Non-Asymptotic and Asymptotic Analyses on Markov Chains in Several Problems

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1 Non-Asymptotic and Asymptotic Analyses on Markov Chains in Several Problems Masahito Hayashi and Shun Watanabe Graduate School of Mathematics, Nagoya University, Japan, and Centre for Quantum Technologies, National University of Singapore, Singapore. Department of Information Science and Intelligent Systems, University of Tokushima, Japan, and Institute for Systems Research, University of Maryland, College Park. Abstract In this paper, we derive non-asymptotic achievability and converse bounds on the source coding with side-information and the random number generation with side-information. Our bounds are efficiently computable in the sense that the computational compleity does not depend on the block length. We also characterize the asymptotic behaviors of the large deviation regime and the moderate deviation regime by using our bounds, which implies that our bounds are asymptotically tight in those regimes. We also show the second order rates of those problems, and derive single letter forms of the variances characterizing the second order rates. I. INTRODUCTION The non-asymptotic analyses of coding problems are attracting a considerable attention recently ], 2]. In this paper, we further develop the non-asymptotic analyses for the fied length source coding with full) side-information at the decoder and the uniform random number generation with sideinformation. Particularly, we are interested in the cases such that underlying sources are Markov chains. A. Motivation To begin with, we shall eplain motivations of this paper. Although the problems treated in this paper are not the channel coding, we consider the channel coding here to eplain motivations. So far, quite many types of non-asymptotic achievability bounds have been proposed. For eample, Verdú and Han derived a non-asymptotic bound by using the information spectrum approach in order to derive the general formula 5] see also 6]), which we call the information-spectrum bound. One of the authors and Nagaoka derived a bound for the classical-quantum channel) by relating the error probability to the binary hypothesis testing 7, Remark 5] see also 8]), which we call the hypothesis testing bound. Polyanskiy et. al. derived the RCU random coding union) bound and the DT dependence testing) bound ] 2. There is also Gallager s bound 9]. The uniform random number generation with side-information is also known as the privacy amplification 3], 4]. 2 A bound slightly looser coefficients are worse) than the DT bound can be derived from the hypothesis testing bound of 7]. Although it is not stated eplicitly in any literatures, we believe that there are two important criteria for non-asymptotic bounds: Computational compleity, and Asymptotic optimality. Let us first consider the first criterion, i.e., the computational compleity. For the BSC, the computational compleity of the RCU bound is On 2 ) and that of the DT bound is On) 0]. However, the computational compleities of these bounds is much larger for general DMCs or channels with memory. It is known that the hypothesis testing bound can be described as a linear programming eg. see ], 2] 3 ), and can be efficiently computed under certain symmetry. However, the number of variables in the linear programming grows eponentially in the block length, and it is difficult to compute in general. The computation of the information-spectrum bound depends on the evaluation of a tail probability. The information-spectrum bound is less operational than the hypothesis testing bound in the sense of the hierarchy introduced in ], and the computational compleity of the former is much smaller than that of the latter. However the computation of a tail probability is still not so easy unless the channel is a DMC. For DMCs, computational compleity of Gallager s bound is O) since the Gallager function is additive quantity for DMCs. However, this is not the case if there is a memory 4. Consequently, there is no bound that is efficiently computable for the Markov chain so far. The situation is the same for the source coding with side-ifnromation. Let us now move to achievability bounds on the random number generation with side-infromation, i.e., the privacy amplification. Renner derived a bound by using the leftover hash lemma and the smooth min-entropy 4], which we call the smooth min-entropy bound. By combining this bound and the information-spectrum approach method, Tomamichel and one of the authors derived another bound ], which we call the inf-spectral entropy bound. One of the authors also 3 In the case of quantum channel, the bound is described as a semi-definite programming. 4 The Gallager bound for finite states channels was considered in 3, Section 5.9], but a closed form epression for the eponent was not derived.

