TY - GEN
T1 - Separating the power of monotone span programs over different fields
AU - Beimel, A.
AU - Weinreb, E.
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 - Conference contribution
AN - SCOPUS:84943422766
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
T2 - 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003
Y2 - 11 October 2003 through 14 October 2003
ER -