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Sensitive computations are now routinely delegated to thirdparties. In response, Confidential Computing technologies are being introduced to microprocessors, offering a protected processing environment, which we generically call an isolate, providing confidentiality and integrity guarantees to code and data hosted within-even in the face of a privileged attacker. Isolates, with an attestation protocol, permit remote third-parties to establish a trusted "beachhead" containing known code and data on an otherwise untrusted machine. Yet, the rise of these technologies introduces many new problems, including: how to ease provisioning of computations safely into isolates; how to develop distributed systems spanning multiple classes of isolate; and what to do about the billions of "legacy" devices without support for Confidential Computing?Tackling the problems above, we introduce Veracruz, a framework that eases the design and implementation of complex privacy-preserving, collaborative, delegated computations among a group of mutually mistrusting principals. Veracruz supports multiple isolation technologies and provides a common programming model and attestation protocol across all of them, smoothing deployment of delegated computations over supported technologies. We demonstrate Veracruz in operation, on private in-cloud object detection on encrypted video streaming from a video camera. In addition to supporting hardware-backed isolates-like AWS Nitro Enclaves and Arm ® Confidential Computing Architecture Realms-Veracruz also provides pragmatic "software isolates" on Armv8-A devices without hardware Confidential Computing capability, using the high-assurance seL4 microkernel and our IceCap framework.

Given a field [Formula: see text] and [Formula: see text], we say that a polynomial [Formula: see text] has newly reducible [Formula: see text]th iterate over [Formula: see text] if [Formula: see text] is irreducible over [Formula: see text], but [Formula: see text] is not (here [Formula: see text] denotes the [Formula: see text]th iterate of [Formula: see text]). We pose the problem of characterizing, for given [Formula: see text], fields [Formula: see text] such that there exists [Formula: see text] of degree [Formula: see text] with newly reducible [Formula: see text]th iterate, and the similar problem for fields admitting infinitely many such [Formula: see text]. We give results in the cases [Formula: see text] as well as for [Formula: see text] when [Formula: see text]. In particular, we show that for all these [Formula: see text] pairs, there are infinitely many monic [Formula: see text] of degree [Formula: see text] with newly reducible [Formula: see text]th iterate over [Formula: see text]. Curiously, the minimal polynomial [Formula: see text] of the golden ratio is one example of [Formula: see text] with newly reducible third iterate; very few other examples have small coefficients. Our investigations prompt a number of conjectures and open questions.

Given a field K and n > 1, we say that a polynomial f ∈ K[x] has newly reducible nth iterate over K if f n−1 is irreducible over K, but f n is not (here f i denotes the ith iterate of f ). We pose the problem of characterizing, for given d, n > 1, fields K such that there exists f ∈ K[x] of degree d with newly reducible nth iterate, and the similar problem for fields admitting infinitely many such f . We give results in the cases (d, n) ∈ {(2, 2), (2, 3), (3, 2), (4, 2)} as well as for (d, 2) when d ≡ 2 mod 4. In particular, we show that for all these (d, n) pairs, there are infinitely many monic f ∈ Z[x] of degree d with newly reducible nth iterate over Q. Curiously, the minimal polynomial x 2 − x − 1 of the golden ratio is one example of f ∈ Z[x] with newly reducible third iterate ; very few other examples have small coefficients. Our investigations prompt a number of conjectures and open questions.

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