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Efficiency, Uniqueness, and Robustness

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There are (at least) three primary considerations when designing a sketch-based authentication system: computational complexity, security, and tolerance. In this chapter, using the generalized Simple K-Space (SKS) approach, the efficiency of model construction is studied. There are two algorithmic implementations to construct this model: the direct/forward approach (or primal algorithm) or the indirect/backward (or dual algorithm). The efficiency of constructing the SKS model is important because the model is most definitely the computational bottleneck of this method. Here, the dual algorithm is shown to enable the use of a model approximation that significantly improves the efficiency for computing the high dimensional model representation. Also, the tradeoff between security and usability is analyzed using SKS. In particular, it is demonstrated that as smoothing parameters of the SKS model asymptotically approach zero, the model is unique and provides “perfect security.” However, we also demonstrate the model is generally not unique, but more robust, when a sufficient amount of smoothing occurs over the model. Therefore, SKS is shown to explicitly provide fine-tuned controllability of the security/usability tradeoff.


  • Efficiency
  • Uniqueness
  • Robustness
  • Tolerance
  • Security
  • Usability

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  • DOI: 10.1007/978-3-319-13629-5_4
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Fig. 4.1


  1. 1.

    Since Σ is a positive definite matrix, not all elements can be equal to zero. Therefore, the elements tend toward zero while maintaining the positive definite property of the matrix. One way for this to occur is by letting a asymptotically approach 0 and \(\Sigma=a\Sigma_0\) for some initial positive definite matrix Σ0.

  2. 2.

    In the case of a 2D model, consider a perfect circle. All points have the same distance and curvature pairs (assuming the reference is the center of the circle).

  3. 3.

    Here, uniqueness implies a one-to-one mapping from a class of equivalent sketch-based passwords to models.

  4. 4.

    Equality, here, means either the ordered or unordered set of features are exactly the same if order is preserved or ignored respectively. If order of the features matters, then model accounts for this order by included order as a feature. If order does not matter, as with the original 2D SKS model, then order is remove d during model construction.

  5. 5.

    The subject of this text does not focus on template security. That is an entirely different subject

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Correspondence to Benjamin S. Riggan .

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Riggan, B., Snyder, W., Wang, C. (2014). Efficiency, Uniqueness, and Robustness. In: Fundamentals of Sketch-Based Passwords. SpringerBriefs in Computer Science. Springer, Cham.

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  • Print ISBN: 978-3-319-13628-8

  • Online ISBN: 978-3-319-13629-5

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