2 derived another bound by using the leftover hash lemma, the approimate smoothing of the Rényi entropy of order 2, and the large deviation technique 5], which we call the eponential bound. Further, the authors compared the infspectral entropy bound and the eponential bound 6]. It turned out that the former is tighter than the latter when the required security level ε is rather large, and the latter is tighter than the former when ε is rather small. A bound that interpolates both the bounds was also derived in 6], which we call the hybrid bound. For the computational compleity issue of the privacy amplification, the situation is the same as the coding problems, i.e., there is no bound that is efficiently computable for the Markov chain. The smooth min-entropy bound can be computed by using the linear programming for rather small block length, but it is difficult to compute in general. The computation of the inf-spectral entropy bound depends on the evaluation of a tail probability, and it is also difficult to compute in general. The eponential bound is described by the Gallager function, and thus can be easily computed provided that a source is memoryless. As described above, there is no bound that is efficiently computable for the Markov chain, and the first purpose of this paper is to derive non-asymptotic bounds that are efficiently computable. Net, let us consider the second criterion, i.e., asymptotic optimality. So far, three kinds of asymptotic regimes have been studied in the information theory ], 2], 7], 8], 9], 20], 2]: The large deviation regime in which the error probability ε asymptotically behaves like e nr for some r > 0, The moderate deviation regime in which ε asymptotically behaves like e n 2tr for some r > 0 and t 0, /2), and The second order regime in which ε is a constant. We shall claim that a good non-asymptotic bound should be asymptotically optimal at least one of the above mentioned three regimes. In fact, the information spectrum bound, the hypothesis testing bound, and the DT bound are asymptotically optimal in the moderate deviation regime and the second order regime; the Gallager bound is asymptotically optimal in the large deviation regime; and the RCU bound is asymptotically optimal in all the regimes 5. B. Main Contribution for Non-Asymptotic Analysis To derive non-asymptotic achievability bounds on the problems, we basically use the eponential type bounds 6 for the single shot setting. For the source coding with side-information and the random number generation with side-infromation, we consider two assumptions on transition matrices see Assumption and Assumption 2 of Section II). Although a computable form of the conditional entropy rate is not known in general, 5 The Gallager bound and the RCU bound are asymptotically optimal in the large deviation regime only up to the critical rate. 6 For the channel coding, it corresponds to the Gallager bound. Assumption, which is less restrictive than Assumption 2, enables us to derive a computable form of the conditional entropy rate. In the problems with side-information, eponential type bounds are described by conditional Rényi entropies. There are several definitions of conditional Rényi entropies see 22], 23] for etensive review), and we use the one defined in 24] and the one defined by Arimoto 25]. We shall call the former one the lower conditional Rényi entropy cf. 2)) and the latter one the upper conditional Rényi entropy cf. 7)). To derive non-asymptotic bounds, we need to evaluate these information measures for the Markov chain. For this purpose, under Assumption, we introduce the lower conditional Rényi entropy for transition matrices cf. 2)). Then, we evaluate the lower conditional Rńyi entropy for the Markov chain in terms of its transition matri counterpart. This evaluation gives non-asymptotic bounds for the coding and random number generation problems under Assumption. Under more restrictive assumption, i.e., Assumption 2, we also introduce the upper conditional Rényi entropy for a transition matri cf. 26)). Then, we evaluate the upper Rényi entropy for the Markov chain in terms of its transition matri counterpart. This evaluate gives non-asymptotic bounds that are tighter than those obtained under Assumption. We also derive converse bounds for every problem by using the change of measure argument developed by the authors in the accompanying paper on information geometry 26]. To derive converse bounds for the problems with sideinformation, we further introduce two-parameter conditional Rényi entropy and its transition matri counterpart cf. 4) and 30)). This novel information measure includes the lower conditional Rényi entropy and the upper conditional Rényi entropy as special cases. Here, we would like to remark on terminologies. There are a few ways to epress eponential type bounds. In statistics or the large deviation theory, we usually use the cumulant generating function CGF) to describe eponents. In information theory, we use the Gallager function or the Rényi entropies. Although these three terminologies are essentially the same and are related by change of variables, the CGF and the Gallager function are convenient for some calculations since they have good properties such as conveity. However, they are merely mathematical functions. On the other hand, the Rényi entropies are information measures including Shannon s information measures as special cases. Thus, the Rényi entropies are intuitively familiar in the field of information theory. The Rényi entropies also have an advantage that two types of bounds eg. 82) and 82)) can be epressed in a unified manner. For these reasons, we state our main results in terms of the Rényi entropies while we use the CGF and the Gallager function in the proofs.

