12 research outputs found
Notes on Two Fully Homomorphic Encryption Schemes Without Bootstrapping
Recently, IACR ePrint archive posted two fully homomorphic encryption schemes without bootstrapping. In this note, we show that these schemes are trivially insecure. Furthermore, we also show that the encryption schemes of Liu and Wang in CCS 2012 and the encryption scheme of Liu, Bertino, and Xun in ASIACCS 2014 are insecure either
Fully Homomorphic Encryption with Isotropic Elements
In previous work I proposed a fully homomorphic encryption without bootstrapping which has the weak point in the enciphering function. In this paper I propose the fully homomorphic encryption scheme with non-zero isotropic octonions. I improve the previous scheme by adopting the non-zero isotropic octonions so that the βm and -m attackβ is not useful because in proposed scheme many ciphertexts exist where the plaintext m is not zero and the norm is zero. The improved scheme is based on multivariate algebraic equations with high degree or too many variables while the almost all multivariate cryptosystems proposed until now are based on the quadratic equations avoiding the explosion of the coefficients. The improved scheme is against the GrΓΆbner basis attack
Fully homomorphic public-key encryption with small ciphertext size
In previous work I proposed a fully homomorphic encryption without bootstrapping which has the large size of ciphertext. This tme I propose the fully homomorphic public-key encryption scheme on non-associative octonion ring over finite field with the small size of ciphertext. In this scheme the size of ciphertext is one-third of the size in the scheme proposed before. Because proposed scheme adopts the medium text with zero norm, it is immune from the βp and -p attackβ. As the proposed scheme is based on computational difficulty to solve the multivariate algebraic equations of high degree, it is immune from the GrΓΆbner basis attack, the differential attack, rank attack and so on
Fully Homomorphic Encryption without bootstrapping
Gentryβs bootstrapping technique is the most famous method of obtaining fully homomorphic encryption. In this paper I propose a new fully homomorphic encryption scheme on non-associative octonion ring over finite field without bootstrapping technique. The security of the proposed fully homomorphic encryption scheme is based on computational difficulty to solve the multivariate algebraic equations of high degree while the almost all multivariate cryptosystems proposed until now are based on the quadratic equations avoiding the explosion of the coefficients. Because proposed fully homomorphic encryption scheme is based on multivariate algebraic equations with high degree or too many variables, it is against the GrΓΆbner basis attack, the differential attack, rank attack and so on.
The key size of this system and complexity for enciphering/deciphering become to be small enough to handle
Enhanced fully homomorphic encryption scheme using modified key generation for cloud environment
Fully homomorphic encryption (FHE) is a special class of encryption that allows performing unlimited mathematical operations on encrypted data without decrypting it. There are symmetric and asymmetric FHE schemes. The symmetric schemes suffer from the semantically security property and need more performance improvements. While asymmetric schemes are semantically secure however, they pose two implicit problems. The first problem is related to the size of key and ciphertext and the second problem is the efficiency of the schemes. This study aims to reduce the execution time of the symmetric FHE scheme by enhancing the key generation algorithm using the Pick-Test method. As such, the Binary Learning with Error lattice is used to solve the key and ciphertext size problems of the asymmetric FHE scheme. The combination of enhanced symmetric and asymmetric algorithms is used to construct a multi-party protocol that allows many users to access and manipulate the data in the cloud environment. The Pick-Test method of the Sym-Key algorithm calculates the matrix inverse and determinant in one instance requires only n-1 extra multiplication for the calculation of determinant which takes 0(N3) as a total cost, while the Random method in the standard scheme takes 0(N3) to find matrix inverse and 0(N!) to calculate the determinant which