Abstract: Several require truly bit sequences, whereas consider a gene - PDF document

Abstract: Several require truly bit sequences, whereas consider a general model for random sequences shall prove be used considering quasi-random type described to extract n bit is efficient and a real-time also prove our method tion achieves optimal to consider number generators generators BM, Ya]. Blum [Bl] a fair source generate this seed crucial because number generator dependence or they can number generators, without weakening functional statistical statistical where [0,1] denotes the unit interval. We are given a source some probability for every for every functional a strengthening strengthening Instead of evaluating a pseudo- random number generator on a few statistical tests (as number generator that the difference between this reason, number generator for any ensures this security. ensure this respect to for any a length n theorems illustrate them to replace truly random truly random nuniber generators. time statistical function from n-length pseudo-random truly random number generator. seeds generated perfect (passes time statistical tests). basic idea number generator number generator formally: Let be any polynomial poly perfect, and seeds is perfect. a desired distribution induces a the area theorem can effective bounds: compute n the error (the area truly random than the This value random variates, not required also very producing some not been analyzed, and may any function density induced n length strings sity induced n length is a a biased coin, where is determined basic principle an adversary. bias can flip, with that the small function is quasi-random. for every is quasi For any n-length . . conditional probability coin flips is that the bias is between any functional source described sources has coin flips. effectively exploited in in to construct a perfect pseudo-random number generator, from any one that the parity function . . by feeding n-length sequence. defined above this source "high quality" quality" 2-( 1-26)m, I/ 2+( 1-26)~]: i.e. IPr(Yi=olY,, ' , . ,:Yi-l=u) - Pr(yi=l Iyl, . . . , yi-l=u) I (1-2deIt~)~. We introduce the following notation, for l: = Pr(X,i+...+"~=olyl~ . . yi-l'u) , rk .i sk,i = fi(%,+,+...+xk$=l Iy1 ' . . prove by semi-random source. ' ' ' input sequences that the size block one quasi-random general a m bits faster than achieve quasi-randomness. m distinct be any boolean . . m-1 adver- these stra- this case chooses bias bias - Pr[ 11 � (1-6)A - 6A = (1-26)A 2 ( 1-26)m * Case 2: p,(O) ()In this case case 11 - Pr[O] 2 (1-6)A - 6A = (1-26)A 2 (1-26)m. Q.E.D. It is somewhat surprising that the bound of Theorem 5 is exactly the same as the in Theorem source outputs that that single slightly- bit-compression is allowed. Let any boolean source output into that the bit is any bit source output. complete binary height m the tree. tree: the b between corresponding 0-branch in the biases on subtree be The following strategy guarantees node, label a is or a k bit . . reaching a consecutively from left strategy, such adversary source. that the strategy ensures the theorem theorem f(x) = 1j1~ Zrn-l, so the value of the whole tree is atleast 1-6. The idea of the inductive step is to show that if A and B are sons of C in the tree, and v(a) 2 v(b), then V(C) zz v(a)(l-6) + v(b)6. By the inductive assumption, the adversary can force probability atleast v(a) of reaching a he can reaching a also for to detect in helping bound proofs. some very useful discussions. discussions. M. Blum, "Coin Flipping by Telephone," IEEE COMPCON (1982). [B12] M. Blum, "Independent Unbiased Coin Flips From a Correlated Biased Source: a Finite State Markov Chain," to appear. [BBS] L. Blum, M. Blum and M. Shub, "A Sim- ple Secure Pseudo-Random Number Strong Sequences Encryption and Play Mental Dekker, Inc. Neumann, 'Various Digits," Notes B. Schmeiser, "Random Variate Genera- Genera- A. Yao, "Theory general approach