Topics. Key Generation. Applying Cryptography

Transcription

1 Applying Cryptography Topics 1. Key Generation 2. Randomness and Information Theory 3. PRNGs 4. Entropy Gathering 5. Key Storage 6. Cryptographic APIs 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 Computer generated numbers are pseudo-random, that is, generated by an algorithm. Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin. John vonneumann 1

2 How can be 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 log 2 (M), where M is # results. U = log 2 M 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. 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, log 2 N bits needed. I = log 2 N which is the same formula as for uncertainty, so U = I. 2

3 Information and Probability If the probability of each of the N messages being sent is equal, then p = 1/N. I = log 2 1/p What does 1/p really mean? Let's see: Message: "Coin comes up heads or tails" Probability p=1 Information I = log 2 1/1 = log 2 1 = 0 Message: "Coin comes up heads" Probability p=0.5 Information I = log 2 1/0.5 = log 2 2 = 1 What if probabilities aren t equal? Given a message of N symbols, such that each symbol occurs N i times i=1..m N i = N Our average uncertainty for the string is i=1..m N i (-log 2 (P i )) / i=1..m N i which can be rewritten as - i=1..m N i /N log 2 (P i ) Information I = - i=1..m P i log 2 (P i ) Reduces to original formula if all symbols equiprobable, i.e., P i = 1/M: I = - i=1..m 1/M log 2 (1/M) = 1/M log 2 (M) i=1..m 1 = log 2 (M) 3

4 Information Content of English For random English letters, log bits/letter For large samples of English text, 1.3 bits/letter For bzipped English text, bits/letter Testing for Randomness A byte stream is random if I is approximately 8 bits/byte Compression is a good randomizing function. Cryptography is a good randomizing function. Statistical tests for randomness Zeros occur about as often as ones. Pairs of zeros occur about half as often as single zeros and as often as pairs of ones. PRNGs 1. Determinism and Randomness 2. Seeding the PRNG 3. Linear Congruential 4. CSPNRGs 5. Blum-Blum-Shub 6. Tiny 7. Attacks on PNRGs 4

5 Determinism Computers are deterministic Can t produce true random numbers. Pseudo-random numbers appear to be random to certain statistical tests. Tests can be derived from compression. If you can compress sequence, it s not random. Software generated pseudo-random sequences are periodic and predictable. Seeds Input used to generate initial PR number. Should be computationally infeasible to predict Generate seed from random, not PR, data. Large seed: 32 bits too small; only 2 32 combinations. Sequence is still periodic, but starts from different point for each different seed. Identical sequences produced for identical seeds. Period needs to be large for security. Linear Congruential Generator n k = (an k 1 + b) mod m m Modulus (a large prime integer), maximum period a Multiplier (integer from 2..m-1) b Increment n 0 Sequence initializer (seed) 5

6 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) >./prng.py , 4, 8, 2, 11, 4, 8, 2, 11, 4 >./prng.py 6 2 0, 1, 7, 4, 12, 8, 10, 9, 3, 6 Modulus Multiplier Seed Linear Congruential Generator Choice of a critical Many choices of a do not produce a full period. Sequence is permutation of integers 1..m-1 Ex: 2, 6, 7, 11 for m=13 For production LCGs, m= common a = is well studied full period multiplier LCGs are statistically random but predictable, giving away LCG state with each number. LCGs are not cryptographically useful. Secure PRNGs Cryptographically Secure PRNGs: 1. Statistically appear random. 2. Difficult to predict next member of sequence from previous members. 3. Difficult to extract internal state of PRNG from observing output. Similar to stream ciphers. May be re-seeded at runtime, unlike PRNGs. 6

7 Blum Blum Shub x n+1 = x n 2 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. x 0 = x 2 mod M Blum Blum Shub Random Output: LSB of x n+1 Can safely use log 2 M 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. Strong Mixing Functions Strong mixing function: function of 2 or more inputs with each bit of output depending on some nonlinear function of all input bits. Examples: AES, DES, SHA-1, SHA-2 Use on UNIX-based systems: (date; ps gaux) md5 where ps gaux lists all information about all processes on system. 7

8 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. 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 ASF On-line Gambling Re-seeded PRNG before each shuffle always start with ordered deck. Shuffling Fair: combinations 32-bit seed: 2 32 combinations ms seed: 86,400,000 combinations synchronize time: 200,000 combinations Predict deck based on 5 known cards. ASF PRNG Flaws 1. PRNG algorithm used small seed (32 bits.) 2. Non-cryptographic PRNG used. 3. Seed generated by poor source of randomness. 8

9 Entropy Collection 1. Hardware Solutions 2. Software Solutions 3. Poor Entropy Collection 4. Entropy Estimation Hardware Sources Radioactive Decay Hotbits: 256 bits/s Thermal or Electrical Noise Comscire QNG Model J1000KU, 1 Mbit/s Via Padlock RNG on Via x86 CPUs kbits/s LavaRnd SGI used LavaLite; LavaRnd uses lenscapped digicam up to 200 kbits/s Software Sources Less Secure, More Convenient Software systems can be sufficiently complex to be almost impossible to predict. 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. 9

10 Software Sources: /dev/random Idea: use multiple random software sources. Store randomness in pool for user requests. Use hash functions (i.e., strong mixing functions) to distill data from multiple sources. /dev/random can use random sources such as CPU load disk seeks kernel interrupts keystrokes network packet arrival times /dev/audio sans microphone Software 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. 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. 10

11 Random Number APIs Windows Java rand() insecure PRNG, uses LCG CryptGenRandom() CSRNG CryptGenKey() to securely generate keys 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 Key Storage Source Code Can use strings 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. Registry Attacker can access with regedit. External Device, e.g. smartcards, smartphones, etc. Attacker can obtain PINs or use power analysis attackers to extract keys from device. Key Points 1. Keys generated must be truly random. 1. Test for randomness by measuring information. 2. I = - i=1..m P i log 2 (P i ) 2. Secure PRNGs must have the following qualities: 1. Statistically appear random. 2. Difficult to predict next member of sequence from previous members. 3. Difficult to extract internal state of PRNG from observing output. 3. Algorithmic PRNG techniques: 1. Linear congruential generators: non-crypto. 2. Blum Blum Shub cryptographic PRNG. 4. Computer RNGs: 1. Hardware RNGs: thermal noise, decays. 2. Software RNGs: disk seeks, interrupts. 11

12 References 1. Matt Bishop, Introduction to Computer Security, Addison-Wesley, D. Eastlake, Randomness Recommendations for Security, RFC 1750, Ian Goldberg and David Wagner, Randomness and the Netscape Browser, Doctor Dobbs Journal, Michael Howard and David LeBlanc, Writing Secure Code, 2 nd edition, Microsoft Press, Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, Handbook of Applied Cryptography, CRC Press, S. K. Park, K. W. Miller, Random number generators: good ones are hard to find, Communications of the ACM, Volume 31 Issue 10, October Tom Schneider, Information Theory Primer, Bruce Schneier, Applied Cryptography, 2 nd edition, Wiley, John Viega and Gary McGraw, Building Secure Software, Addison-Wesley, Joss Visser, Kernel based random number generation in HP-UX 11.00, David Wheeler, Secure Programming for UNIX and Linux HOWTO, HOWTO/index.html,