White Paper 002
How the Adaptive Key is Numerically Guaranteed
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This paper summerizes the numerical properties of the adaptive key. Those properties are obtained through a theoretical/numerical study of a mathematical model of the adaptive key. The model and theory consists of the following*:
- a mathematical formulation of file systems
- a function, d(x,y), to calculate the difference of given 2 file systems x and y
- a proof of "semi" triangle inequality for the function
- a mathematical formulation of user's actions on file systems
- a parametric model of the probability distribution of user's actions
- the derivation of the probality distribution of d(x,y) from the model
- the evaluation of the curve d(x(t), x(0)), using the semi-triangle inequality
- an analysis of the curve, applying the central limit theorem
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Numerical results for 3 typical cases
Case I. The total number of files (in the file system) = 105
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E(d( x(t+1), x(t) ))
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0.0000167
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σ(d( x(t+1), x(t) ))
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0.0001953
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i.e. the difference in the file system created by an action of a user is
about 0.0017% on average, with the standard deviation of 0.02%.
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Allowance level (%)
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5
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10
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20
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30
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50
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70
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# of actions that increases the
probalibity to reject a right user
from 0% to 0.02%
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752.7
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2821.3
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7503.5
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12491.9
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22887.0
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33581.8
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Probalibity to accept a wrong user (%)
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0.002
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0.002
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0.427
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21.876
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43.112
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69.764
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Case II. The total number of files (in the file system) = 104
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E(d( x(t+1), x(t) ))
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0.0001206
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σ(d( x(t+1), x(t) ))
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0.0008819
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i.e. the difference in the file system created by an action of a user is
about 0.0121% on average, with the standard deviation of 0.09%.
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Allowance level (%)
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5
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10
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20
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30
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50
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70
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# of actions that increases the
probalibity to reject a right user
from 0% to 0.02%
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N/A
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92.1
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616.8
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1211.1
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2498.4
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3855.205
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Probalibity to accept a wrong user (%)
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0.000
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0.000
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0.002
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0.002
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0.002
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0.002
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Case III. The total number of files (in the file system) = 103
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E(d( x(t+1), x(t) ))
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0.0010973
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σ(d( x(t+1), x(t) ))
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0.0040453
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i.e. the difference in the file system created by an action of a user is
about 1.1% on average, with the standard deviation of 4.0%.
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Allowance level (%)
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5
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10
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20
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30
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50
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70
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# of actions that increases the
probalibity to reject a right user
from 0% to 0.02%
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N/A
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N/A
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8.1
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60.0
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180.2
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312.0
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Probalibity to accept a wrong user (%)
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0.000
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0.000
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0.000
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0.000
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0.000
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0.000
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Conclusion
This study discovered a general relation between the desirable allowance-level and the size of file systems: the bigger a file system grows, the less the desirable allowance-level becomes.
For an example, when combined with the key-update of every 100 actions, adjustment of the allowance level to 50%, 30%, and 10% for the file-sytem sizes of 103, 104, and 105 respectively ensures
- the probability to reject a right user (i.e. so-called type-I error) < 0.0002 (i.e. 0.02%)
- the probability to accept a wrong user (i.e. so-called type-II error) < 0.00002 (i.e. 0.002%)
In short, the adaptive key works with great precision, under the appropriate control over the allowance level and the frequency of key-update.
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