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CSC 482/582: Computer Security

Learn about the applications of randomness in computer security, including key generation, cryptographic use of randomness, measuring randomness, information content, and testing for randomness.

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CSC 482/582: Computer Security

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  1. CSC 482/582: Computer Security Applying Cryptography CSC 482/582: Computer Security

  2. Topics • Applications of Randomness • Defining and Evaluating Randomness • Pseudo-Random Number Generators (PRNGs) • Cryptographically Secure PRNGs (CSPRNGs) • Attacks on PRNGs • Entropy Gathering • Storing Secrets CSC 482/582: Computer Security

  3. Key Generation Goal: generate difficult to guess keys Given set of K potential keys, choose one randomly. • Equivalent to selecting a random number between 0 and K–1 inclusive. Difficulty: generating random numbers • Software generated numbers are pseudo-random, that is, generated by an algorithm. • If you start with the same random seed, then software random number generators will produce the same sequence of numbers each time. CSC 482/582: Computer Security

  4. Cryptographic Use of Randomness • Generation of session keys. • Seeds for routines that generate large prime numbers for public key ciphers like RSA. • Salts for password hashing. • Initialization vectors for block cipher chaining modes. • Nonces for cryptographic protocols. CSC 482/582: Computer Security

  5. How can we measure randomness? For a fair coin flip, our uncertainty is 2 results. It could be either heads or tails. • The uncertainty of 2 flips would be 2*2 = 4. Logarithmic measure of uncertainty. • We feel uncertainties should add, not multiply. • Measures uncertainties in bits, instead of raw #s. • Uncertainty is log2(M), where M is # results. U = log2 M CSC 482/582: Computer Security

  6. Information The amount of information in a message is the minimal number of bits needed to encode all possible meanings. Example: day of the week (7 possibilities) • Encode in 3 bits • 000 Sunday to 110 Saturday, with 111 unused • ASCII strings “Sunday” through “Saturday” use more bits, but don’t encode more information. CSC 482/582: Computer Security

  7. Information The amount of information in a message is the minimal number of bits needed to encode all possible meanings. Example: day of the week • Encode in <3 bits • 000 Sunday to 110 Saturday, with 111 unused • ASCII strings “Sunday” through “Saturday” use more bits, but don’t encode more information. In general, if N possible items, log2 N bits needed. I = log2 N which is the same formula as for uncertainty, so U = I. CSC 482/582: Computer Security

  8. Information and Probability If the probability of each of the N messages being sent is equal, then p = 1/N. I = log2 1/p What does 1/p really mean? Let's see: Message: "Coin comes up heads or tails"  Probability p=1  Information I = log21/1 = log21 = 0 Message: "Coin comes up heads"  Probability p=0.5  Information I = log21/0.5 = log22 = 1 CSC 482/582: Computer Security

  9. What if probabilities aren’t equal? Given a message of N symbols, such that each symbol occurs Ni times i=1..MNi = N Our average uncertainty for the string is i=1..MNi (-log2(Pi))/i=1..MNi which can be rewritten as -i=1..MNi/N log2(Pi) CSC 482/582: Computer Security

  10. Information I = -i=1..MPi log2(Pi) Reduces to original formula if all symbols equiprobable, i.e., Pi = 1/M: I = -i=1..M1/M log2(1/M) = 1/M log2(M) i=1..M1 = log2(M) CSC 482/582: Computer Security

  11. Information Content of English For random English letters, log2(26) = 4.7 bits/letter For large samples of English text, 1.3 bits/letter For bzipped English text, 7.95+ bits/letter CSC 482/582: Computer Security

  12. Testing for Randomness A byte stream is random if • Iis approximately 8 bits/byte This means that • Compression is a good randomizing function. • Encryption is a good randomizing function too. Statistical tests for randomness provide precise checks. CSC 482/582: Computer Security

  13. Statistical Tests of Randomness Frequency (Monobits) Test: proportion of 1s in a bit sequence is approximately ½. Runs Test: A run is an uninterrupted sequence of identical bits. This test checks runs of various lengths to see if they appear in approximately the proportion that would be expected for a random sequence. Serial Test: Determine whether number of occurrences of the 2m m-bit overlapping patterns is approximately the same as would be expected for a random sequence. NIST SP 800-22 describes a statistical test suite for PRNGs. CSC 482/582: Computer Security

  14. PRNGs • Seeding the PRNG • Linear Congruential • CSPNRGs • Blum-Blum-Shub • Tiny • Attacks on PNRGs CSC 482/582: Computer Security

  15. Seeds Input used to generate initial pseudo-random (PR) numbers. Seeds should be computationally infeasible to predict • Generate seed from random, not PR, data. • Size: 32 bits too small; only 232 combinations. Sequence is periodic, but starts from different point for each different seed. • Identical sequences produced for identical seeds. • Period needs to be large for security. CSC 482/582: Computer Security

