TY - JOUR

T1 - Locally Computable UOWHF with Linear Shrinkage

AU - Applebaum, Benny

AU - Moses, Yoni

N1 - Publisher Copyright:
© 2016, International Association for Cryptologic Research.

PY - 2017/7/1

Y1 - 2017/7/1

N2 - We study the problem of constructing locally computable universal one-way hash functions (UOWHFs) H: { 0 , 1 } n→ { 0 , 1 } m. A construction with constant output locality, where every bit of the output depends only on a constant number of bits of the input, was established by Applebaum et al. (SIAM J Comput 36(4):845–888, 2006). However, this construction suffers from two limitations: (1) it can only achieve a sublinear shrinkage of n- m= n1 - ϵ and (2) it has a super-constant input locality, i.e., some inputs influence a large super-constant number of outputs. This leaves open the question of realizing UOWHFs with constant output locality and linear shrinkage of n- m= ϵn, or UOWHFs with constant input locality and minimal shrinkage of n- m= 1. We settle both questions simultaneously by providing the first construction of UOWHFs with linear shrinkage, constant input locality and constant output locality. Our construction is based on the one-wayness of “random” local functions—a variant of an assumption made by Goldreich (Studies in Complexity and Cryptography, 76–87, 2011; ECCC 2010). Using a transformation of Ishai et al. (STOC, 2008), our UOWHFs give rise to a digital signature scheme with a minimal additive complexity overhead: signing n-bit messages with security parameter κ takes only O(n+ κ) time instead of O(nκ) as in typical constructions. Previously, such signatures were only known to exist under an exponential hardness assumption. As an additional contribution, we obtain new locally computable hardness amplification procedures for UOWHFs that preserve linear shrinkage.

AB - We study the problem of constructing locally computable universal one-way hash functions (UOWHFs) H: { 0 , 1 } n→ { 0 , 1 } m. A construction with constant output locality, where every bit of the output depends only on a constant number of bits of the input, was established by Applebaum et al. (SIAM J Comput 36(4):845–888, 2006). However, this construction suffers from two limitations: (1) it can only achieve a sublinear shrinkage of n- m= n1 - ϵ and (2) it has a super-constant input locality, i.e., some inputs influence a large super-constant number of outputs. This leaves open the question of realizing UOWHFs with constant output locality and linear shrinkage of n- m= ϵn, or UOWHFs with constant input locality and minimal shrinkage of n- m= 1. We settle both questions simultaneously by providing the first construction of UOWHFs with linear shrinkage, constant input locality and constant output locality. Our construction is based on the one-wayness of “random” local functions—a variant of an assumption made by Goldreich (Studies in Complexity and Cryptography, 76–87, 2011; ECCC 2010). Using a transformation of Ishai et al. (STOC, 2008), our UOWHFs give rise to a digital signature scheme with a minimal additive complexity overhead: signing n-bit messages with security parameter κ takes only O(n+ κ) time instead of O(nκ) as in typical constructions. Previously, such signatures were only known to exist under an exponential hardness assumption. As an additional contribution, we obtain new locally computable hardness amplification procedures for UOWHFs that preserve linear shrinkage.

KW - Input locality

KW - NC0

KW - Output locality

KW - Universal one-way hash functions

UR - http://www.scopus.com/inward/record.url?scp=85024497353&partnerID=8YFLogxK

U2 - 10.1007/s00145-016-9232-x

DO - 10.1007/s00145-016-9232-x

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AN - SCOPUS:85024497353

SN - 0933-2790

VL - 30

SP - 672

EP - 698

JO - Journal of Cryptology

JF - Journal of Cryptology

IS - 3

ER -