Remark. The formal proof could be assigned on an examor as homework.Question. Why study random distributions, especiallyif they do not exist in finite, discrete systems?Remark. Interestingly enough, the computation of uniformly random distributions remains one of the key goals of number theory and cryptology. The use of randomness to obscure properties of a message that are useful to a cryptanalyst (i.e., an adversary) is a key problem in existing cryptographictheory and practice. Therefore, we emphasize discussionof various techniques for computing "uniformly random"distributions from nonrandom processes or data. Later in this class we will consider the problem of semanticciphers or steganography, in which properties of plaintext are disguised in the ciphertext so as to appear innocuous. In the absence of techniques for rigorously producing random processes,such camouflage methods may be a more operationally feasible approach to disguising plaintext than attempting to simulate randomness. Observation. One needs random bits (or values) for several cryptographic purposes, of which the two most common are (a) generation of cryptographic keys (or passwords) and (b) concealmentof values in certain protocols. Several definitions of randomness are employed in cryptology. However, there is the following basic, implementational criterion for a random source. Assumption. Let an adversary have(i) full knowledge of an encryption site's software/ hardware, (ii) the money to build and run a matching cryptosystem for exhaustive attack, and (iii) the ability to compromise the site'sphysical facilities (i.e., wiretapping, planting bugs, etc.) This capable adversary must not be able to predict the next bitthe site's "random" generator produces, even if he knows all the bits produced at the site thus far. Definition. Implementationally, random sources can be classified as random or pseudo-random. Since wehave shown that no discrete system can be random, the randomnumber (RN) generation techniques we shall discuss are algorithms or physical processes that imitate true randomness. Random sources can be considered unconditionally unguessable, even byan adversary with infinite computing resources. In contrast, pseudo-random sources useful only in the presence of attacksconstrained by finite computational resources. Observation. Random bits are typically obtained via the following process:Gather a physically-generated bitstream. One first gathers bits unknown to and unguessable by the adversary. Such information is typically generated by a high-entropy (nearly random) physical device connected to one'scryptographic equipment by a secure I/O line. For example, thefollowing techniques have been suggested or employed:Radioactive Source that emits particles to a completely absorbing counter. The counter must be absorbingin order to conceal the bitstream. Several commercially available radioactive monitors have RS232 output. Hence, this is a practical method. Quantum Effects in Semiconductors (e.g., noisy diodes). Several electronic sources of nearly-random bitstreams use noisy diodes or noisy resistors, which can be cost effective. As in the case ofa radioactive source, the semiconductor must be completely shielded. Photon Polarization Detection This is a source of quantum uncertainty that is highly experimental,and will be discussed later in this class. Isolate Monoaural Microphone In some cases, an analog-to-digitalconverter whose input is an unplugged microphone yields noise thatexhibits moderate entropy levels.Note that the microphone input must be fullyshielded to avoid corruption by correlated (real-world) soundssuch as voice, keyboard clicks, 60Hz electrical hum, etc. Air Turbulence within a Dedicated Sealed Disk Drivehas been reported recently [Dav94]. This mechanism shows promise, given a dedicated secure disk drive and special system software to measure disk performance parameters without affectingdisk behavior. If a non-dedicated drive and customary softwareare employed, then one can measure I/O completion times for a disk in normal use, which are known to be Poisson-distributed. However, this method does not hold if the disk drive is notproperly balanced or is defective, since correlated noise canresult, for example, from spindle "tramping" due to mechanical imbalance. Additional methods for producing random bits that yield pseudorandom output include:Differenced Stereo Microphone Output, where a noisy room with moving sound sources can yield stereo disparity betweentwo microphones that fluctuates pseudorandomly. If the microphoneamplification is normalized to minimize the difference signal, it is difficult for an adversary to reconstruct the resultantbit stream, whether or not the room is bugged with a monoauralmicrophone. If the room is bugged with a stereomicrophone, or is a realistic environment having correlatedsignals, then this method is highly vulnerable to correlationsbetween noise detected by the adversary and noise used to generatethe bitstream. Monaural Microphone in a secure (unbugged) room can detect certain random signals. However, the randomnesswill be submerged in a large amount of correlated noise. Additionally, one cannot guarantee if the room is bugged. Timing between Keystrokes, where a user types nonsenseand the key value and activation time are collected. Keying times are quantized by the system clock, thus requiringpredictive modelling of expected inter-keying delay and quantization error values for each implementation. Due to temporal quantization, such systems are insecure within several orders of magnitude greater than the temporal quantization interval. For fasttypists, this constraint may be prohibitive. Additionally, unshielded keyboards or audio listening devices placed in the room that contains the keyboard can facilitate compromise of this method. Mouse Stroke Timing, derived from a user signinghis name with a mouse or joystick. Although this islikely the most efficient human-driven sources of entropyit can be easily compromised by poorly shielded equipmentor bugs placed