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A proposal for: Functionality classes for random number generators
18 September 2011 AIS 20 / AIS 31 page 1
Wolfgang Killmann
T-Systems GEI GmbH, Bonn
Werner Schindler
Bundesamt für Sicherheit in der Informationstechnik (BSI), Bonn
A proposal for:
Functionality classes for random number generators
1
Version 2.0
18 September 2011
1
The authors wish to express their thanks for the numerous comments, suggestions and notes that have been
incorporated into this document.
A proposal for: Functionality classes for random number generators
18 September 2011 AIS 20 / AIS 31 page 2
Table of contents
1. Introduction ............................................................................................................................... 7
1.1. Motivation ................................................................................................................................... 7
1.2. Abbreviations .............................................................................................................................. 8
1.3. Common Criteria (Abbreviations)............................................................................................... 8
1.4. Terminology ................................................................................................................................ 9
1.5. Symbols ..................................................................................................................................... 16
2. Basic Concepts ......................................................................................................................... 18
2.1. Randomness .............................................................................................................................. 18
2.1.1. Concept of Randomness and Random Experiments .............................................. 18
2.1.2. Random number generators (RNGs) ..................................................................... 19
2.2. Random Numbers in IT Security............................................................................................... 21
2.2.1. Usage of Random Numbers in IT Security ............................................................ 21
2.2.2. Basic considerations for RNG types ...................................................................... 23
2.2.3. Design Description of RNG ................................................................................... 24
2.3. Mathematical Background ........................................................................................................ 28
2.3.1. Random variables .................................................................................................. 28
2.3.2. Entropy and Guess Work ....................................................................................... 31
2.3.3. Random mappings ................................................................................................. 34
2.4. Stochastics and Statistical Analysis of Physical RNGs ............................................................. 36
2.4.1. Stochastic model .................................................................................................... 36
2.4.2. Overview of Statistical Tests ................................................................................. 41
2.4.3. Standard Statistical Tests ....................................................................................... 44
2.4.4. Test procedures ...................................................................................................... 54
2.4.5. Additional Statistical Tests .................................................................................... 57
3. Security Functional Requirements - Family FCS_RNG ...................................................... 61
3.1. Definition of FCS_RNG ............................................................................................................ 61
3.2. Security capabilities of RNG types ........................................................................................... 62
3.3. Rationale for definition of the extended component ................................................................. 66
4. Pre-defined RNG Classes ........................................................................................................ 67
4.1. Overview of pre-defined RNG classes ...................................................................................... 67
4.2. General Remarks (Exemplary applications, side-channel attacks, fault attacks) ...................... 71
4.3. Class PTG.1 ............................................................................................................................... 71
4.3.1. Security functional requirements for the RNG class PTG.1 .................................. 71
A proposal for: Functionality classes for random number generators
18 September 2011 AIS 20 / AIS 31 page 3
4.3.2. Application notes ................................................................................................... 72
4.4. Class PTG.2 ............................................................................................................................... 74
4.4.1. Security functional requirements for the RNG class PTG.2 .................................. 74
4.4.2. Application notes ................................................................................................... 75
4.4.3. Further aspects ....................................................................................................... 77
4.5. Class PTG.3 ............................................................................................................................... 79
4.5.1. Security functional requirements for the RNG class PTG.3 .................................. 79
4.5.2. Application notes ................................................................................................... 80
4.5.3. Further aspects ....................................................................................................... 82
4.6. Class DRG.1 .............................................................................................................................. 84
4.6.1. Security functional requirements for the RNG class DRG.1 ................................. 84
4.6.2. Application notes ................................................................................................... 84
4.6.3. Further aspects ....................................................................................................... 87
4.7. Class DRG.2 .............................................................................................................................. 88
4.7.1. Security functional requirements for the RNG class DRG.2 ................................. 88
4.7.2. Application notes ................................................................................................... 89
4.7.3. Further aspects ....................................................................................................... 89
4.8. Class DRG.3 .............................................................................................................................. 90
4.8.1. Security functional requirements for the RNG class DRG.3 ................................. 90
4.8.2. Application notes ................................................................................................... 91
4.8.3. Further aspects ....................................................................................................... 91
4.9. Class DRG.4 .............................................................................................................................. 91
4.9.1. Security functional requirements for the RNG class DRG.4 ................................. 91
4.9.2. Application notes ................................................................................................... 92
4.9.3. Further aspects ....................................................................................................... 93
4.10. Class NTG.1 .............................................................................................................................. 93
4.10.1. Security functional requirements for the NPTRNG class NTG.1 .......................... 93
4.10.2. Application notes ................................................................................................... 94
5. Examples .................................................................................................................................. 96
5.1. Guesswork for binomial distributed data .................................................................................. 96
5.2. Contingency tables .................................................................................................................... 99
5.3. Forward and backward secrecy ............................................................................................... 103
5.4. Examples of post-processing algorithms ................................................................................. 107
5.4.1. Von Neumann unbiasing ..................................................................................... 107
5.4.2. Xoring of non-overlapping segments of independent bits ................................... 108
A proposal for: Functionality classes for random number generators
18 September 2011 AIS 20 / AIS 31 page 4
5.4.3. Two sources ......................................................................................................... 108
5.4.4. Uniformly distributed input data for random mappings ...................................... 109
5.5. Examples of online test, tot test, and start-up test ................................................................... 111
5.5.1. An online test of the internal random numbers .................................................... 111
5.5.2. A straightforward online test ............................................................................... 112
5.5.3. A more sophisticated online test procedure ......................................................... 113
5.6. Examples of RNG designs ...................................................................................................... 116
5.6.1. PTRNG with two noisy diodes ............................................................................ 116
5.6.2. Examples of DRNGs ........................................................................................... 120
5.6.3. NPTRNG ............................................................................................................. 127
6. Literature ............................................................................................................................... 130
A proposal for: Functionality classes for random number generators
18 September 2011 AIS 20 / AIS 31 page 5
Tables
Table 1: Attack potential, guessing probability and security bits .................................................... 22
Table 2: Attack potential and guessing passwords ............................................................................ 22
Table 3: Statistics of random mappings ............................................................................................. 34
Table 4: Statistics of random permutations ....................................................................................... 35
Table 5: Brief overview of error types of statistical tests .................................................................. 43
Table 6: Typical values of -distribution with 1 degree of freedom ............................................ 45
Table 7: Typical values of -distribution with degree of freedom d ............................................ 46
Table 8: Typical values of -distribution for runs ......................................................................... 47
Table 9: Typical values of Normal (Gaussian) N(0,1) for a two-sided test of
autocorrelation ........................................................................................................................ 50
Table 10: Parameters for entropy test ................................................................................................ 53
Table 11: Recommended parameter settings for the NIST test suite .............................................. 57
Table 12: Attack potential, Min-entropy, and recommended length of the internal
state ........................................................................................................................................... 85
Table 13: Requirements for the parameters in (DRG.1.3) depending on claimed
attack potential ........................................................................................................................ 87
Table 14: Work factor and work factor defect for uniform mappings with
equidistributed input ............................................................................................................. 111
Table 15: Probability for a noise alarm within a test suite and the expected number
of noise alarms per year for different distributions of the das-random
numbers .................................................................................................................................. 115
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