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1.1 The adversarial streaming model The adversarial streaming model, in various forms, was considered by [MNS11,GHR+12,GHS+12, AGM12a,AGM12b,HW13,BY19,BJWY20,HKM+20,KMNS21,BHM+21]. We give here the formu- lation presented by Ben-Eliezer et al. [BJWY20]. The adversarial setting is modeled by a two-player game between a (randomized) StreamingAlgorithm and an Adversary. At the beginning, we fix a function : X∗ R. Then the game proceeds in rounds, where in the ith round: F → 1. The Adversary chooses an update u X for the stream, which can depend, in particular, i ∈ on all previous stream updates and outputs of StreamingAlgorithm.

2. The StreamingAlgorithm processes the new update ui and outputs its current response z R. i ∈

The goal of the Adversary is to make the StreamingAlgorithm output an incorrect response zi at some point i in the stream. For example, in the distinct elements problem, the adversary’s goal is that at some step i, the estimate zi will fail to be a (1 + α)-approximation of the true current number of distinct elements. In this work we present a new framework for transforming an oblivious streaming algorithm into an adversarially-robust streaming algorithm. Before presenting our framework, we first elaborate on the existing literature and the currently available frameworks.

1.2 Existing framework: Ben-Eliezer et al. [BJWY20] To illustrate the results of [BJWY20], let us consider the distinct elements problem, in which the function counts the number of distinct elements in the stream. Observe that, assuming that there F are no deletions in the stream, this quantity is monotonically increasing. Furthermore, since we are aiming for a multiplicative (1 α) error, even though the stream is large (of length m), the number ± 1 of times we actually need to modify the estimates we release is quite small (roughly α log m times). Informally, the idea of [BJWY20] is to run several independent copies of an oblivious algorithm (in parallel), and to use each algorithm to release answers over a part of the stream during which the estimate remains constant. In more detail, the generic transformation of [BJWY20] (applicable not only to the distinct elements problem) is based on the following definition.

Definition 1.2 (Flip number [BJWY20]). Given a function , the (α, m)-flip number of , de- F F noted as λ (or simply λ in short), is the maximal number of times that the value of can α,m F change (increase or decrease) by a factor of (1 + α) during a stream of length m.

The generic construction of [BJWY20] for a function is as follows. F 1. Instantiate λ independent copies of an oblivious streaming algorithm for the function , and F set j = 1.

2 2. When the next update ui arrives:

(a) Feed ui to all of the λ copies. (b) Release an estimate using the jth copy (rounded to the nearest power of (1+ α)). If this estimate is different than the previous estimate, then set j j + 1. ← Ben-Eliezer et al. [BJWY20] showed that this can be used to transform an oblivious streaming algorithm for into an adversarially robust streaming algorithm for . In addition, the overhead F F in terms of memory is only λ, which is small for many problems of interest. The simple, but powerful, observation of Ben-Eliezer et al. [BJWY20], is that by “using every copy at most once” we can break the dependencies between the internal randomness of our algorithm and the choice for the elements in the stream. Intuitively, this holds because the answer is always computed using a “fresh copy” whose randomness is independent from the choice of stream items.

1.3 Existing framework: Hassidim et al. [HKM+20] Hassidim et al. [HKM+20] showed that, in fact, we can use every copy of the oblivious algorithm much more than once. In more detail, the idea of Hassidim et al. is to protect the internal random- ness of each of the copies of the oblivious streaming algorithm using differential privacy [DMNS06]. Hassidim et al. showed that this still suffices in order to break the dependencies between the in- ternal randomness of our algorithm and the choice for the elements in the stream. Informally, the framework