# Quantitative Examination of Multiple Authenticator Deployment

It appears that there are so many security professionals who pay no attention to the exactly opposite effects of 'multi-layer' and 'multi-entrance deployments. ‘Multi-Layer’ is also represented by ‘In-Series’, ‘In-Addition-To’, ‘All/BothAnd’ and ‘Conjunction’ in logic, while

‘Multi-Entrance’ by ‘In-Parallel’, ‘In-Stead-Of’, ‘EitherOr’ and ‘Disjunction’. Let me offer a quantitative examination of multiple authenticators deployed in two different ways.

Vulnerability (attack surface) of an authenticator is generally presented as a figure between 0 and 1. The larger the figure is, the larger the attack surface is, i.e., the more vulnerable. Assume, for instance, as just a thought experiment, that the vulnerability of the PKI-enabled token (x) be 1/10,000 and that of the password (y) be 10 times more vulnerable, say. 1/1,000. When the two are deployed in ‘multi-layer’ method, the total vulnerability (attack surface) is the product of the two, say, (x) and (y) multiplied. The figure of 1/10,000,000 means it is 1,000 times more secure than (x) alone.

On the other hand, when the two are deployed in ‘multi-entrance’ method, the total vulnerability (attack surface) is obtained by (x) + (y) – (xy), approximately 0.0011. It is about 11 times less secure than (x) alone.

So long as the figures are below 1, whatever figures are given to (x) and (y), deployment of 2 authenticators in ‘multi-layer’ method brings higher security while ‘multi-entrance’ deployment brings lower security. As such ‘multi-layer’ and ‘multi-entrance’ must be distinctly separated when talking about security effects of multiple authenticators.

**Remark**: Some people may wonder why (xy) is deducted from the sum of (x)+(y).

When (x) and (y) is very small, the (xy) is very close to 0, which we can practically ignore. But we should not ignore it when the figures are considerably large.

Imagine a case that both the two authenticators are deployed in an extremely careless manner, for instance, that the attack surfaces of (x) and (y) reach 70% (0.7) and 60% (0.6) respectively. If simply added, the figure would be 130% (1.3). It conflicts with the starting proposition the figures being between 0 and 1.

Imagine a white surface area. Painting 70% of it in black leaves 30% white surface. Painting 60% of the remaining 30% in black will result in 88% black and 12% white surfaces. Painting 60% first in black and then painting 70% of the remaining 40% brings the same result of 88% black and 12% white. So does “(x)+(y)-(xy)”. The overall vulnerability (attack surface) is 0.88 (88%) in this case.

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