Universal Fuzzy Statistical Test for Pseudo Random Number Generators (UFST-PRNG)

Transcription

1 Universal Fuzzy Statistical Test for Pseudo Random Number Generators (UFST-PRNG) Raad A. Muhajjar, Ph.D. ICCR Scholar, Dept. of Computer Science, Dr. S. Kazim Naqvi, Sr. System Analyst, Centre for IT,

2 Outline Basic Information Operating System PRNGs (Linux & Windows) The solution UFST-PRNG Limitations & Directions for Future Work 2

3 Basic Information Pseudorandom or random bit generator is a device or algorithm which outputs a sequence of statistically independent and unbiased binary digits. It also can be used to generate (uniformly distributed) random numbers [51]. Pseudorandom number generators are important in cryptographic applications. For real security, PRNG output must to be unpredictable for adversaries. Many protocols rely on the ability to have random looking numbers.

4 Basic Information Weak random values increase the chances for adversaries to break the system. Breaking the Netscape implementation of SSL [4] and predicting Java session-ids [5] are two well documented instances. Although the theory of random number generation is well researched, building a secure RNG in practice has proven to be quite difficult. As all operating systems provide RNGs, most cryptographic applications take their output quality to be good for granted. Unfortunately, this is not true. The basic weakness with seeds and keys can be effectively exploited by hackers/adversaries to break the system.

5 Basic Information To alleviate problem with PRNG sources, a new practical test A Universal Statistical Test for Pseudorandom Numbers (UFST-PRNG) for investigating the randomness of cryptographic RNGs has been proposed and implemented. This new practical test is based on statistical tests, the next-bit test theory, and fuzzy logic. The new proposed test has the property that, if the bit string passes the test, it will definitely pass the standard tests. In addition it will pass, yet another test dependent on the idea of predict next bit tests [8]. 5

6 Outline Basic Information Operating System PRNGs (Linux & Windows) The solution UFST-PRNG Limitations & Directions for Future Work Footer 6

7 Linux Operating System o Linux, Mac OS, FreeBSD and OpenBSD provide access to PRNGs through device nodes /dev/random and /dev/urandom. o The Linux pseudorandom number generator (LPRNG) is part of the kernel of all Linux distributions and the outputs of this generator is used for almost every security protocol. 7

8 Analysis of Linux PRNG First, a function was developed to capture the Linux PRNG state into a file. Then these random sequences were analyzed using standard statistical tests, NIST test suite, and a new universal test for bit strings. 8

9 Program segment for extracting seq. from LPRNG

10 Few Sequences generated from /dev/random

11 Results of The NIST Statistical Test Suite NIST Statistical Test P.Value Assessment Frequency SUCCESS Cusum-Forward SUCCESS Cusum-Reverse SUCCESS Runs SUCCESS Long Runs of Ones SUCCESS Block Frequency (m=128 ) SUCCESS Spectral DFT SUCCESS Overlapping Templates m= SUCCESS Non-Overlapping Templates (m= 9, B= ) SUCCESS Approximate Entropy m= SUCCESS Lempel Ziv Complexity SUCCESS Linear Complexity (M=368) SUCCESS Serial (m=5) SUCCESS SUCCESS Table 6.5 Result NIST statistical test suite on Sequence No.1 of Table

12 Result of Standard statistical Tests Run Test : The input to the Run test is the sequence No.1 ( Bi e i ) 2 Z= + ei i= 1 k (Gi ei ) 2 i= 1 e i k the result of Run test has been got as: Z=

13 Result of Standard statistical Tests Autocorrelation Test: sequence output from The test on the first /dev/random has been conducted. n d F = 2( A(d ) )/ n d 2 From equation above, when the value of d =10, the P-value for the Autocorrelation test has been got as: F=

14 Result of A New Universal Test for Bit Strings For Local non-random behavior : As shown in the below Table, it is possible to predict the last colored bits if the first colored bits are known. 14

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17 Local non-random behavior 17

18 For Local non-random behavior The result shows that more than L+1 bits can be predicted using L-1 bits. Thus, there is some local non-randomness in the sequence No.1 generated by /dev/random and as such the test fails. 18

19 For overall non-random behavior The curve resulting from the number of predictable bits after them for sequence No.1 is compared to another curve resulting from De_bruijn sequence [4]. Both have been shown in the following Figure. 19

