Defending Against Intelligent Attackers at Large Scales

TL;DR

AI-enabled cyber threats can scale to attack large, distributed, learning adversaries. The paper develops a mathematical defense-in-depth framework with blockade and delay strategies, capturing breach probability via $L = 1 - (1 - p^n)^N$ and delay dynamics via $L_I = e^{-\ au \lambda n}$, while also addressing learning attackers through negative-binomial perspectives. It shows that small increases in defense count $n$, per-defense hardness $p$, or detection $d$ can offset exponential growth in attackers $N_A$, and provides a combined blockade-delay model with learning, summarized by the criterion $\frac{nd}{p} - \ln(N_A) > 1$. The results offer a scalable design guide for aggregated defenses in an AI-augmented threat landscape, highlighting where modest defense investments yield outsized security gains.

Abstract

We investigate the scale of attack and defense mathematically in the context of AI's possible effect on cybersecurity. For a given target today, highly scaled cyber attacks such as from worms or botnets typically all fail or all succeed. Here, we consider the effect of scale if those attack agents were intelligent and creative enough to act independently such that each attack attempt was different from the others or such that attackers could learn from their successes and failures. We find that small increases in the number or quality of defenses can compensate for exponential increases in the number of independent attacks and for exponential speedups.

Figures (4)

• Figure 1: The number of independent attacks that would be needed to overcome a set of defenses increases exponentially with the number of defenses. Parts a, b, and c are for a constant probability (0.001) that at least one attack succeeds against all defenses. Part d shows the likelihood of at least one breach while holding the probability of individual defense failure ($p$) at 0.43. It shows linear spacing between exponentially increasing numbers of attacks.
• Figure 2: Harder individual defenses can compensate for exponentially increasing numbers of attacks. The probability of all defenses failing at least once, $L$, was held constant at 0.001 in the figure.
• Figure 3: An exponential increase in the number of attacks or speedups can be compensated for by a linear increase in the number of defenses. For parts a and b, the likelihood of at least one breach ($L$) is held constant at 0.001.
• Figure 4: The number of defenses and the number of attacks are sensitive to the ratio of defensive and offensive rates detection and defense bypass respectively.