3 C. Main Contribution for Asymptotic Analysis For asymptotic analyses of the large deviation and the moderate deviation regimes, we derive the characterizations 7 by using our non-asymptotic achievability and converse bounds, which implies that our non-asymptotic bounds are tight in the large deviation regime and the moderate deviation regime. We also derive the second order rate. It is also clarified that the reciprocal coefficient of the moderate deviation regime and the variance of the second order regime coincide. Furthermore, a single letter form of the variance is clarified 8. D. Organization of Paper In Section II, we introduce information measures that will be needed in later sections. Then, we consider the source coding with side-information and the uniform random number generation with side-information in Section III and Section IV respectively. Omitted results and proofs can be found in 29]. II. INFORMATION MEASURES In this section, we introduce information measures that will be used in later sections. A. Information measures for Single-Shot Setting In this section, we introduce conditional Rényi entropies for the single-shot setting. For more detailed review of conditional Rényi entropies, see 23]. For a correlated random variable X, Y ) on X Y with probability distribution P XY and a marginal distribution Q Y on Y, we introduce the conditional Rényi entropy of order + relative to Q Y as H P XY Q Y ) := log,y P XY, y) Q Y y), ), 0) 0, ). The conditional Rényi entropy of order relative to Q Y is defined by the it with respect to. One of important special cases of H P XY Q Y ) is the case with Q Y = P Y. We shall call this special case the lower conditional Rényi entropy of order + and denote 9 H X Y ) := H P XY P Y ) 2) = log,y The following property holds. Lemma We have P XY, y) P Y y).3) 0 H X Y ) = HX Y ) 4) 7 For the large deviation regime, we only derive the characterizations up to the critical rates. 8 An alternative way to derive a single letter characterization of the variance for the Markov chain was shown in 27, Lemma 20]. It should be also noted that a single letter characterization can be derived by using the fundamental matri 28]. The single letter characterization of the variance in 7, Section VII] and 2, Section III] has an error, which is corrected in this paper. 9 This notation was first introduce in 30]. and VX Y ) := Var log 2 = 0 ] P X Y X Y ) ] HX Y ) H X Y ) 5). 6) The other important special cases of H P XY Q Y ) is the measure maimized over Q Y. We shall call this special case the upper conditional Rényi entropy of order and denote 0 H X Y ) 7) := ma H P XY Q Y ) 8) Q Y PY) = H P XY P ) Y ) 9) = + log ] P Y y) P X Y y), y 0) P ) Y y) := P XY, y) ] y P XY, y ) ]. ) For this measure, we also have properties similar to Lemma. Lemma 2 We have and 0 H X Y ) = HX Y ) 2) ] 2 HX Y ) H X Y ) = VX Y ). 3) 0 When we derive converse bounds on the source coding with side-information or the random number generation with sideinformation, we need to consider the case such that the order of the Rényi entropy and the order of conditioning distribution defined in ) are different. For this purpose, we introduce two-parameter conditional Rényi entropy: H, X Y ) 4) := H P XY P ) Y ) 5) = log ] P Y y) P X Y y) 6) y ] P X Y y) + + H X Y ). 7) 0 For < < 0, 9) can be proved by using the Hölder inequality, and, for 0 <, 9) can be proved by using the reverse Hölder inequality 3, Lemma 8].

4 B. Information Measures for Transition Matri Let {W, y, y )},y),,y )) X Y) 2 be an ergodic and irrecucible transition matri. The purpose of this section is to introduce transition matri counter parts of those measures in Section II-A. For this purpose, we first need to introduce some assumptions on transition matrices: Assumption Non-Hidden) We say that a transition matri W is non-hidden if W, y, y ) = W y y ) 8) for every X and y, y Y. Assumption 2 Strongly Non-Hidden) We say that a transition matri W is strongly non-hidden if, for every, ) and y, y Y, W y y ) := W, y, y ) 9) is well defined, i.e., the right hand side of 9) is independent of. Assumption requires 9) to hold only for = 0, and thus Assumption 2 implies Assumption. However, Assumption 2 strictly stronger condition than Assumption. For eample, let consider the case such that the transition matri is a product form, i.e., W, y, y ) = W )W y y ). In this case, Assumption is obviously satisfied. However, Assumption 2 is not satisfied in general. First, we introduce information measures under Assumption. In order to define a transition matri counterpart of 2), let us introduce the following tilted matri: W, y, y ) := W, y, y ) W y y ). 