results in 0(N4) as a total cost. Furthermore, the implementation results show that the proposed key generation algorithm based on the pick-test method could be used as an alternative to improve the performance of the standard FHE scheme. The secret key in the Binary-LWE FHE scheme is selected from {0,1}n to obtain a minimal key and ciphertext size, while the public key is based on learning with error problem. As a result, the secret key, public key and tensored ciphertext is enhanced from logq , 0(n2log2q) and ((n+1)n2log2q)2log q to n, (n+1)2log q and (n+1)2log q respectively. The Binary-LWE FHE scheme is a secured but noise-based scheme. Hence, the modulus switching technique is used as a noise management technique to scale down the noise from e and c to e/B and c/B respectively thus, the total cost for noise management is enhanced from 0(n3log2q) to 0(n2log q) . The Multi-party protocol is constructed to support the cloud computing on Sym-Key FHE scheme. The asymmetric Binary-LWE FHE scheme is used as a small part of the protocol to verify the access of users to any resource. Hence, the protocol combines both symmetric and asymmetric FHE schemes which have the advantages of efficiency and security. FHE is a new approach with a bright future in cloud computing
A Survey on Homomorphic Encryption Schemes: Theory and Implementation
Legacy encryption systems depend on sharing a key (public or private) among
the peers involved in exchanging an encrypted message. However, this approach
poses privacy concerns. Especially with popular cloud services, the control
over the privacy of the sensitive data is lost. Even when the keys are not
shared, the encrypted material is shared with a third party that does not
necessarily need to access the content. Moreover, untrusted servers, providers,
and cloud operators can keep identifying elements of users long after users end
the relationship with the services. Indeed, Homomorphic Encryption (HE), a
special kind of encryption scheme, can address these concerns as it allows any
third party to operate on the encrypted data without decrypting it in advance.
Although this extremely useful feature of the HE scheme has been known for over
30 years, the first plausible and achievable Fully Homomorphic Encryption (FHE)
scheme, which allows any computable function to perform on the encrypted data,
was introduced by Craig Gentry in 2009. Even though this was a major
achievement, different implementations so far demonstrated that FHE still needs
to be improved significantly to be practical on every platform. First, we
present the basics of HE and the details of the well-known Partially
Homomorphic Encryption (PHE) and Somewhat Homomorphic Encryption (SWHE), which
are important pillars of achieving FHE. Then, the main FHE families, which have
become the base for the other follow-up FHE schemes are presented. Furthermore,
the implementations and recent improvements in Gentry-type FHE schemes are also
surveyed. Finally, further research directions are discussed. This survey is
intended to give a clear knowledge and foundation to researchers and
practitioners interested in knowing, applying, as well as extending the state
of the art HE, PHE, SWHE, and FHE systems.Comment: - Updated. (October 6, 2017) - This paper is an early draft of the
survey that is being submitted to ACM CSUR and has been uploaded to arXiv for
feedback from stakeholder
Fully Homomorphic Encryption on Octonion Ring
In previous work(2015/474 in Cryptology ePrint Archive), I proposed a fully homomorphic encryption without bootstrapping which has the weak point in the enciphering function. In this paper I propose the improved fully homomorphic encryption scheme on non-associative octonion ring over finite field without bootstrapping technique. I improve the previous scheme by (1) adopting the enciphering function such that it is difficult to express simply by using the matrices and (2) constructing the composition of the plaintext p with two sub-plaintexts u and v. The improved scheme is immune from the βp and -p attackβ. The improved scheme is based on multivariate algebraic equations with high degree or too many variables while the almost all multivariate cryptosystems proposed until now are based on the quadratic equations avoiding the explosion of the coefficients. The improved scheme is against the GrΓΆbner basis attack.
