each key k, Fk is a random permutation on {0,1}n

The ideal-cipher model This is more than assuming F is a PRP Fk random even when k is known! No weak keys No related-key attacks Formally, similar to the RO model In particular, the only way to evaluate F is via explicit oracle queries Attacker allowed to query F and F-1

Building a hash function Two-stage approach Build a compression function (from a block cipher) I.e., collision-resistant hash function for fixed-length inputs Build a full-fledged hash function from a compression function Other approaches are possible

Building a compression function (Davies-Meyer) h(k, m) = Fk(m)  m m k F 

Proof of security Claim: attacker making q queries finds a collision with probability  q2/2n (optimal) Proof Each query to F/F-1 reveals one value hi = h(ki ,mi) Moreover, each hi is (essentially) uniform and independent of all previous outputs So probability of finding a collision is (essentially) the same as for a birthday attack

Building a hash function Assume as a primitive a compression function h for fixed input lengths Construct a hash function H for arbitrary-length inputs from a compression function h Prove collision-resistance of H based on collision-resistance of h (Could look at other properties also)

Merkle-Damgard transform mt |M| z1 … h h h h Note: M is padded with 0s if necessary

Merkle-Damgard transform Claim: if h is collision-resistant, than so is H Proof: Collision in H  collision in h Say H(m1, …, mt) = H(m’1, …, m’t’) |M|  |M’|, obvious |M| = |M’|, look at largest i with (zi, mi)  (z’i, m’i) m1 m2 mt |M| = mt+1 z1 z2 z3 zt+1 zt+2 … h h h h

(Computational) number theory

Why now? We have not needed any number theory or “advanced math” until now Practical private-key cryptography is based on stream ciphers, block ciphers, and hash functions Lots of non-trivial crypto can be done without any number theory!

Why now? Reason 1: Culmination of “top-down” approach For most cryptography, we ultimately need to assume some problem is hard The “lowest-level” assumptions we can make relate to problems in number theory These problems have often also been studied a long time

Why now? Reason 2: The public-key setting Public-key cryptography requires number theory (in some sense)