A Lower Bound for Sampling Disjoint Sets

Suppose Alice and Bob each start with private randomness and no other input, and they wish to engage in a protocol in which Alice ends up with a set x subseteq[n] and Bob ends up with a set y subseteq[n], such that (x,y) is uniformly distributed over all pairs of disjoint sets. We prove that for some constant beta0 of the uniform distribution over all pairs of disjoint sets of size sqrt{n}

Affine Extractors and AC0-Parity

We study a simple and general template for constructing affine extractors by composing a linear transformation with resilient functions. Using this we show that good affine extractors can be computed by non-explicit circuits of various types, including AC0-Xor circuits: AC0 circuits with a layer of parity gates at the input. We also show that one-sided extractors can be computed by small DNF-Xor circuits, and separate these circuits from other well-studied classes. As a further motivation for studying DNF-Xor circuits we show that if they can approximate inner product then small AC0-Xor circuits can compute it exactly - a long-standing open problem

Randomness Condensers for Efficiently Samplable, Seed-Dependent Sources

We initiate a study of randomness condensers for sources that are efficiently samplable but may depend on the seed of the con- denser. That is, we seek functions Cond : {0, 1}n ×{0, 1}d → {0, 1}m such that if we choose a random seed S ← {0,1}d, and a source X = A(S) is generated by a randomized circuit A of size t such that X has min- entropy at least k given S, then Cond(X;S) should have min-entropy at least some k′ given S. The distinction from the standard notion of ran- domness condensers is that the source X may be correlated with the seed S (but is restricted to be efficiently samplable). Randomness extractors of this type (corresponding to the special case where k′ = m) have been implicitly studied in the past (by Trevisan and Vadhan, FOCS ‘00). We show that: – Unlike extractors, we can have randomness condensers for samplable, seed-dependent sources whose computational complexity is smaller than the size t of the adversarial sampling algorithm A. Indeed, we show that sufficiently strong collision-resistant hash functions are seed-dependent condensers that produce outputs with min-entropy k′ = m − O(log t), i.e. logarithmic entropy deficiency. – Randomness condensers suffice for key derivation in many crypto- graphic applications: when an adversary has negligible success proba- bility (or negligible “squared advantage” [3]) for a uniformly random key, we can use instead a key generated by a condenser whose output has logarithmic entropy deficiency. – Randomness condensers for seed-dependent samplable sources that are robust to side information generated by the sampling algorithm imply soundness of the Fiat-Shamir Heuristic when applied to any constant-round, public-coin interactive proof system.Engineering and Applied Science

How to Subvert Backdoored Encryption: Security Against Adversaries that Decrypt All Ciphertexts

In this work, we examine the feasibility of secure and undetectable point-to-point communication when an adversary (e.g., a government) can read all encrypted communications of surveillance targets. We consider a model where the only permitted method of communication is via a government-mandated encryption scheme, instantiated with government-mandated keys. Parties cannot simply encrypt ciphertexts of some other encryption scheme, because citizens caught trying to communicate outside the government\u27s knowledge (e.g., by encrypting strings which do not appear to be natural language plaintexts) will be arrested. The one guarantee we suppose is that the government mandates an encryption scheme which is semantically secure against outsiders: a perhaps reasonable supposition when a government might consider it advantageous to secure its people\u27s communication against foreign entities. But then, what good is semantic security against an adversary that holds all the keys and has the power to decrypt? We show that even in the pessimistic scenario described, citizens can communicate securely and undetectably. In our terminology, this translates to a positive statement: all semantically secure encryption schemes support subliminal communication. Informally, this means that there is a two-party protocol between Alice and Bob where the parties exchange ciphertexts of what appears to be a normal conversation even to someone who knows the secret keys and thus can read the corresponding plaintexts. And yet, at the end of the protocol, Alice will have transmitted her secret message to Bob. Our security definition requires that the adversary not be able to tell whether Alice and Bob are just having a normal conversation using the mandated encryption scheme, or they are using the mandated encryption scheme for subliminal communication. Our topics may be thought to fall broadly within the realm of steganography. However, we deal with the non-standard setting of an adversarially chosen distribution of cover objects (i.e., a stronger-than-usual adversary), and we take advantage of the fact that our cover objects are ciphertexts of a semantically secure encryption scheme to bypass impossibility results which we show for broader classes of steganographic schemes. We give several constructions of subliminal communication schemes under the assumption that key exchange protocols with pseudorandom messages exist (such as Diffie-Hellman, which in fact has truly random messages)