IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 56, NO. 12, DECEMBER 2010 6501 Deterministic Extractors for Independent-Symbol Sources Chia-Jung Lee, Chi-Jen Lu, and Shi-Chun Tsai Abstract—In this paper, we consider the task of deterministically extracting randomness from sources consisting of a sequence of independent symbols from . The only randomness guar- antee on such a source is that the whole source has min-entropy . We give an explicit deterministic extractor which extract bits with error , for any , , and . For sources with a larger min-entropy, we can extract even more randomness. When , for any constant , we can extract bits with any error . When , for some constant , we can extract bits with any error . Our results gen- eralize those of Kamp and Zuckerman and Gabizon et al. which only work for bit-fixing sources (with and each bit of the source being either fixed or perfectly random). Moreover, we show the existence of a nonexplicit deterministic extractor which can ex- tract bits whenever . Finally, we show that even to extract from bit-fixing sources, any extractor, seeded or not, must suffer an entropy loss . This generalizes a lower bound of Radhakrishnan and Ta-Shma on extracting from general sources. Index Terms—Independent-symbol sources, min-entropy, pseudo-randomness, randomness extractors. I. INTRODUCTION R ANDOMNESS has become a useful tool in computer sci- ence. For many computational problems, the most effi- cient algorithms known are randomized. For some tasks in dis- tributed computing, only randomized solutions are possible. In cryptography, randomness is essential in generating secret keys. However, when using randomness in designing algorithms or protocols, people usually assume the randomness being perfect, and the performance guarantees are based on this assumption. In reality, the random sources we (or computers) have access to are typically not so perfect at all, but only contain some crude randomness. One approach to solve this problem is to construct Manuscript received June 09, 2008; revised July 16, 2010. Date of current version November 19, 2010. The work of C.-J. Lu was supported (in part) by the National Science Council of Taiwan under contract NSC-97-2221-E-001- 012-MY3 and was performed while she was with the Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan. The work of S.-C. Tsai was supported (in part) by the National Science Council of Taiwan under Contracts NSC-97-2221-E-009-064-MY3 and NSC-98-2221-E-009-078-MY3. The material in this paper was presented (in part) at the 33rd International Col- loquium on Automata, Languages, and Programming (ICALP 2006), Venice, Italy, July 2006. C.-J. Lee and C.-J. Lu are with the Institute of Information Science, Academia Sinica, Taipei, Taiwan (e-mail: leecj@iis.sinica.edu.tw; cjlu@iis.sinica.edu.tw). S.-C. Tsai is with the Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan (e-mail: sctsai@csie.nctu.edu.tw). Communicated by T. Fujiwara, Associate Editor for Complexity and Cryp- tography. Digital Object Identifier 10.1109/TIT.2010.2079012 so-called extractors, which can extract almost perfect random- ness from weakly random sources [35], [22]. Extractors turn out to have close connections to other fundamental objects such as pseudorandom generators, hash functions, error-correcting codes, expander graphs, and samplers, and they have found a wide range of applications in areas such as complexity theory, cryptography, data structures, coding theory, distributed com- puting, and combinatorics (e.g., [29], [22], [36], [37], [34], [32], [31], [18], [33]). A nice survey can be found in [27]. We measure the amount of randomness in a source by its min-entropy; a source is said to have min-entropy if every el- ement occurs with probability at most . Given sources with enough min-entropy, one would like to construct an extractor which can extract a string with distribution close to uniform. However, it is well known that one cannot deterministically ex- tract even one bit from an -bit source with min-entropy [6]. In contrast, it becomes possible if we are allowed a few random bits, called a seed, to aid the extraction. Such a pro- cedure is called a seeded extractor. During the past decades, a long line of research has worked on using a shorter seed to ex- tract more randomness (e.g., [22], [21], [24], [11], [26], [32], [30], [28]), and finally an optimal (up to constant factors) con- struction has been given recently [19]. The problem with a seeded extractor is again to get a seed which is perfectly (or almost) random. For some applications, this issue can be taken care of (for example, by enumerating all possible seed values when the seed is short), but for others, we are back to the same problem which extractors are originally asked to solve. This motivates one to consider the possibility of more restricted sources from which randomness can be ex- tracted in a deterministic (seedless) way. One line of research studies the case with multiple indepen- dent sources. The goal is to have a small number of indepen- dent sources with a low min-entropy requirement on sources, while still being able to extract randomness from them. With two independent sources, the requirement on the min-entropy rate (average min-entropy per bit) stayed slightly above for a long time [6], [8], [16], but this barrier has been broken by a re- cent construction which pushes the requirement slightly below [5]. The requirement on min-entropy rate can be lowered to any constant when there are a constant number of indepen- dent sources [3], and the number of sources has recently been reduced to three [4]. The other line of research considers the case of bit-fixing sources. In an oblivious bit-fixing source, each bit is either fixed (containing no randomness) or perfectly random, and is inde- pendent of other bits. From such a source of length with min-entropy , for any constant , Kamp and 0018-9448/$26.00 © 2010 IEEE