The key size of this scheme and complexity for enciphering /deciphering become to be small enough to handle
Π Π½Π΅ΡΡΠΎΠΉΠΊΠΎΡΡΠΈ Π΄Π²ΡΡ ΡΠΈΠΌΠΌΠ΅ΡΡΠΈΡΠ½ΡΡ Π³ΠΎΠΌΠΎΠΌΠΎΡΡΠ½ΡΡ ΠΊΡΠΈΠΏΡΠΎΡΠΈΡΡΠ΅ΠΌ, ΠΎΡΠ½ΠΎΠ²Π°Π½Π½ΡΡ Π½Π° ΡΠΈΡΡΠ΅ΠΌΠ΅ ΠΎΡΡΠ°ΡΠΎΡΠ½ΡΡ ΠΊΠ»Π°ΡΡΠΎΠ²
ΠΠ΄Π½ΠΎΠΉ ΠΈΠ· Π½Π°ΠΈΠ±ΠΎΠ»Π΅Π΅ Π°ΠΊΡΡΠ°Π»ΡΠ½ΡΡ
Π·Π°Π΄Π°Ρ, ΡΠ²ΡΠ·Π°Π½Π½ΡΡ
Ρ Π·Π°ΡΠΈΡΠΎΠΉ ΠΎΠ±Π»Π°ΡΠ½ΡΡ
Π²ΡΡΠΈΡΠ»Π΅Π½ΠΈΠΉ, ΡΠ²Π»ΡΠ΅ΡΡΡ Π°Π½Π°Π»ΠΈΠ· ΠΊΡΠΈΠΏΡΠΎΡΡΠΎΠΉΠΊΠΎΡΡΠΈ Π³ΠΎΠΌΠΎΠΌΠΎΡΡΠ½ΡΡ
ΡΠΈΡΡΠΎΠ². ΠΠ°Π½Π½Π°Ρ ΡΡΠ°ΡΡΡ ΠΏΠΎΡΠ²ΡΡΠ΅Π½Π° ΠΈΠ·ΡΡΠ΅Π½ΠΈΡ Π²ΠΎΠΏΡΠΎΡΠ° ΠΎ Π·Π°ΡΠΈΡΠ΅Π½Π½ΠΎΡΡΠΈ Π΄Π²ΡΡ
Π½Π΅Π΄Π°Π²Π½ΠΎ ΠΏΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½Π½ΡΡ
Π³ΠΎΠΌΠΎΠΌΠΎΡΡΠ½ΡΡ
ΠΊΡΠΈΠΏΡΠΎΡΠΈΡΡΠ΅ΠΌ, ΠΊΠΎΡΠΎΡΡΠ΅, Π² ΡΠ²ΡΠ·ΠΈ Ρ ΠΈΡ
Π²ΡΡΠΎΠΊΠΎΠΉ Π²ΡΡΠΈΡΠ»ΠΈΡΠ΅Π»ΡΠ½ΠΎΠΉ ΡΡΡΠ΅ΠΊΡΠΈΠ²Π½ΠΎΡΡΡΡ, ΠΌΠΎΠ³ΡΡ Π±ΡΡΡ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½Ρ Π΄Π»Ρ ΡΠΈΡΡΠΎΠ²Π°Π½ΠΈΡ Π΄Π°Π½Π½ΡΡ
Π½Π° ΠΎΠ±Π»Π°ΡΠ½ΡΡ
ΡΠ΅ΡΠ²Π΅ΡΠ°Ρ
