1. Cryptography
Lecture 14

2. Other applications of hash functions

3. Hash functions are ubiquitous
Collision-resistance  “fingerprinting”
Used as a one-way function
Key derivation

4. Fingerprinting E.g., file integrity
Assuming it is possible to get a reliable copy of H(x) for file x
Note: different from integrity in the context of message-authentication codes

5. Outsourced storage How to outsource files to an untrusted server? x x
h=H(x) x
H(x)=?h

6. Outsourced storage x1, …, xn x1, …, xn hi =H(xi) i xi H(xi)=?hi
O(n) client storage!

7. Outsourced storage x1, …, xn x1, …, xn h =H(x1, …, xn) i x1, …, xn
H(x1, …, xn)=?h
O(n|x|) communication!

8. Outsourced storage x1, …, xn x1, …, xn h =H(H(x1), …, H(xn)) i
xi, h1, …, hn
H(h1, …, H(xi), …, hn)=?h
|xi| + O(n) communication!

9. Merkle tree Only store the root! Verify…x1 x2 x2 x3 x4
Only store the root!
Verify…
O(log n) communication/computation!

10. Outsourced storage
Using a Merkle tree, we can solve the outsourcing problem with O(1) client storage and |x| + O(log n) communication

11. Password hashing Server stores H(pw) instead of pw
Requires more than one-wayness of H…
See later discussion on random oracles
Salting…

12. 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

13. 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 entropy

14. 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., uniformly distributed) cryptographic key?

15. Exam review

16. Comments on the exam Q1-Q3 should have been immediate
Q1/Q3 were straight from the book
Q2 was similar to problems done in class, and on the optional exercises
Q4 required some thought (16 points)
Q5 was based on HW4, but required some thought (14 points)

17. Requests for re-grades
Mistaken calculation
Happy to fix
Incorrectly graded/scored
Scores followed a consistent rubric for all exams
Make sure you understand the correct answer
If you still want a re-grade:
Submit your exam along with a written explanation for why you are requesting a re-grade
Entire exam will be re-graded