  16. Linear Congruential Generator nk = (ank–1 + b) mod m m Modulus (a large prime integer), maximum period a Multiplier (integer from 2..m-1) bIncrement n0 Sequence initializer (seed) CSC 482/582: Computer Security

  17. LCG Example in Python #!/usr/bin/env python import sys def lcg(x): return a*x % 13 i = 0; li=[] a, x = map(int, sys.argv[1:3]) while(i < 10): x = lcg(x) li.append(str(x)) i += 1 print ", ".join(li) Modulus Multiplier Seed >./prng.py 5 2 11, 4, 8, 2, 11, 4, 8, 2, 11, 4 >./prng.py 6 2 0, 1, 7, 4, 12, 8, 10, 9, 3, 6 CSC 482/582: Computer Security

  18. LCG Period The period of an LCG is at most m, the modulus. • Modulus only allows numbers 0 .. m-1 to be produced. • An LCG with a period of m is aid to have a full period. An LCG will have a full period for all seeds if and only if • b and m are relatively prime, • a-1 is divisible by all prime factors of m, • a-1 is a multiple of 4 if m is a multiple of 4 For production LCGs, m=232-1 common • a = 16807 is well studied full period multiplier LCGs are predictable, and thus not secure for crypto • Knowing just one LCG output allows prediction of next. CSC 482/582: Computer Security

  19. Secure PRNGs Cryptographically Secure PRNGs (CSPRNGs) must: • Statistically appear random. • Difficult to predict next member of sequence from previous members. • Difficult to extract internal state of PRNG from observing output. May be re-seeded at runtime, unlike PRNGs. CSC 482/582: Computer Security

  20. Classes of CSPRNGs • Designs based on cryptographic primitives • Based on block cipher in counter mode or • Use a secure hash of a counter. • Number theoretic designs • Based on hard mathematical problems. • Example: Blum BlumShub • Special purpose designs • May introduce extra entropy when available. • Example: Yarrow (FreeBSD, Mac OS X) CSC 482/582: Computer Security

  21. Block cipher-based CSPRNG Operate block cipher in counter mode. • Choose a random key. • Nonce is a random initialization vector. • Plaintext is a predictable sequence, produced by incrementing by 1 or by any aperiodic function. CSC 482/582: Computer Security

  22. Blum Blum Shub xn+1 = xn2 mod M Blum Number M • Product of two large primes, p and q • p mod 4 = 3, q mod 4 = 3 Seed • Choose random integer x, relatively prime to M. • x0 = x2 mod M CSC 482/582: Computer Security

  23. Blum Blum Shub Random Output: • Least significant bit of xn+1 • Can safely use log2M bits. Provably secure • Distinguishing output bits from random bits is as difficult as factoring M for large M. Slow • Requires arbitrary precision software math libraries. CSC 482/582: Computer Security

  24. Yarrow Yarrow is named after plant whose leaves are used in I Ching divination. • Used for /dev/random in FreeBSD and Mac OS X. Steps • Accumulates entropy from system sources. • Pools are SHA-1 hash contexts, 160 bits maximum. • Reseeds generator with key made from pool entropy to limit state compromise attacks. • Generates numbers using Triple-DES in counter mode. CSC 482/582: Computer Security

  25. Attacks on PNRGs Direct Cryptanalytic • Distinguish between PRNG output and random output with better than 50% accuracy. Input-Based • Use knowledge of PRNG input to predict output, or • Insert input into PRNG to control output. State Compromise Extension • Extend previously successful attack that has recovered internal state to recover either or both: • past unknown PRNG outputs • future PRNG outputs after additional inputs given to PRNG CSC 482/582: Computer Security

  26. ASF On-line Gambling Re-seeded PRNG before each shuffle • always start with ordered deck. Shuffling • Fair: 52! @ 2226 combinations • 32-bit seed: 232 combinations • ms seed: 86,400,000 combinations • synchronize time: 200,000 combinations Predict deck based on 5 known cards. CSC 482/582: Computer Security

  27. Entropy Collection • Hardware Solutions • Software Solutions • Poor Entropy Collection • Entropy Estimation CSC 482/582: Computer Security

  28. Hardware Sources Radioactive Decay • Hotbits: 256 bits/s • http://www.fourmilab.ch/hotbits/ Thermal or Electrical Noise • ComscireQNG Model J1000KU, 1 Mbit/s • Digital RNG (DRNG) on Ivy Bridge and later Intel CPUs LavaRnd • SGI used LavaLite; LavaRnd uses lenscappeddigicam • http://www.lavarnd.org/ • up to 200 kbits/s CSC 482/582: Computer Security

