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Intro To Encryption Exercise 6. Problem. Is every (weak) CRHF also a OWF. Solution. No!!! Counter Example: Suppose h is a weakly CRHF Let h’ (x)=x|| h (x) No collisions in h’ , clearly not a OWF. What about h’ :{0,1} n {0,1} l l <n. Exercise . Problem.
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Problem • Is every (weak) CRHF also a OWF
Solution • No!!! • Counter Example: • Suppose h is a weakly CRHF • Let h’(x)=x||h(x) • No collisions in h’, clearly not a OWF. • What about h’:{0,1}n{0,1}ll<n. • Exercise
Problem • Show a OWHF and distribution of passwords s.t. both unix and S/Key fail
Solution • Let h(x) be a OWHF. • Let h’(x) return: • 0 if 3 final chars of x end with AAA • h(x) otherwise • What kinds of an attack should ADV use? • For Unix Password Scheme • For S/Key password scheme
Problem • Lets assume a SALT mechanism is proposed for the previous problem. • How should you implement it using the proposed h’ without changing its internal design?
Solution • h’’(x)=h’(x||salt) with salt being != AAA.
Problem • Why does brute-force attack on Target Collision Resistant takesO(2n) guesses (not O(2n/2)-from bithday paradox)
Solution • Since ADV picks x, x’ he may be able to find a collision with O(2n/2). • BUT!!! ADV does not know key k prior to his choice. The key is chose AFTER he chose. • So? • So ADV can’t efficiently calculate hashes for x, x’ because he does not know which hash function The user may choose. • In other words for some key kf(x)=f(x’) but for other key kf(x)!=f(x’)
Problem • Computer viruses modify executable program files to `infect` them. • One common protection against viruses is to maintain, in read-only storage, a list containing a short `fingerprint` of each executable file, allowing the antivirus program to validate that an executable was not modified. • Which of the hash function properties are necessary for computing the fingerprint?
Solution • We need collision resistance features. • Do we need Weakly, Strong or target collision resistance requirements?
Problem • We wish to build hash functions from block ciphers. • We wish Same function as WCRHF that is constructed as:h(x)=Ex(0) [if we use a block cipher which allows arbitrary long keys] • Does this construction provides WCRHF?
Solution • No!!! • Assume Ek(x) is a block cipher. • Assume E’k1,k2(x)=k1Ek2(X). • Is this still a block cipher??? Prove!!! • Let X=X1||X2(without the limitation of generality) • Let h(x1||x2)=E’x1,x2(0)=x1Ex2(0)
Solution • Assume X=10011100 -> X1=1001, X2=1100 • Assume E1100(0)=1101 • Let h(x)=10011101=0101 • Let ADV A find a collision to X=10011100 with h(x)=0101 • Let there be Y=10001110 -> Y1=1000,Y2=1110 • Y2 ‘=Ey2(0)=0011. • Y1 ‘=h(x) Ey2’(0)=0101 0011=0110
Solution • For our construction: • Let h(Y1’||Y2)=Y1’Ey2(0).i.e. h(01101110). • Y1’Ey2(0)=(h(x) Ey2’(0)) Ey2(0)=h(x).i.e.: 0110 0011 = 0101.
Problem • Alice and Bob communicate by phone. • Assume they can identify each other’s voice, but a hacker, Eve, may eavesdrop on their communication. • Alice wants to send a shared key to Bob, carried by Charlie, a completely reliable and trustworthy courier, which is unfortunately not known to Bob. • We want Charlie to know some secret so it can prove his identity to Bob by exposing this secret • We also don’t want Eve to impersonate as Charlie. • Show how Alice can establish such a secret using (only) a one-way hash function.
Solution • Alice hands charlie a voice print. She may say: “Hello Bob”. • Charlie uses h as the OWF and uses:h(“Hello Bob”Alice||”Hello Bob from Alice”charlie)To establish a key.