20) Let λ be the Perron-Frobenius eigenvalue and P,XY be its normalized eigenvector. Then, we define the lower conditional Rényi entropy for W by X Y ) := log λ, 2), 0) 0, ). For = 0, we define the lower conditional Rényi entropy for W by H W X Y ) = X Y ) 22) := X Y ), 23) 0 and we just call it the conditional entropy for W. As a counterpart of 6), we also define ] 2 H W X Y ) V W X Y ) X Y ) :=. 24) 0 Net, we introduce information measures under Assumption 2. In order to define a transition matri counterpart of 7), let us introduce the following Y Y matri: K y y ) := W y y ), 25) W is defined by 9). Let κ be the Perron-Frobenius eigenvalue. Then, we define the upper conditional Rényi entropy for W by H,W + X Y ) := log κ, 26), 0) 0, ). We have the following properties. Lemma 3 We have 0 H,W X Y ) = HW X Y ) 27) and ] 2 H W X Y ) H,W X Y ) = V W X Y ). 28) 0 Now, let us introduce a transition matri counterpart of 4). For this purpose, we introduce the following Y Y matri: N, y y ) := W y y )W y y ). 29) Let ν, be the Perron-Frobenius eigenvalue of N,. Then, we define the two-parameter conditional Rényi entropy by H W, X Y ) := log ν, + + H,W X Y ). 30) For the information measures introduced in this section, we have the following property. Lemma 4 ) The function X Y ) is a concave function of, and it is strict concave iff. V W X Y ) > 0. 2) X Y ) is a monotonically decreasing function of. 3) The function H,W X Y ) is a concave function of, and it is strict concave iff. V W X Y ) > 0. 4) H,W X Y ) is a monotonically decreasing function of. 5) For every, 0) 0, ), we have X Y ) H,W X Y ). 6) For fied, the function H, W X Y ) is a concave function of, and it is strict concave iff. V W X Y ) > 0. 7) For fied, H, W X Y ) is a monotonically decreasing function of. 8) We have 9) We have H,X Y W ) = X Y ). 3) H,X Y W ) = H,W X Y ). 32) 0) For every, 0) 0, ), H, W X Y ) is maimized at =. Net, we consider the etreme cases of X Y ). Let λ be the Perro-Frobenius eigenvalue of W, y, y ) > 0]W y y ). 33)

5 Then, we define 0 X Y ) := log λ. 34) On the other hand, let G W = X Y, E W ) be the graph such that, y ),, y)) E W iff. W, y, y ) > 0. Then, for each, y) X Y, let C,y) be the set of all Hamilton cycle from, y) to itself. Then, we define X Y ) := log ma Lemma 5 We have H,W H,W,ȳ) X Y,y ),,y)) c ma 35) c C,ȳ) W, y, y) / c 36). X Y ) = H,W 0 X Y ), 37) X Y ) = H,W X Y ). 38) Net, we consider the etreme cases of H,W X Y ). When W satisfies Assumption 2, we note that Sy y ) := suppw, y, y)), 39) T y y ) := ma W, y, y) 40) are well defined, i.e., the right hand sides of 39) and 40) are independent of. Let G W = Y, E W ) be the graph such that y, y) E W iff. W y y ) > 0. Then, for each y Y, let C y be the set of all Hamilton cycle from y to itself. Then, we define / c H,W 0 X Y ) := log ma Sy y ). 4) ma ȳ Y c Cȳ y,y) c On the other hand, let κ be the Perro-Frobenius eigenvalue of W y y )T y y ). Then, we define Lemma 6 We have H,W X Y ) := log κ. 42) H,W H,W X Y ) = H,W 0 X Y ), 43) X Y ) = H,W X Y ). 44) From Statement of Lemma 4, dh,w X Y )] d is monotonically decreasing. Thus, we can define the inverse function a) of dh,w X Y )] d by X Y )] d = a 45) =a) d dh for a < a a, a :=,W X Y )] dh,w X Y )] d. Let d and a := Ra) := + a)) a) a) X Y ). 