Some Public Key Cryptography Schemes Based On Quartic Residues

With the rapid development of network, we cannot do business withoutnetwork. The emergence and development of the network technology, whichgreatly facilitates our work and study, brings a lot of security problems atthe same time. In view of the Internet is open, information is vulnerable toeavesdropping, tampering and other attacks, and cases of huge losses due tosecurity breaches and hacking are increasing. Network security problemsbecome increasingly important and urgent. As the technology to ensuresecurity of information on the network, public key cryptography can provideusers with information confidentiality, information integrity protection, useridentity and information for source certification.Although traditional public key cryptography signature schemes havewithstood the trial of all kinds of attacks and been widely applied ineveryday life, there is still an inadequacy that the verifier needs to obtain thepublic key through a public key infrastructure before he verifies a signature.And the interaction process is cumbersome. Signature schemes based onidentity (also known as identity-based signature schemes) simplify theinteraction process. Because in identity-based schemes, the signer’s publickey can be calculated directly from his identity information, without goingthrough a public key infrastructure. Therefore identity-based signatureschemes attract lots of attention of researchers.Up to now, the proposed identity-based signature schemes are based onpairing operation, which consists of a large number of high complexityoperations and seriously reduce the efficiency of the schemes. That is alsoone of the reasons why identity-based signature schemes have incomparableadvantages over traditional signature schemes but were not widely applied. Therefore identity-based signature schemes without pairing operationbecome a research topic of cryptography.Quadratic residue theory is famous in number theory. Identity-basedsignature schemes based on quadratic residues have made fruitful researchresults. Then those schemes based on quartic residues are an emerging fieldof study. Compared with quadratic residues, quartic residues have similarproperties such as the Jacobi symbol, quartic reciprocity law and so on, butthe latter can express1more bit of information than the former. Hence it ispractical to construct identity-based signature schemes based on quarticresidues.The main work of this article is to study quartic residues and identity-based signature schemes, then design effective and secure signature schemes.The article first introduced the theoretical basis, definitions and relatedproperties of quartic residues. Then a new algorithm was proposed tocalculate the fourth root of quartic residue integers in a Gaussian ring, andbased on which a new identity-based signature scheme using quarticresidues theory was constructed. The security of the scheme is based on thedifficult problem of factorization in a Gaussian ring. By detailed securityanalysis, the article formally proves that the scheme can avoid existentialforgery on the adaptive chosen message and identity attacks. It should bepointed out that the scheme is the first identity based signature schemewhich is based on quartic residues. In order to withdraw the signing privilege,the article proposes the second identity based signature scheme based onquartic residues. The basic idea is to use intermediaries to complete the finalsignature. If you would like to revoke signer permissions, you just make theintermediate institutions refuse to sign.This research develops the theory and application of high times powerresidues as well as enriches the identity-based cryptographic schemeswithout pairing operation and fills the blank of construction of identity-based signature schemes using quartic residues.