. ΠΠ±Π΅ ΠΊΡΠΈΠΏΡΠΎΡΠΈΡΡΠ΅ΠΌΡ ΠΎΡΠ½ΠΎΠ²Π°Π½Ρ Π½Π° ΡΠΈΡΡΠ΅ΠΌΠ°Ρ
ΠΎΡΡΠ°ΡΠΎΡΠ½ΡΡ
ΠΊΠ»Π°ΡΡΠΎΠ², ΡΡΠΎ ΠΏΠΎΠ·Π²ΠΎΠ»ΡΠ΅Ρ ΡΠ°ΡΡΠΌΠΎΡΡΠ΅ΡΡ ΠΈΡ
Ρ Π΅Π΄ΠΈΠ½ΡΡ
ΠΏΠΎΠ·ΠΈΡΠΈΠΉ. ΠΠΌΠ΅Π½Π½ΠΎ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΠ΅ ΡΠΈΡΡΠ΅ΠΌ ΠΎΡΡΠ°ΡΠΎΡΠ½ΡΡ
ΠΊΠ»Π°ΡΡΠΎΠ² Π΄Π΅Π»Π°Π΅Ρ ΠΏΡΠΈΠΌΠ΅Π½Π΅Π½ΠΈΠ΅ ΡΡΠΈΡ
ΠΊΡΠΈΠΏΡΠΎΡΠΈΡΡΠ΅ΠΌ Π² ΡΠ΅Π°Π»ΡΠ½ΡΡ
ΠΏΡΠΈΠ»ΠΎΠΆΠ΅Π½ΠΈΡΡ
Π·Π°ΠΌΠ°Π½ΡΠΈΠ²ΡΠΌ Ρ ΡΠΎΡΠΊΠΈ Π·ΡΠ΅Π½ΠΈΡ ΡΡΡΠ΅ΠΊΡΠΈΠ²Π½ΠΎΡΡΠΈ ΠΏΠΎ ΡΡΠ°Π²Π½Π΅Π½ΠΈΡ Ρ Π΄ΡΡΠ³ΠΈΠΌΠΈ Π³ΠΎΠΌΠΎΠΌΠΎΡΡΠ½ΡΠΌΠΈ ΡΠΈΡΡΠ°ΠΌΠΈ, ΡΠ°ΠΊ ΠΊΠ°ΠΊ ΠΏΠΎΡΠ²Π»ΡΠ΅ΡΡΡ Π²ΠΎΠ·ΠΌΠΎΠΆΠ½ΠΎΡΡΡ Π»Π΅Π³ΠΊΠΎ ΡΠ°ΡΠΏΠ°ΡΠ°Π»Π»Π΅Π»ΠΈΡΡ Π²ΡΡΠΈΡΠ»Π΅Π½ΠΈΡ. ΠΠ΄Π½Π°ΠΊΠΎ ΠΈΡ
ΠΊΡΠΈΠΏΡΠΎΡΡΠΎΠΉΠΊΠΎΡΡΡ Π½Π΅ Π±ΡΠ»Π° Π² Π΄ΠΎΡΡΠ°ΡΠΎΡΠ½ΠΎΠΉ ΠΌΠ΅ΡΠ΅ ΠΈΠ·ΡΡΠ΅Π½Π° Π² Π»ΠΈΡΠ΅ΡΠ°ΡΡΡΠ΅ ΠΈ Π½ΡΠΆΠ΄Π°Π΅ΡΡΡ Π² Π°Π½Π°Π»ΠΈΠ·Π΅.
ΠΡΠΌΠ΅ΡΠΈΠΌ, ΡΡΠΎ ΡΠ°Π½Π΅Π΅ ΠΏΡΠ΅Π΄ΡΠ΅ΡΡΠ²Π΅Π½Π½ΠΈΠΊΠ°ΠΌΠΈ Π±ΡΠ»Π° ΡΠ°ΡΡΠΌΠΎΡΡΠ΅Π½Π° ΠΊΡΠΈΠΏΡΠΎΡΠΈΡΡΠ΅ΠΌΠ° ΠΏΠΎΡ