46) Since R a) = + a)), 47) Ra) is a monotonic increasing function of a < a < Ra). Thus, we can define the inverse function ar) of Ra) by + ar)))ar) ar)) ar)) X Y ) = R 48) for Ra) < R < 0 X Y ). For H,W X Y ), by the same reason, we can define the inverse function a) by dh,w X Y )] d = a, 49) =a) and the inverse function ar) of by Ra) := + a))a a)h,w a) X Y ) 50) + ar)))ar) ar))h,w ar)) X Y ) = R, 5) for Ra) < R < H,W 0 X Y ). C. Information Measures for Markov Chain Let X, Y) be the Markov chain induced by transition matri W and some initial distribution P XY. Now, we show how information measures introduced in Section II-B are related to the conditional Rényi entropy rates. First, we introduce the following lemma, which gives finite upper and lower bounds on the lower conditional Rényi entropy. Lemma 7 Suppose that transition matri W satisfies Assumption. Let v be the eigenvector of W T with respect to the Perron-Frobenius eigenvalue λ such that min,y v, y) =. Let w, y) := P XY, y) P Y y). Then, we have n ) X Y ) + δ) 52) H Xn Y n ) 53) n ) X Y ) + δ), 54) δ) := log v w + log ma,y v, y), 55) δ) := log v w. 56) From Lemma 7, we have the following. Theorem Suppose that transition matri W satisfies Assumption. For any initial distribution, we have n n H Xn Y n ) = X Y ), 57) n n HXn Y n ) = H W X Y ), 58) n n H 0 Xn Y n ) = 0 X Y ), 59) n n H X n Y n ) = H,W X Y ). 60)

6 We also have the following asymptotic evaluation of the variance. Theorem 2 Suppose that transition matri W satisfies Assumption. For any initial distribution, we have n n VXn Y n ) = V W X Y ). 6) Theorem 2 is practically important since the it of the variance can be described by a single letter characterized quantity. A method to calculate V W X Y ) can be found in 32]. Net, we show the lemma that gives finite upper and lower bound on the upper conditional Rényi entropy in terms of the upper conditional Rényi entropy for the transition matri. Lemma 8 Suppose that transition matri W satisfies Assumption 2. Let v be the eigenvector of K T with respect to the Perro-Frobenius eigenvalue κ such that min y v y) =. Let w be the Y -dimensional vector defined by w y) := Then, we have P XY, y) ]. 62) n ) + H,W X Y ) + ξ) 63) + H Xn Y n ) 64) n ) + H,W X Y ) + ξ), 65) ξ) := log v w + log ma v y), 66) y ξ) := log v w. 67) From Lemma 8, we have the following. Theorem 3 Suppose that transition matri W satisfies Assumption 2. For any initial distribution, we have n n H Xn Y n ) = H,W X Y ), 68) n n H 0 Xn Y n ) = H,W 0 X Y ), 69) n n H X n Y n ) = H,W X Y ). 70) Finally, we show the lemma that gives finite upper and lower bounds on the two-parameter conditional Rényi entropy in terms of the two-parameter conditional Rényi entropy for the transition matri. Lemma 9 Suppose that transition matri W satisfies Assumption 2. Let v, be the eigenvector of N, T with respect to the Perro-Frobenius eigenvalue ν, such that min y v, y) =. Let w, be the Y -dimensional vector defined by w, y) 7) ] ] := P XY, y) P XY, y) 72). Then, we have n )H W, X Y ) + ζ, ) 73) H, X n Y n ) 74) n )H W, X Y ) + ζ, ), 75) ζ, ) := log v, w, + log ma v, y) + ξ ), y ζ, ) := log v, w, + ξ ) for > 0 and ζ, ) := log v, w, + log ma v, y) + ξ ), y ζ, ) := log v, w, + ξ ) for < 0 From Lemma 9, we have the following. Theorem 4 Suppose that transition matri W satisfies Assumption 2. For any initial distribution, we have n n H, Xn Y n ) = H, W X Y ). 76) III. SOURCE CODING WITH FULL SIDE-INFORMATION A. Problem Formulation A code Ψ = e, d) consists of one encoder e : X {,..., M} and one decoder d : {,..., M} Y X. The decoding error probability is defined by P e Ψ) := Pr{X dex), Y )}. 77) For notational convenience, we introduce the infimum of error probabilities under the condition that the message size is M: P e M) := inf Ψ P eψ) 78) When we construct a source code, we often use a twouniversal hash family F and a random function F on F. Then, we bound the error probability P e ΨF )) averaged over the random function by only using the property of twouniversality. For this reason, it is convenient to introduce the quantity: P e M) := sup EP e ΨF ))], 79) F the supremum is taken over all two-universal hash family from X to {,..., M}. From the definition, we obviously have P e M) P e M). When we consider n-fold A family of functions is said to be universal-two if Pr{F ) = F )} M for any district and 33].

