Eyal Kushilevitz and
The key to our approach is a new primitive of independent interest, which we call an Exposure-Resilient Function (ERF) -- a deterministic function whose output appears random (in a perfect, statistical or computational sense) even if almost all the bits of the input are known. ERF's by themselves efficiently solve the partial key exposure problem in the setting where the secret is simply a random value, like in private-key cryptography. They can also be viewed as very secure pseudorandom generators, and have many other applications.
To solve the general partial key exposure problem, we use the (generalized) notion of an All-Or-Nothing Transform (AONT), an invertible (randomized) transformation T which, nevertheless, reveals ``no information'' about x even if almost all the bits of T(x) are known. By applying an AONT to the secret key of any cryptographic system, we obtain security against partial key exposure. To date, the only known security analyses of AONT candidates were made in the random oracle model.
We show how to construct ERF's and AONT's with nearly optimal parameters. Our computational constructions are based on any one-way function. We also provide several applications and additional properties concerning these notions.