ΠΎΠΆΠ°Ρ Π½Π° ΠΎΠ΄ΠΈΠ½ ΠΈΠ· ΡΠΈΡΡΠΎΠ², ΠΊΡΠΈΠΏΡΠΎΡΡΠΎΠΉΠΊΠΎΡΡΡ ΠΊΠΎΡΠΎΡΠΎΠ³ΠΎ ΠΈΡΡΠ»Π΅Π΄ΡΠ΅ΡΡΡ. ΠΡΠ»Π° ΠΏΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½Π° ΠΈΠ΄Π΅Ρ Π°Π΄Π°ΠΏΡΠΈΠ²Π½ΠΎΠΉ Π°ΡΠ°ΠΊΠΈ ΠΏΠΎ Π²ΡΠ±ΡΠ°Π½Π½ΡΠΌ ΠΎΡΠΊΡΡΡΡΠΌ ΡΠ΅ΠΊΡΡΠ°ΠΌ Π½Π° ΡΡΡ ΠΊΠΎΠ½ΡΡΡΡΠΊΡΠΈΡ ΠΈ Π΄Π°Π½Π° ΠΎΡΠ΅Π½ΠΊΠ° Π½Π΅ΠΎΠ±Ρ
ΠΎΠ΄ΠΈΠΌΠΎΠ³ΠΎ Π΄Π»Ρ ΡΠ°ΡΠΊΡΡΡΠΈΡ ΠΊΠ»ΡΡΠ° ΠΊΠΎΠ»ΠΈΡΠ΅ΡΡΠ²Π° ΠΏΠ°Ρ >. ΠΠ΄Π΅ΡΡ ΠΏΡΠΎΠ²ΠΎΠ΄ΠΈΡΡΡ Π°Π½Π°Π»ΠΈΠ· ΡΡΠΎΠΉ Π°ΡΠ°ΠΊΠΈ ΠΈ ΠΏΠΎΠΊΠ°Π·ΡΠ²Π°Π΅ΠΌ, ΡΡΠΎ ΠΈΠ½ΠΎΠ³Π΄Π° ΠΎΠ½Π° ΠΌΠΎΠΆΠ΅Ρ ΡΠ°Π±ΠΎΡΠ°ΡΡ Π½Π΅ΠΊΠΎΡΡΠ΅ΠΊΡΠ½ΠΎ. Π’Π°ΠΊΠΆΠ΅ ΠΎΠΏΠΈΡΡΠ²Π°Π΅ΡΡΡ Π±ΠΎΠ»Π΅Π΅ ΠΎΠ±ΡΠΈΠΉ Π°Π»Π³ΠΎΡΠΈΡΠΌ Π°ΡΠ°ΠΊΠΈ Ρ ΠΈΠ·Π²Π΅ΡΡΠ½ΡΠΌΠΈ ΠΎΡΠΊΡΡΡΡΠΌΠΈ ΡΠ΅ΠΊΡΡΠ°ΠΌΠΈ. ΠΡΠΈΠ²ΠΎΠ΄ΡΡΡΡ ΡΠ΅ΠΎΡΠ΅ΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΠΎΡΠ΅Π½ΠΊΠΈ Π²Π΅ΡΠΎΡΡΠ½ΠΎΡΡΠΈ ΡΡΠΏΠ΅ΡΠ½ΠΎΠ³ΠΎ ΡΠ°ΡΠΊΡΡΡΠΈΡ ΡΠ΅ΠΊΡΠ΅ΡΠ½ΠΎΠ³ΠΎ ΠΊΠ»ΡΡΠ° Ρ Π΅Π³ΠΎ ΠΏΠΎΠΌΠΎΡΡΡ ΠΈ ΠΏΡΠ°ΠΊΡΠΈΡΠ΅ΡΠΊΠΈΠ΅ ΠΎΡΠ΅Π½ΠΊΠΈ ΡΡΠΎΠΉ Π²Π΅ΡΠΎΡΡΠ½ΠΎΡΡΠΈ, ΠΏΠΎΠ»ΡΡΠ΅Π½Π½ΡΠ΅ Π² Ρ
ΠΎΠ΄Π΅ Π²ΡΡΠΈΡΠ»ΠΈΡΠ΅Π»ΡΠ½ΠΎΠ³ΠΎ ΡΠΊΡΠΏΠ΅ΡΠΈΠΌΠ΅Π½ΡΠ°.
