Constrained Pseudorandom Functions: Verifiable and Delegatable

Constrained pseudorandom functions (introduced independently by Boneh and Waters (CCS 2013), Boyle, Goldwasser, and Ivan (PKC 2014), and Kiayias, Papadopoulos, Triandopoulos, and Zacharias (CCS 2013)), are pseudorandom functions (PRFs) that allow the owner of the secret key $k$ to compute a constrained key $k_f$, such that anyone who possesses $k_f$ can compute the output of the PRF on any input $x$ such that $f(x) = 1$ for some predicate $f$. The security requirement of constrained PRFs state that the PRF output must still look indistinguishable from random for any $x$ such that $f(x) = 0$. Boneh and Waters show how to construct constrained PRFs for the class of bit-fixing as well as circuit predicates. They explicitly left open the question of constructing constrained PRFs that are delegatable - i.e., constrained PRFs where the owner of $k_f$ can compute a constrained key $k_{f\u27}$ for a further restrictive predicate $f\u27$. Boyle, Goldwasser, and Ivan left open the question of constructing constrained PRFs that are also verifiable. Verifiable random functions (VRFs), introduced by Micali, Rabin, and Vadhan (FOCS 1999), are PRFs that allow the owner of the secret key $k$ to prove, for any input $x$, that $y$ indeed is the output of the PRF on $x$; the security requirement of VRFs state that the PRF output must still look indistinguishable from random, for any $x$ for which a proof is not given. In this work, we solve both the above open questions by constructing constrained pseudorandom functions that are simultaneously verifiable and delegatable

Performance Analysis of Sparse Recovery Based on Constrained Minimal Singular Values

The stability of sparse signal reconstruction is investigated in this paper. We design efficient algorithms to verify the sufficient condition for unique $\ell_1$ sparse recovery. One of our algorithm produces comparable results with the state-of-the-art technique and performs orders of magnitude faster. We show that the $\ell_1$-constrained minimal singular value ($\ell_1$-CMSV) of the measurement matrix determines, in a very concise manner, the recovery performance of $\ell_1$-based algorithms such as the Basis Pursuit, the Dantzig selector, and the LASSO estimator. Compared with performance analysis involving the Restricted Isometry Constant, the arguments in this paper are much less complicated and provide more intuition on the stability of sparse signal recovery. We show also that, with high probability, the subgaussian ensemble generates measurement matrices with $\ell_1$-CMSVs bounded away from zero, as long as the number of measurements is relatively large. To compute the $\ell_1$-CMSV and its lower bound, we design two algorithms based on the interior point algorithm and the semi-definite relaxation

A Generic Approach to Constructing and Proving Verifiable Random Functions

Verifiable Random Functions (VRFs) as introduced by Micali, Rabin and Vadhan are a special form of Pseudo Random Functions (PRFs) wherein a secret key holder can also prove validity of the function evaluation relative to a statistically binding commitment. Prior works have approached the problem of constructing VRFs by proposing a candidate under specific number theoretic setting --- mostly in bilinear groups --- and then grapple with the challenges of proving security in the VRF environments. These constructions achieved different results and tradeoffs in practical efficiency, tightness of reductions and cryptographic assumptions. In this work we take a different approach. Instead of tackling the VRF problem as a whole we demonstrate a simple and generic way of building Verifiable Random Functions from more basic and narrow cryptographic primitives. Then we can turn to exploring solutions to these primitives with a more focused mindset. In particular, we show that VRFs can be constructed generically from the ingredients of: (1) a 1-bounded constrained pseudo random function for a functionality that is ``admissible hash friendly , (2) a non-interactive statistically binding commitment scheme (without trusted setup) and (3) a non-interactive witness indistinguishable proofs or NIWIs. The first primitive can be replaced with a more basic puncturable PRF constraint if one is willing to settle for selective security or assume sub-exponential hardness of assumptions. In the second half of our work we support our generic approach by giving new constructions of the underlying primitives. We first provide new constructions of perfectly binding commitments from the Learning with Errors (LWE) and Learning Parity with Noise (LPN) assumptions. Second, we give give two new constructions of 1-bounded constrained PRFs for admissible hash friendly constructions. Our first construction is from the \nddh assumption. The next is from the $\phi$ hiding assumption