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Foundations of Cryptography Lecture 7 Lecturer:Danny Harnik

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Maurer ’ s Bounded Storage Model Most Cryptographic tasks are only possible when parties are known to be bounded. “ Mainstream Cryptography ” : Assume parties are time bounded (run in polynomial time). Maurer ’ s model: Assume parties have bounded storage. Remark: Bounded Storage ≠ Bounded Space. Measures only the storage capacity at one point of the process.

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The bounded storage model: The setting A long random string R is transmitted. Honest parties store small portions of R. Parties interact. Protocol is secure even against dishonest parties which store almost all of R. A long random string R of length N Alice Bob Malicious party Stores ¾N bits Stores N ½ (Arbitrary function of R)

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Example: Key-Agreement Alice and Bob interact over a public channel (with no initial secret key). They want to agree on a secret key. Alice Bob Eavesdropper public channel key ??

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A long random string R of length N Protocol: Key-Agreement [CM97] A long random string R is transmitted. Alice and Bob store random subsets of size ~N ½. Send position of subsets and agree on content of intersection. Next, we show that an eavesdropper which stores ¾N bits has a lot of entropy on the key. Alice Bob Eavesdropper Stores N ½ key Does not know the key!

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¾N bits key The view of the adversary Simplifying assumption: The adversary stores a subset bits of R of size ¾N. The sets chosen by the players are random. The set which defines the key is a random set. The adversary does not remember ~ ¼N bits. Eavesdropper random set From my point of view the key is a high- entropy source! * This holds even when the adversary stores an arbitrary function of R [NZ93]. ¾ known ¼ unknown

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Randomness Extractors [NZ93] Extract randomness from arbitrary distributions which contain sufficient (min)-entropy. Use a short seed of truly random bits. Output is (close to) uniform even when the adversary knows the seed. Relation to BSM pointed out by [Lu02,Vad03] high entropy distribution Extractor seed random output

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A long random string R of length N Key-Agreement using extractors A long random string R is transmitted. Alice and Bob store random subsets of size ~N ½. Send position of subsets and agree on content of intersection. Alice randomly chooses a seed and sends it to Bob. Both apply an extractor To receive the key. Alice Bob Stores N ½ Extractor seed random key

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Further Improvements Instead of random subsets, Alice & Bob remember pairwise independent locations Eavesdropper still has high min-entropy [NZ]. Saves communication when finding the intersection of both sides. Can further use better “ Samplers ” to choose these locations. Only need to send seed to the sampler in order to agree on intersection.

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The Secret Key Setting Seed to sampler is used as the secret key. Alice & Bob only store the bits at the locations the sampler chooses. Can use small set for Alice and Bob. For the Eavesdropper this set is a high min-entropy source. By applying extractor, receive a long key that is close to uniform from Eavesdropper ’ s point of view. Best result so far for message of length m [Vad03]: Alice & Bob store only O(m + log 1/ ε ) Secret Key length: O(log N + log 1/ ε )

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The bounded storage model Practical? Depends on ratio between price of memory and speed of broadcast. Most of the research so far focused on: Key agreement [Mau93,CM97]. Secret-key encryption [Mau93,CM97,AR99,ADR02,DR02,DM02,Lu02,Vad03]. Advantages: Clean model. Security does not require unproven assumptions. Everlasting security: The security is guaranteed even if at a later stage the adversary gains more memory.

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