ΠΠ°ΡΠΈΡΠ΅Π½Π½ΠΎΡΡΡ Π²ΡΠΎΡΠΎΠΉ ΠΊΡΠΈΠΏΡΠΎΡΠΈΡΡΠ΅ΠΌΡ Π½Π΅ Π±ΡΠ»Π° ΠΈΡΡΠ»Π΅Π΄ΠΎΠ²Π°Π½Π° ΡΠ°Π½Π΅Π΅ Π² Π»ΠΈΡΠ΅ΡΠ°ΡΡΡΠ΅. ΠΠ·ΡΡΠ΅Π½Π° Π΅Ρ ΡΡΠΎΠΉΠΊΠΎΡΡΡ ΠΊ Π°ΡΠ°ΠΊΠ΅ Ρ ΠΈΠ·Π²Π΅ΡΡΠ½ΡΠΌΠΈ ΠΎΡΠΊΡΡΡΡΠΌΠΈ ΡΠ΅ΠΊΡΡΠ°ΠΌΠΈ. ΠΡΠΎΠ°Π½Π°Π»ΠΈΠ·ΠΈΡΠΎΠ²Π°Π½Π° Π·Π°Π²ΠΈΡΠΈΠΌΠΎΡΡΡ Π½Π΅ΠΎΠ±Ρ
ΠΎΠ΄ΠΈΠΌΠΎΠ³ΠΎ Π΄Π»Ρ Π²Π·Π»ΠΎΠΌΠ° ΠΊΠΎΠ»ΠΈΡΠ΅ΡΡΠ²Π° ΠΏΠ°Ρ > ΠΎΡ ΠΏΠ°ΡΠ°ΠΌΠ΅ΡΡΠΎΠ² ΠΊΡΠΈΠΏΡΠΎΡΠΈΡΡΠ΅ΠΌΡ ΠΈ Π΄Π°Π½Ρ ΡΠ΅ΠΊΠΎΠΌΠ΅Π½Π΄Π°ΡΠΈΠΈ, ΠΊΠΎΡΠΎΡΡΠ΅ ΠΌΠΎΠ³ΡΡ ΠΏΠΎΠΌΠΎΡΡ ΡΠ»ΡΡΡΠΈΡΡ ΠΊΡΠΈΠΏΡΠΎΡΡΠΎΠΉΠΊΠΎΡΡΡ.
ΠΡΠΎΠ³ ΠΏΡΠΎΠ²Π΅Π΄Π΅Π½Π½ΠΎΠ³ΠΎ Π°Π½Π°Π»ΠΈΠ·Π° Π·Π°ΠΊΠ»ΡΡΠ°Π΅ΡΡΡ Π² ΡΠΎΠΌ, ΡΡΠΎ ΠΎΠ±Π΅ ΠΊΡΠΈΠΏΡΠΎΡΠΈΡΡΠ΅ΠΌΡ ΡΠ²Π»ΡΡΡΡΡ ΡΡΠ·Π²ΠΈΠΌΡΠΌΠΈ ΠΊ Π°ΡΠ°ΠΊΠ΅ Ρ ΠΈΠ·Π²Π΅ΡΡΠ½ΡΠΌΠΈ ΠΎΡΠΊΡΡΡΡΠΌΠΈ ΡΠ΅ΠΊΡΡΠ°ΠΌΠΈ. ΠΠΎΡΡΠΎΠΌΡ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°ΡΡ ΠΈΡ
Π΄Π»Ρ ΡΠΈΡΡΠΎΠ²Π°Π½ΠΈΡ ΠΊΠΎΠ½ΡΠΈΠ΄Π΅Π½ΡΠΈΠ°Π»ΡΠ½ΡΡ
Π΄Π°Π½Π½ΡΡ
ΠΌΠΎΠΆΠ΅Ρ Π±ΡΡΡ Π½Π΅Π±Π΅Π·ΠΎΠΏΠ°ΡΠ½ΠΎ.
