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Foundations of Cryptography Lecture 3 Lecturer: Moni Naor

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Recap of last weeks lecture One-way functions are essential to the two guard identification problem. –Important idea: simulation Examples of one-way functions –Subset sum, discrete log, factoring Weak one-way functions –Constructing strong one-way functions from weak one-way functions –Important idea: reduction –Finish the two repetition case from lecture 2

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Is there an ultimate one-way function? If f 1 :{0,1} * {0,1} * and f 2 :{0,1} * {0,1} * are guaranteed to: –Be polynomial time computable –At least one of them is one-way. then can construct a function g:{0,1} * {0,1} * which is one-way: g(x 1, x 2 )= (f 1 (x 1 ),f 2 (x 2 )) If an 5n 2 time one-way function is guaranteed to exist, can construct an O(n 2 log n) one-way function g : – Idea: enumerate Turing Machine and make sure they run 5n 2 steps g(x 1, x 2,…, x log (n) )=M 1 (x 1 ), M 2 (x 2 ), …, M log n (x log (n) ) If a one-way function is guaranteed to exist, then there exists a 5n 2 time one-way: – Idea: concentrate on the prefix 1/p(n)

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Conclusions Be careful what you wish for Problem with resulting one-way function: –Cannot learn about behavior on large inputs from small inputs –Whole rational of considering asymptotics is eroded Construction does not work for non-uniform one- way functions

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Homework Show that if probabilistic one-way functions exist then so do deterministic ones

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Identification - many times Alice would want to send an `approve message to Bob many times. They want to prevent Eve from interfering –Bob should be sure that Alice indeed approved each time. How to specify? Alice Bob Eve

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Specification of the Problem Alice and Bob communicate through a channel C Bob has an external counter C (# of times Alice approved) Eve completely controls the channel Requirements: CIf Alice wants to approve and Eve does not interfere – Bob increases the counter C CThe number of times Alice approves is a bound the value of counter C CIf Alice wants to approve and Eve does interfere - no requirements from the counter C until there is a quiescent period – A time that Alice wants to approve and Eve does not interfere Not the only possible specification! Can mandate that an approval was sent since the last time counter increased

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Solution to the password problem Assume that – f: {0,1} n {0,1} n is a (t,ε) one-way function –Adversaries run times is bounded by t –Let k be an upper bound on the number of identifications Setup phase: Alice chooses x {0,1} n, computes y=f (k) (x) and gives Bob and Charlie y When Alice wants to approve the i th time – she sends special symbol $ followed by i and y i =f (k-i) (x) If Bob gets a $ followed any symbols on channel – call them (j,z) ; compute y=f (j) (z) and compare to y C –If equal moves counter C to state j C –If not equal do nothing to counter C

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Is it secure? Need care in choosing f Should be difficult to invert any one of the iterated instances of f

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One-way on its iterates A function f: {0,1} n {0,1} n is called one-way on its iterates, if f is a polynomial-time computable function for every probabilistic polynomial-time algorithm A, every polynomial p(.), and all sufficiently large n s and all k p(n) Prob[A[f (k) (x)] f -1 (f (k) (x)) ] 1/p(n) Where x is chosen uniformly in {0,1} n and the probability is also over the internal coin flips of A From homework: not all one-way functions are one-way on their iterates Every one-way permutation is one-way on its iterates Subset sum function one-way on its iterates

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Example: the squaring function (Rabin) f(x,N)= (x 2 mod N,N) Quadratic residue mod a prime: If s and r satisfy s=r 2 mod P then s is called a quadratic residue modulo P If P is a prime then: – s=r 2 mod P has exactly two solutions mod P if 0~~
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Finding Square-roots and factoring are equivalent If know the factorization of N=PQ, then can compute square-roots If there is a procedure that computes square-roots correctly for non- negligible fraction – can boost it –Random self reducibility If we know (r,t) such that – s=r 2 =t 2 mod N –r =t mod P –r t mod Q Then we can factor by computing GCD(t-r,N) Homework: show how to use a square-root computing routine to factor while preserving the probability of success.

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A one-way on its iterates function To fully specify the function – need a starting procedure for generating – N=PQ where P,Q=3 mod 4 –Easy to specify given deterministic primality testing (even probabilistic is sufficient) density of primes –A quadratic residue mod N Easy by generating a random square Resulting function – one-way on its iterates

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Security of scheme If scheme can be broken then there is a j k where when Alice approved only j-1 times Eve convinced Bob to accepts j times with probability at least 1/kp(n) For this j can break the (k-j) th iterate of f with probability at least 1/kp(n) – Given y j =f (k-j) (x) compute y=f (j) (y j ) and simulate the adversary for j rounds –Adversary sees exactly the same distribution as in real life Hence probability adversary succeeds in forgery at step j (i.e. inverts y j ) is at least 1/kp(n)

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Problems with the scheme Need to know an upper bound k on the number of identifications Need to perform work proportional to k before first identification (what if it flops) Total work (in all k sessions) by Alice: O(k 2 ) –For Bob, if stores last value: O(k) –If Alice stores all k values y j : total work (in all k sessions) only O(k) – Homework : how can Alice store O(log k) values and perform amortized O(log k) work More problems: –need to maintain state. –Synchronization if both Bob and Charlie interleave as verifiers and the sum of their counters should be the number of times Alice identified.

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Want a scheme with unlimited use If we have a function that only Alice can compute but both Bob and Charlie can verify Alice can compute for session number i the value f(i) Problem: interleaving of verifiers – can replay Solution: challenge response –Verifier chooses a random nonce r and asks to see f(r)

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