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21 For overall non-random behavior It can be easily seen that the graph of Sequence No.1 is quite dissimilar to the required graph for Debruijn sequence. This means that the random sequence No.1 extracted from the /dev/random generator shows overall non-random behavior. 21

22 Summary for LPRNG The architecture of LPRNG /dev/random in Red Hat Enterprise Linux EL has been presented. To study the quality of randomness in the LPRNG, a number of sequences were generated 22

23 Summary for LPRNG The test using /dev/random function then NIST, standard statistical tests and A New Universal test for bit string were applied. It was found that 100% of the sequences generated from /dev/random generator passed the NIST test suite and standard statistical tests but 30% of these sequence failed to pass tests based on the next-bit theory. 23

24 Windows Operating System All Windows versions since Windows 95 have a Crypto API function CryptGenRandom for generating cryptographically secure random numbers. The PRNG of Windows 2000 & Windows XP were examined. The CryptGenRandom function was used to generate the random sequences from the Windows Pseudo Random Number Generator (WPRNG). 24

25 Analysis of Windows PRNG First, a function was developed to capture the Windows PRNG state into a file. Then these random sequences were analyzed using standard statistical tests, NIST test suite, and a new universal test for bit strings. 25

26 Program segment for extracting seq. from WPRNG

27 Few Seq. generated from WPRNG 27

28 Results of The NIST Stat. Test Suite 28

29 Result of Standard stat. Tests The Result of Run Test is : Z= The Result of Autocorrelation Test is : F=

30 A new universal test for bit strings 30

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32 Local non-random behavior 32

33 Overall non-random behavior 33

34 Summary for WPRNG Study the quality of randomness in the WPRNG, a number of sequences have been generated using Crypto API function CryptGenRandom then applied NIST suite, a new universal test for bit strings and other standard statistical tests 34

35 Summary for WPRNG This work found that the random sequences generated by WPRNG are weaker than those generated by /dev/random. 55% of the random sequences generated using WPRNG failed to pass the NIST test suite and standards statistical tests. 72% of the sequences fail in tests based on the next-bit theory such as A new universal test for bit strings. 35

36 Thus.. In this paper, an extensive statistical analytic investigation for PRNGs of Linux and Windows OS was made. This enabled us to conclude that the random bit strings generated by LPRNG and WPRNG are weak. If a hacker is able to predict few-bits he can potentially recover the key, which can jeopardize the entire security.

37 Outline Basic Information Operating System PRNGs (Linux & Windows) The solution UFST-PRNG Limitations & Directions for Future Work Footer 37

38 The Solution The above study and results on two most popular operating systems viz. Windows and Linux shows that the security in cryptographic applications can be compromised because of weaknesses in the PRNG algorithms of the OS loaded on the interacting machines. 38

39 The Solution There are two possible ways to deal with this issue; first; - change the PRNG algorithms in the OS systems; second; - develop a way which can bridge the requirements of producing quality random numbers on the existing machines and OS. 39

40 The Solution-UFST-PRNG Framework In this paper, I have proposed to implement the later solution for feasibility reasons. The proposed solution (UFST-PRNG Framework) requires that as soon as the OS generates a random sequence, it is captured and put it on rigorous standard statistical tests and tests based on next-bit theory for checking the quality of randomness. 40

41 The Solution-UFST-PRNG Framework Once the UFST-PRNG test determines that a sequence is weak, it is not forwarded to any application rather, the system will be requested to generate a new random sequence for consumption of the requesting application. 41

42 Requirement for implemental UFST-PRNG A test that can efficiently identity if a pseudorandom sequences is good or not for security purposes?

43 Outline Basic Information Operating System PRNGs (Linux & Windows) The solution UFST-PRNG Limitations & Directions for Future Work Footer 43

44 A Universal Fuzzy Statistical Test for Pseudorandom Number Generators (UFST-PRNG) To alleviate problem with PRNG sources, in this paper an attempt to design and implement a new practical test for bit string generators was made based on statistical tests, the next-bit test theory, and fuzzy logic. 44

45 The General Architecture of UFST-PRNG Test

46 The General Architecture of UFST-PRNG Test Fuzzification: The Fuzzification job is to transform the crisp valued inputs Z, F, and R into fuzzy sets depending on the trapezoidal membership functions 46