7 etension, the source code and related quantities are denoted with subscript n. Instead of evaluating the error probability P e M n ) or P e M n )) for given M n, we are also interested in evaluating for given 0 ε. Mn, ε) := inf{m n : P e M n ) ε}, 80) Mn, ε) := inf{m n : P e M n ) ε} 8) B. Finite Bounds for Markov Source Under Assumption, we have the following achievability and converse bounds. Theorem 5 Suppose that transition matri W satisfies Assumption. Let R := n log M n. Then we have log P e M n ) sup 0 nr + n )H X Y ) + δ) ]. Theorem 6 Suppose that transition matri W satisfies Assumption. Let R := n log M n. For any H W X Y ) < R < 0 X Y ), we have log P e M n ) inf s>0 < <ar)) { n ) + s) + s) log + X Y ) H,W ++s) X Y ) } + δ 2e n )ER, )+δ 2 ) ] /s, ER, ) := ar)) )ar) ar)) ar)),w X Y ) + H X Y ), + a) and ar) are the inverse functions defined by 45) and 48) respectively, and δ := + s)δ ) δ + s) ), δ 2 := ar)) )R + )δar))) + + ar)))δ ). + ar)) Net, we derive tighter achievability and converse bounds under Assumption 2. Theorem 7 Suppose that transition matri W satisfies Assumption 2. Let R := n log M n. Then we have log P e M n ),W nr + n )H sup X Y ) + ξ) Theorem 8 Suppose that transition matri W satisfies Assumption 2. Let R := n log M n. For any H W X Y ) < R < H,W 0 X Y ), we have log P e M n ) n ) + s) inf s>0 < <ar)) { H W +,ar)) X Y ) } H W X Y ) + δ ++s),ar)) ) ] + s) log 2e n )ER, )+δ 2 /s, ER, ) := ar)) )ar) ar))h,w ar)) X Y ) + H W X Y )] +,ar)) a) and ar) are the inverse functions defined by 49) and 5) respectively, and δ := + s)ζ, ar))) ζ + s), ar))), δ 2 := ar)) )R + )ζar)), ar))) C. Large Deviation + ar)) + + ar)))ζ, ar))) + ar)) From Theorem 5 and Theorem 6, we have the following. Theorem 9 Suppose that transition matri W satisfies Assumption. For H W X Y ) < R, we have inf n n log P e e nr ) sup 0 R + X Y )]. On the other hand, for H W X Y ) < R < 0 X Y ), we have sup n n log P e e nr ) ar))ar) + ar)) ar)) X Y ). Under Assumption 2, from Theorem 7 and Theorem 8, we have the following tighter bound. Theorem 0 Suppose that transition matri W satisfies Assumption 2. For H W X Y ) < R, we have inf n n log P e e nr ),W R + H sup X Y )

8 On the other hand, for H W X Y ) < R < H,W 0 X Y ), we have sup n n log P e e nr ) ar))ar) + ar))h,w ar)) X Y ). Remark For R R cr, cf. 50) for the definition of Ra)) dh,w ) X Y )] R cr := R 82) d = 2 is the critical rate, we can rewrite the lower bound in 82) as,w R + H sup X Y ) = ar))ar) + ar))h,w ar)) X Y ). Thus, the lower bound and the upper bound coincide up to the critical rate. D. Moderate Deviation From Theorem 5 and Theorem 6, we have the following. Theorem Suppose that transition matri W satisfies Assumption. For arbitrary t 0, /2) and δ > 0, we have n n 2t log P e e ) nhw X Y )+n t δ = n n 2t log P ) e e nhw X Y )+n t δ δ 2 = 2V W X Y ). E. Second Order By applying the central it theorem to information spectrum type bounds, and by using Theorem 2, we have the following. Theorem 2 Suppose that transition matri W satisfies Assumption. For arbitrary ε 0, ), we have log Mn, ε) nh W X Y ) n n log = Mn, ε) nh W X Y ) n n = V W X Y )Φ ε). IV. UNIFORM RANDOM NUMBER GENERATION WITH SIDE-INFORMATION In this section, we consider the uniform random number generation, which is also known as the privacy amplification. A. Problem Formulation The privacy amplification is conducted by a function f : X {,..., M}. The security of the generated key is evaluated by f) := 2 P fx)y P Ū P Y, 83) Ū is the uniform random variable on {,..., M} and is the variational distance. For notational convenience, we introduce the infimum of the security criterion under the condition that the range size is M: M) := inf f). 