Multifactor Authentication

4. Enabling Flexible MFA Operation

4.1. Conventional Approach

One of the approaches considered within the scope of this work is based on utilizing Lagrange polynomials for secret sharing. The system secret S is usually "split" and distributed among a set of key holders. It could be recovered later on, as described in and numerous other works, as

$\begin{array}{l} f(x)=S+a_{1} x+a_{2} x^{2}+\cdots+a_{l-1} x^{l-1} \\ f(0)=S \end{array}$

where $a_{i}$ are the generated polynomial indexes and $x$ is a unique identification factor $F_{i}$ . In such systems, every key holder with a factor ID obtains its own unique key share $S_{I D}=f(I D)$

In conventional systems, it is required to collect any $/$ shares $\left\{S_{I D_{1}}, S_{I D_{2}}, \ldots, S_{I D_{l}}\right\}$ of the initial secret to unlock the system, while the curve may offer $n>l$ points, as it is shown in Figure 5. The basic principle behind this approach is to specify the secret $S$ and use the generated curve based on the random coefficients $a_{i}$ to produce the secret shares $S_{i}$ . This methodology is successfully utilized in many secret sharing systems that employ the Lagrange interpolation formula.

Figure 5. Lagrange secret sharing scheme.

Unfortunately, this approach may not be applied for the MFA scenario directly, since the biometric parameters are already in place, i. e ., we can neither assign a new

$S_{i}$ to a user nor modify them. On the one hand, the user may set some of the personal factors, such as password, PIN-code, etc. On the other hand, some of them may be unchangeable (biometric parameters and behavior attributes). In this case, an inverse task where the shares of the secret

$S_{I D_{i}}$ are known as factor values

$S_{i}$ is to be solved. Basically,

$S_{i}$ are fixed and become unique

$\left\{S_{1}, S_{2}, \ldots, S_{l}\right\}$ when set for a user. In this case,

$S$ is the secret for accessing the system and should be acquired with the user factor values. A possible solution based on the reversed Lagrange interpolation formula is proposed in the following subsection.