Abstract: This paper studies All-or-Nothing Transforms (AONTs), which have been proposed by Rivest as a mode of operation for block ciphers. An AONT is an unkeyed, invertible, randomized transformation, with the property that it is hard to invert unless all of the output is known. Applications of AONTs include improving the security and efficiency of symmetric-key and public-key encryption. We give several formal definitions of security for AONTs that are stronger than the original ones and are more suited to practical applications. We then prove that Optimal Asymmetric Encryption Padding (OAEP), which was originally introduced by Bellare and Rogaway in a different context, satisfies these definitions (in the random oracle model). This is the first construction of an AONT that has been proven secure in the strong sense. The adversary's advantage in getting information about the input of the OAEP is shown to be inversely exponential in the number of bits removed from the output. Our bound is nearly optimal, in the sense that no adversary can do substantially better against the OAEP than by exhaustive search. We also show that no AONT can achieve substantially better security than OAEP.

Abstract: We study the problem of partial key exposure. Standard cryptographic definitions and constructions do not guarantee any security even if a tiny fraction of the secret key is compromised. We show how to build cryptographic primitives, in the standard model (without random oracles), that remain secure even when an adversary is able to learn almost all of the secret key.

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.