  29. Software Sources Less Secure, More Convenient • Software systems can be sufficiently complex to be almost impossible to predict. • Example: time between user keystrokes or mouse events. User Input: Push, don’t Pull • Record time stamp when keystroke or mouse event occurs. • Don’t poll most recent user input every .1s • Far fewer possible timestamps. UNIX systems provide via /dev/random • User inputs, network inputs, disk seeks, etc. with an algorithm like Yarrow to aggregate entropy and reseed. CSC 482/582: Computer Security

  30. Linux Sources: /dev/random /dev/random • each bit is truly random. • blocks unless enough random bits are available. /dev/urandom • supplies requested number of bits immediately. • reuses current state of pool—lower quality randomness. CSC 482/582: Computer Security

  31. Poor Entropy: Netscape 1.1 SSL encryption • generates random 40- or 128-bit session key • Netscape 1.1 seeded PRNG with • time of day • PID and PPID • All visible to attacker on same machine. Remote attack broke keys in 30 seconds • guessed limited randomness in PID/PPID. • packet sniffing can determine time of day. CSC 482/582: Computer Security

  32. Random Number APIs Windows • rand() – insecure PRNG, uses LCG • CryptGenRandom() – CSRNG • CryptGenKey() – to securely generate keys Java • java.util.Random – insecure PRNG • java.security.SecureRandom– CSRNG • Relies on OS, so SecureRandom can fall back to insecure Random if OS does not provide /dev/random or similar CSC 482/582: Computer Security

  33. Key Storage Source Code • Can use strings command to extract from binary. File on Disk • Attacker can search disk for files with high entropy, which are likely to contain keys. • Encryption of file adds another layer of difficulty, but there must be a key someplace. Many languages provide APIs for storing keys or certificates in encrypted files. Registry • Attacker can access with regedit. External Device, e.g. smartcards, smartphones, remote server, … • Attacker can obtain PINs or use power analysis attackers to extract keys from device. • Remote servers can be compromised too. Store parts of key in different places • Break up key, then store part in source, part in file, part in db, etc. CSC 482/582: Computer Security

  34. Lifetime of 64MB of freed memory CSC 482/582: Computer Security

  35. Key Storage in Memory • Minimize time spent holding secrets. • Load only when needed. • Erase when not needed any longer. • Prevent pages with secrets from being written to disk. • mlock() and munlock() in UNIX • VirtualLock() and VirtualUnLock() in MS Windows • Erase secrets securely. • Use memset() to overwrite secret with zeros. • Prevent unnecessary duplication. • Avoid realloc() in C. If your threat model includes attacks on secrets in memory, then you cannot use a garbage-collected language like Java or Python. CSC 482/582: Computer Security

  36. Key Points • Measuring randomness • Measure information (entropy) content. • Statistical tests: frequency of 1s, bit sequences, etc. • CSPRNGs must have the following qualities: • Statistically appear random. • Difficult to predict next member of sequence from previous members. • Difficult to extract internal state of PRNG from observing output. • Algorithmic PRNG techniques: • Linear congruentialgeneratorsare insecure. • CSPRNG types: cipher-based, algorithmic, special designs. • Computer sources of randomness: • Hardware RNGs: thermal noise, radioactive decay. • Software RNGs: disk seeks, interrupts, time btw keystrokes. • Securely storing keys: • Permanent: disk, db, registry, hardware device. • In memory: minimize time holding secrets, erase securely. CSC 482/582: Computer Security

  37. References • Brian Chess and Jacob West. Secure programming with static analysis. Pearson Education, 2007. • D. Eastlake, “Randomness Recommendations for Security,” RFC 1750, http://www.ietf.org/rfc/rfc1750.txt, 1994. • Ian Goldberg and David Wagner,“Randomness and the Netscape Browser,” Doctor Dobbs’ Journal, 1996. http://www.cs.berkeley.edu/~daw/papers/ddj-netscape.html • John Kelsey, Bruce Schneier, and Niels Ferguson. "Yarrow-160: Notes on the design and analysis of the yarrow cryptographic pseudorandom number generator." Selected Areas in Cryptography. Springer Berlin Heidelberg, 2000. • Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, Handbook of Applied Cryptography, http://www.cacr.math.uwaterloo.ca/hac/, CRC Press, 1996. • S. K. Park, K. W. Miller, “Random number generators: good ones are hard to find,”  Communications of the ACM,  Volume 31 Issue 10 , October 1988. • John R. Pierce, An Introduction to Information Theory, Dover Press, 1980. • Tom Schneider, “Information Theory Primer,” http://www.lecb.ncifcrf.gov/~toms/paper/primer/, 2000. • Bruce Schneier, Applied Cryptography, 2nd edition, Wiley, 1996. CSC 482/582: Computer Security

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