84) f When we construct a function for the privacy amplification, we often use a two-universal hash family F and a random function F on F. Then, we bound the security criterion averaged over the random function by only using the property of two-universality. For this reason, it is convenient to introduce the quantity: M) := sup E F )], 85) F the supremum is taken over all two-universal hash families from X to {,..., M}. From the definition, we obviously have M) M). 86) When we consider n-fold etension, the security criteria are denoted by M n ) or M n ). As in Section III, we also introduce the quantities Mn, ε) and Mn, ε). B. Finite Bounds for Markov Source Under Assumption, we have the following achievability and converse bounds. Theorem 3 Suppose that transition matri W satisfies Assumption. Let R := n log M n. Then we have log M n ) sup 0 nr + n ) X Y ) + δ) log3/2). + Theorem 4 Suppose that transition matri W satisfies Assumption. Let R := n logm n/2). For any a < R < H W X Y ), we have log M n ) inf s>0 >a) { n ) + s) + s) log + X Y ) H,W ++s) X Y ) } + δ e n )ER, )+δ 2 ) ] /s +, ER, ) := R) )R R),W R) X Y ) + H X Y ) +

9 a) is the inverse function defined by 45), and δ := + s)δ ) δ + s) ), δ 2 := R) )R δr)) + δ ). Net, we derive tighter achievability and converse bounds under Assumption 2. Theorem 5 Suppose that transition matri W satisfies Assumption 2. Let R := n log M n. Then we have log M n ) sup 0 nr + n )H,W X Y ) + Theorem 6 Suppose that transition matri W satisfies Assumption 2. Let R be such that n )R + { + ar)))ar) ξar)))) } = logm n /2). If Ra) < R < H W X Y ), then we have log M n ) n ) + s) inf s>0 >ar)) { H W +,ar)) X Y ) } H W X Y ) + δ ++s),ar)) ) ] + s) log e n )ER, )+δ 2 /s + 2, ER, ) := ar)) )ar)) ar))h,w ar)) X Y ) + H W X Y )] +,ar)) a) and ar) are the inverse functions defined by 49) and 5) respectively, δ := + s)ζ, ar))) ζ + s), ar))), δ 2 := ar)) )ar)) ζar)), ar))) C. Large Deviation +ζ, ar))). From Theorem 3 and Theorem 4, we have the following. Theorem 7 Suppose that transition matri W satisfies Assumption. For R < H W X Y ), we have inf n n log e nr) sup 0 R + Remark 2 For R cr R, cf. 50) for the definition of Ra)) + ξ) log3/2). dh,w ) X Y )] R cr := R d X Y ) + On the other hand, for a < R < H W X Y ), we have sup n n log e nr) R)R + R) R) X Y ). Under Assumption 2, from Theorem 5 and Theorem 6, we have the following tighter bound.. Theorem 8 Suppose that transition matri W satisfies Assumption 2. For R < H W X Y ), we have inf n n log e nr) sup 0 R + H,W X Y ) + On the other hand, for Ra) < R < H W X Y ), we have sup n n log e nr) ar))ar) + ar))h,w ar)) X Y ). = is the critical rate, we can rewrite the lower bound in 87) as sup 0 R + H,W X Y ) + = ar))ar) + ar))h,w ar)) X Y ). Thus, the lower bound and the upper bound coincide up to the critical rate. D. Moderate Deviation From Theorem 3 and Theorem 4, we have the following. Theorem 9 Suppose that transition matri W satisfies Assumption. For arbitrary t 0, /2) and δ > 0, we have ) n n 2t log e nhw X Y ) n t δ = log e ) nhw X Y ) n t δ n n 2t δ 2 = 2V W X Y ). E. Second Order By applying the central it theorem to information spectrum bounds, and by using Theorem 2, we have the following. Theorem 20 Suppose that transition matri W satisfies Assumption. For arbitrary ε 0, ), we have log Mn, ε) nh W X Y ) n n log = Mn, ε) nh W X Y ) n n = V W X Y )Φ ε). ACKNOWLEDGMENT HM is partially supported by a MEXT Grant-in-Aid for Scientific Research A) No He is partially supported by the National Institute of Information and Communication Technology NICT), Japan. The Centre for Quantum Technologies is funded by the Singapore Ministry of Education and the National Research Foundation as part of the Research Centres of Ecellence programme..

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