TY - GEN

T1 - Constructions of low-degree and error-correcting ε-biased generators

AU - Shpilka, Amir

PY - 2006

Y1 - 2006

N2 - In this work we give two new constructions of ε-biased generators. Our first construction answers an open question of Dodis and Smith [DS05], and our second construction significantly extends a result of Mossel et al. [MST03]. In particular we obtain the following results: 1. We construct a family of asymptotically good binary codes such that the codes in our family are also ε-biased sets for an exponentially small e. Our encoding and decoding algorithms run in polynomial time in the block length of the code. This answers an open question of Dodis and Smith [DS05]. 2. For every k = o(log n) we construct a degree k ε-biased generator G: {0, 1}m → {0, 1}n (namely, every output bit of the generator is a degree k polynomial in the input bits). For k constant we get that n = Ω(m/log(1/ε))k, which is nearly optimal. Our result also separates degree k generators from generators in NCk0, showing that the stretch of the former can be much larger than the stretch of the latter. The problem of constructing degree k generators was introduced by Mossel et al. [MST03] who gave a construction only for the case of degree 2 generators.

AB - In this work we give two new constructions of ε-biased generators. Our first construction answers an open question of Dodis and Smith [DS05], and our second construction significantly extends a result of Mossel et al. [MST03]. In particular we obtain the following results: 1. We construct a family of asymptotically good binary codes such that the codes in our family are also ε-biased sets for an exponentially small e. Our encoding and decoding algorithms run in polynomial time in the block length of the code. This answers an open question of Dodis and Smith [DS05]. 2. For every k = o(log n) we construct a degree k ε-biased generator G: {0, 1}m → {0, 1}n (namely, every output bit of the generator is a degree k polynomial in the input bits). For k constant we get that n = Ω(m/log(1/ε))k, which is nearly optimal. Our result also separates degree k generators from generators in NCk0, showing that the stretch of the former can be much larger than the stretch of the latter. The problem of constructing degree k generators was introduced by Mossel et al. [MST03] who gave a construction only for the case of degree 2 generators.

UR - http://www.scopus.com/inward/record.url?scp=34247550388&partnerID=8YFLogxK

U2 - 10.1109/CCC.2006.15

DO - 10.1109/CCC.2006.15

M3 - פרסום בספר כנס

AN - SCOPUS:34247550388

SN - 0769525962

SN - 9780769525969

T3 - Proceedings of the Annual IEEE Conference on Computational Complexity

SP - 33

EP - 45

BT - Proceedings - Twenty-First Annual IEEE Conference on Computational Complexity, CCC 2006

Y2 - 16 July 2006 through 20 July 2006

ER -