Generalized powering functions and their application to digital signatures

Hisayoshi Sato, Tsuyoshi Takagi, Satoru Tezuka, Kazuo Takaragi

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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 p2g-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.

Original languageEnglish
Pages (from-to)434-451
Number of pages18
JournalLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume2894
Publication statusPublished - 2003 Dec 1
Externally publishedYes

Fingerprint

Electronic document identification systems
Digital Signature
Generalized Functions
Cryptography
Factoring
Encryption
Modulus
Provable Security
Modular Functions
Infeasibility
Random Oracle Model
Hashing
Randomisation
Division
Inversion
Efficient Algorithms
Computing

Keywords

  • Digital signature
  • Factoring
  • Modular powering function
  • RSA

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Cite this

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N2 - 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 p2g-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.

AB - 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 p2g-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.

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