TY - GEN

T1 - Separating the power of monotone span programs over different fields

AU - Beimel, Amos

AU - Weinreb, Enav

N1 - Publisher Copyright:
© 2003 IEEE.

PY - 2003/1/1

Y1 - 2003/1/1

N2 - Monotone span programs are a linear-algebraic model of computation. They are equivalent to linear secret sharing schemes and have various applications in cryptography and complexity. A fundamental question is how the choice of the field in which the algebraic operations are performed effects the power of the span program. In this paper we prove that the power of monotone span programs over finite fields of different characteristics is incomparable; we show a super-polynomial separation between any two fields with different characteristics, answering an open problem of Pudlák and Sgall (1998). Using this result we prove a super-polynomial lower bound for monotone span programs for a function in uniform - NC2 (and therefore in P), answering an open problem of Babai, Wigderson, and Gál (1999). Finally, we show that quasi-linear schemes, a generalization of linear secret sharing schemes introduced in Beimel and Ishai (2001), are stronger than linear secret sharing schemes. In particular, this proves, without any assumptions, that non-linear secret sharing schemes are more efficient than linear secret sharing schemes.

AB - Monotone span programs are a linear-algebraic model of computation. They are equivalent to linear secret sharing schemes and have various applications in cryptography and complexity. A fundamental question is how the choice of the field in which the algebraic operations are performed effects the power of the span program. In this paper we prove that the power of monotone span programs over finite fields of different characteristics is incomparable; we show a super-polynomial separation between any two fields with different characteristics, answering an open problem of Pudlák and Sgall (1998). Using this result we prove a super-polynomial lower bound for monotone span programs for a function in uniform - NC2 (and therefore in P), answering an open problem of Babai, Wigderson, and Gál (1999). Finally, we show that quasi-linear schemes, a generalization of linear secret sharing schemes introduced in Beimel and Ishai (2001), are stronger than linear secret sharing schemes. In particular, this proves, without any assumptions, that non-linear secret sharing schemes are more efficient than linear secret sharing schemes.

KW - Arithmetic

KW - Circuits

KW - Combinatorial mathematics

KW - Computational complexity

KW - Computational modeling

KW - Computer science

KW - Cryptography

KW - Galois fields

KW - Linear algebra

KW - Vectors

UR - http://www.scopus.com/inward/record.url?scp=84943422766&partnerID=8YFLogxK

U2 - 10.1109/SFCS.2003.1238216

DO - 10.1109/SFCS.2003.1238216

M3 - פרסום בספר כנס

AN - SCOPUS:0345412672

T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

SP - 428

EP - 437

BT - Proceedings - 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003

PB - Institute of Electrical and Electronics Engineers

Y2 - 11 October 2003 through 14 October 2003

ER -