ΠΡΠ½ΠΎΠ²Π½ΡΠΌ Π°Π»Π³ΠΎΡΠΈΡΠΌΠΎΠΌ, ΠΈΡΠΏΠΎΠ»ΡΠ·ΡΠ΅ΠΌΡΠΌ Π² ΠΏΡΠ΅Π΄Π»ΠΎΠΆΠ΅Π½Π½ΡΡ
Π°ΡΠ°ΠΊΠ°Ρ
Π½Π° ΠΊΡΠΈΠΏΡΠΎΡΠΈΡΡΠ΅ΠΌΡ, ΡΠ²Π»ΡΠ΅ΡΡΡ Π°Π»Π³ΠΎΡΠΈΡΠΌ ΠΏΠΎΠΈΡΠΊΠ° Π½Π°ΠΈΠ±ΠΎΠ»ΡΡΠ΅Π³ΠΎ ΠΎΠ±ΡΠ΅Π³ΠΎ Π΄Π΅Π»ΠΈΡΠ΅Π»Ρ. ΠΠ°ΠΊ ΡΠ»Π΅Π΄ΡΡΠ²ΠΈΠ΅, Π²ΡΠ΅ΠΌΡ, Π½Π΅ΠΎΠ±Ρ
ΠΎΠ΄ΠΈΠΌΠΎΠ΅ Π΄Π»Ρ ΡΠ΅Π°Π»ΠΈΠ·Π°ΡΠΈΠΈ Π°ΡΠ°ΠΊ, ΡΠ²Π»ΡΠ΅ΡΡΡ ΠΏΠΎΠ»ΠΈΠ½ΠΎΠΌΠΈΠ°Π»ΡΠ½ΡΠΌ ΠΎΡ ΡΠ°Π·ΠΌΠ΅ΡΠ° Π²Ρ
ΠΎΠ΄Π½ΡΡ
Π΄Π°Π½Π½ΡΡ
Improved fully homomorphic public-key encryption with small ciphertext size
A cryptosystem which supports both addition and multiplication (thereby preserving the ring structure of the plaintexts) is known as fully homomorphic encryption (FHE) and is very powerful. Using such a scheme, any circuit can be homomorphically evaluated, effectively allowing the construction of programs which may be run on ciphertexts of their inputs to produce a ciphertext of their output. Since such a program never decrypts its input, it can be run by an untrusted party without revealing its inputs and internal state. The existence of an efficient and fully homomorphic cryptosystem would have great practical implications in the outsourcing of private computations, for instance, in the context of cloud computing. In previous work I proposed the fully homomorphic public-key encryption scheme with the size of ciphertext which is not small enough. In this paper the size of ciphertext is one-eighth of the size in the previously proposed scheme. Because proposed scheme adopts the medium text with zero norm, it is immune from the the βp and -p attackβ. As the proposed scheme is based on computational difficulty to solve the multivariate algebraic equations of high degree, it is immune from the GrΓΆbner basis attack, the differential attack, rank attack and so on
Improved Fully Homomorphic Encryption without Bootstrapping
Gentryβs bootstrapping technique is the most famous method of obtaining fully homomorphic encryption. In previous work I proposed a fully homomorphic encryption without bootstrapping which has the weak point in the enciphering function. In this paper I propose the improved fully homomorphic public-key encryption scheme on non-associative octonion ring over finite field without bootstrapping technique. The plaintext p consists of two sub-plaintext u and v. The proposed fully homomorphic public-key encryption scheme is immune from the βp and -p attackβ. The cipher text consists of three sub-cipher texts. As the scheme is based on computational difficulty to solve the multivariate algebraic equations of high degree while the almost all multivariate cryptosystems proposed until now are based on the quadratic equations avoiding the explosion of the coefficients. Because proposed fully homomorphic encryption scheme is based on multivariate algebraic equations with high degree or too many variables, it is against the GrΓΆbner basis attack, the differential attack, rank attack and so on