within the mouse pad. /dev/random is a UNIX device available on certain secure systems that gathers bits from system tables and events unavailable to users. If the adversary is a user running a process on your machine, the source remains secure. If an adversary has certain system privileges, then the method can be easily compromised. The following methods are often used as bit sources, but haveserious flaws since they are observable, predictable, or subject toinfluence by a determined adversary, especially on multiprocessmachines: Network Statistics can easily be influencedby a capable adversary with communication-intensive software. Process Statistics are less easily influenced, but canbe easily compromised by determined adversaries executingconcurrent heterogeneous processes. I/O Completion Timing and Statistics are readilyinfluenced by I/O-intensive processes run by an adversary.The following are nearly worthless bit sources that are frequentlyrecommended or used for purposes of convenience: TV or Radio Broadcasts are of little value, sincethe majority of information not contained in local noise is available to an adversary. Published Information on a CD, Tape, or Hardcopy isvulnerable due to wide availability. An exception is a one-timeencoding on a CD of a pseudorandom bitstream in the least significantbit of the soundtrack. This technique will be further discussed inthe section on steganography. System Date and Time have low information content andare available to an adversary. Process Runtime has little information content excepton extremely busy systems, where adversarial influence of theprocess load can corrupt the security of this measure withoutnecessarily being detected by security monitors. Furthermore,the runtime calculator will, in a round-robin scheduler, make thecall to compute the runtime at an approximately uniform time afterprocess startup. This renders the runtime I/O measures predictable by an adversary, especially if he can influence theprocess mix. Multiple, Free-running Cascaded Oscillators yield asequence that appears to be random due to its multi-periodbeat frequencies, but can easily be discerned as periodicthrough application of the Fourier or Cosine transform. Equally or entirely worthless are the following bit sources: Chaotic Systems appear visually complex but are highly structured (i.e., are pseudorandom). Current researchemphasizes determination of this structure.System Library Random Number Generators (RNGs) were never designed to be cryptographically strong and hence havelittle utility in cryptology. Linear-congruential RNGs - the simplest RNG algorithm,which we will analyze later in this class. Chain Addition - another simple and easily broken statistical RNG. E-mail is useful only if the e-mail is so well encryptedthat it cannot be discovered by the adversary. In such cases,however, one would already have an effective RNG in place. Generally,E-mail is as vulnerable as USENET data.Unfortunately, the bits gathered from the foregoing methods arenot necessarily independent. That is, one might be able topredict a bit value with probability greater than 1/2, given all other bits. The adversary might even know entire subsequences of a bitstream, which he could obtain by eavesdropping.The key constraintis that the gathered bits contain information (entropy)that is everywhere unavailable to the adversary. Determine entropy by estimating how many unguessable bits were gathered. Here, one needs to know how many of the bits are independent and unguessable. This number of bitsis usually referred to as source entropy. We will discussthis concept in greater depth when we consider information theory in Appendix C. Reduce to independent bits. As a third step, one can hash the harvested bits to reduce them to independent random bits. The hash function for this stage of operation needs to have each output bit functionally dependent on all input bits butindependent of all other output bits. Without presenting formal analysis at this time, we assume that such hash functions are cryptographically strong. Strong hash functions such as MD5 and SHA will be discussed later in this class. Provided that the hash function meets the required criteria and cannot be guessed by the adversary, the output of Step 3 is a set of independent, "unguessable" bits. These can be used with confidence wherever random bits are required.Remark. When implementing the foregoing method of random number generation, one preserves security by:Testing bit sources for degeneration - if a bitsource becomes less random but feeds a cryptographically strongfunction f, the output of f would not immediately be problematic to the normal user but could facilitate entry to thecryptanalyst. One needs to frequently test the bit source [Mau91] before it is hashed. Mix different sources if one is unsure about tappingor bugging. If multiple sources are used, but tapping of anysource is a probabilistically independent event, then one can reduce the probability that a tap is successful by using multiple independent sources. Each source would have its own bit gatheringprocedure, and the individual bitstreams would be hashed together.The probability of adversary success is then the product of theindividual probabilities of tapping. Clearly, if one source haszero tapping probability (i.e., is secure), then the hashed bits are secure. Avoid discarding known bits but feed all bits to theinitial hash function. Given a cryptographically strong hashfunction, there is no other effective method forfiltering out dependent bits. Definition. Various types of non-uniform distributions will be useful to us in forthcoming development. For purposesof consistency, we assume that µ denotes the mean, denotes standarddeviation, and X denotes a random variable. Note that Pr(X=x) could be computed for a uniformdistribution using the preceding method.Normal Distribution: A discrete random variable X that is normally distributed has a probability function given by e-(x - µ)2/22
Pr(X=x) = .
·(2)1/2
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