47 Three trapezoidal membershi p functions have been used as inputs

48 The General Architecture of UFST-PRNG Test

49 The General Architecture of UFST-PRNG Test Inference System : There are two steps to complete the work of Inference System in UFSTPRNG. They are as the following: Set of IF-THEN Apply the Minimum methods for Fire Rules 49

50 Inference System Set of IF-THEN: The behavior of system to be controlled is described in this component through a set of IF_THEN rules describing inputoutput relations in terms of fuzzy sets. Here Mamdani fuzzy rules was used, each rule is represented as: IF A IS Strong AND B IS Accepted AND C IS Weak THEN D IS Accepted 3 May

51 The General Architecture of UFST-PRNG Test

52 Inference System The inference rules in proposed test are 27 rules as shown in the following Table. 52

53 The General Architecture of UFST-PRNG Test Apply the Minimum methods for Fire Rules: The premises of all rules are compared to the crisp inputs to determine which rules apply to the current situation. This matching process involves determining the certainty that a rule applies (μ premise) which could be calculated by the Minimum methods as: μpremises= Min (μ (Z), μ (F), μ (R)) 3 May

54 The General Architecture of UFST-PRNG Test

55 The General Architecture of UFST-PRNG Test Defuzzification: The defuzzifire takes the output fuzzy sets from the inference system and transforms them into crisp values depending on the triangular membership functions and the centre of gravity method. It is given by: n μ C i i i = 1 Crisp Output = n μ i i=1 55

56 The General Architecture of UFST-PRNG Test Triangular membership functions as output of this system 3 May

57 The General Architecture of UFST-PRNG Test

58 UFST-PRNGs Summary To overcome the inherent problems of PRNGs, which hamper the Cryptographic security, the UFST-PRNG Framework and Test was proposed. The UFST-PRNG test is based on standard statistical tests, the next-bit test theory, and fuzzy logic. 58

59 UFST-PRNGs Summary 1. In this paper, I have demonstrated the use of UFSTPRNG on random sequences generated using Linux and Windows PRNGs respectively. 2. The test has a very useful property that if a random string passes UFST-PRNG, the string will also pass other standard statistical test and tests based on nextbit theory but the inverse may not be always true. 59

60 UFST-PRNGs Summary 1. In addition to test standard statistical properties the UFST-PRNG test also determines both local and overall non-random behaviors of the bit strings. Thus, UFST-PRNG may be used as a single unified test to check the quality of randomness of bit strings. 2. It can be applied to all the strings, from rather small to those containing large number of bits.

61 UFST-PRNGs Summary 1. The UFST-PRNG framework is a model, which uses UFST-PRNG to test the goodness of random bits generated by the PRNGs. 2. It can be equally effective in improving the security of any application which depends on random sequences from operating system. 61

62 UFST-PRNGs Summary 1. As the UFST-PRNG framework is able to solve the root problems related to weak random numbers it will definitely help in boosting the security in cryptographic applications. 62

63 Outline Basic Information Operating System PRNGs (Linux & Windows) The solution UFST-PRNG Limitations & Directions for Future Work Footer 63

64 Limitations & Directions for Future Work The UFST-PRNG framework needs to be made operational as software. All Client machines must support UFST-PRNG framework. In addition, the framework will add some processing overhead on the server as well as on client machines. The impact of UFST-PRNG on performance may further be studied. 64

65 Important References used in the Presentation Bruce Potter, Wireless Hotspots: Petri Dish of Wireless Security, Communications of the ACM June 2006, Vol. 49, No. 6, pp I. Goldberg and D. Wagner, Randomness and the Netscape browser, Dr Dobb s, Jan. 1996, pp P. Gutmann. Software generation of practically strong random numbers. In Proc. of 7th USENIX Security Symposium, An updated version appears in B. Sadeghiyah and J. Mohajeri, A new universal test for bit strings, Lecturer Notes in Computer Science, Springer ACISP "96, pp Raad A. Muhajjar, Rafat Parveen, Nupur Prakash, and S. Kazim Naqvi., A Universal Fuzzy Statistical Test for Bit Strings, Communicated in International Journal of Computation Intelligence and Applications (IJCIA). Ian S. Shaw, Fuzzy Control of Industrial Systems: Theory and Applications, Kluwer Academic Publishers; ISBN: , 1998.

66 THANKS 66

67 Questions? 67