Generalized Powering Functions and Their Application to Digital Signatures

Hisayoshi SATO  Tsuyoshi TAKAGI  Satoru TEZUKA  Kazuo TAKARAGI  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E89-A    No.1    pp.81-89
Publication Date: 2006/01/01
Online ISSN: 1745-1337
DOI: 10.1093/ietfec/e89-a.1.81
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Cryptography and Information Security)
Category: Digital Signature
factoring,  RSA,  modular powering function,  digital signature,  

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This paper investigates some modular powering functions suitable for cryptography. It is well known that the Rabin encryption function is a 4-to-1 mapping and breaking its one-wayness is secure under the factoring assumption. The previously reported encryption schemes using a powering function are variants of either the 4-to-1 mapping or higher n-to-1 mapping, where n > 4. In this paper, we propose an optimized powering function that is a 3-to-1 mapping using a p2q-type modulus. The one-wayness of the proposed powering function is as hard as the infeasibility of the factoring problem. We present an efficient algorithm for computing the decryption for a p2q-type modulus, which requires neither modular inversion nor division. Moreover, we construct new provably secure digital signatures as an application of the optimized functions. In order to achieve provable security in the random oracle model, we usually randomize a message using random hashing or padding. However, we have to compute the randomization again if the randomized message is a non-cubic residue element--it is inefficient for long messages. We propose an algorithm that can deterministically find the unique cubic residue element for a randomly chosen element.