FHPKE based on multivariate discrete logarithm problem

Previously I proposed fully homomorphic public-key encryption (FHPKE) based on discrete logarithm problem which is vulnerable to quantum computer attacks. In this paper I propose FHPKE based on multivariate discrete logarithm assumption. This encryption scheme is thought to withstand to quantum computer attacks. Though I can construct this scheme over many non-commutative rings, I will adopt the FHPKE scheme based on the octonion ring as the typical example for showing how this scheme is constructed. The multivariate discrete logarithm problem (MDLP) is defined such that given f(x), g(x), h(x) and a prime q, final goal is to find m0, m1, n0, n1∈Fq* where h(x)=f ^m0(g^n0(x))+f ^m1(g^n1(x)) mod q over octonion ring

Key distribution system and attribute-based encryption

I propose the new key distribution system and attribute-based encryption scheme on non-commutative ring where the complexity required for enciphering and deciphering is small. As in this system encryption keys and decryption keys involve the attributes of each user, the system is adaptive for cloud computing systems. The security of this system is based on the complexity for solving the multivariate algebraic equations of high degree over finite field, that is, one of NP complete problems. So this system is immune from the Gröbner basis attacks. The key size of this system becomes to be small enough to handle

A Digital Signature Using Multivariate Functions on Quaternion Ring

We propose the digital signature scheme on non-commutative quaternion ring over finite fields in this paper. We generate the multivariate function of high degree F(X) . We construct the digital signature scheme using F(X). Our system is immune from the Gröbner bases attacks because obtaining parameters of F(X) to be secret keys arrives at solving the multivariate algebraic equations that is one of NP complete problems

Key Agreement Protocols Using Multivariate Equations on Non-commutative Ring

In this paper we propose two KAP(key agreement protocols) using multivariate equations. As the enciphering functions we select the multivariate functions of high degree on non-commutative ring H over finite field Fq. Two enciphering functions are slightly different from the enciphering function previously proposed by the present author. In proposed systems we can adopt not only the quaternion ring but also the non-associative octonion ring as the basic ring. Common keys are generated by using the enciphering functions. Proposed systems are immune from the Gröbner bases attacks because obtaining parameters of the enciphering functions to be secret keys arrives at solving the multivariate algebraic equations, that is, one of NP complete problems .Our protocols are also thought to be immune from the differential attacks because of the equations of high degree. We can construct our system on the some non-commutative rings, for example quaternion ring, matrix ring or octonion ring

FHPKE with Zero Norm Noises based on DLA&CDH

In this paper I propose the fully homomorphic public-key encryption(FHPKE) scheme with zero norm noises that is based on the discrete logarithm assumption(DLA) and computational Diffie-Hellman assumption(CDH) of multivariate polynomials on octonion ring. Since the complexity for enciphering and deciphering become to be small enough to handle, the cryptosystem runs fast

Fully Homomorphic Public-Key Encryption with Two Ciphertexts based on Discrete Logarithm Problem

In previous paper I proposed the fully homomorphic public-key encryption based on discrete logarithm problem which may be vulnerable to “m and -m attack”. In this paper I propose improved fully homomorphic public-key encryption (FHPKE) with composite number modulus based on the discrete logarithm assumption (DLA) and computational Diffie–Hellman assumption (CDH) of multivariate polynomials on octonion ring which is immune from “m and -m attack”. The scheme has two ciphertexts corresponding to one plaintext

Improved Fully Homomorphic Encryption with Composite Number Modulus

Gentry’s 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 plaintext. I also proposed a fully homomorphic encryption with composite number modulus which avoids the weak point by adopting the plaintext including the random numbers in it. In this paper I propose another fully homomorphic encryption with composite number modulus where the complexity required for enciphering and deciphering is smaller than the same modulus RSA scheme. In the proposed scheme it is proved that if there exists the PPT algorithm that decrypts the plaintext from the any ciphertexts of the proposed scheme, there exists the PPT algorithm that factors the given composite number modulus. In addition it is said that the proposed fully homomorphic encryption scheme is immune from the “p and -p attack”. Since 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 Gröbner basis attack, the differential attack, rank attack and so on

Fully Homomorphic Encryption with Zero Norm Cipher Text

Gentry’s 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 plaintext. I also proposed fully homomorphic encryptions with composite number modulus which avoid the weak point by adopting the plaintext including the random numbers in it. In this paper I propose another fully homomorphic encryption with zero norm cipher text where zero norm medium text is generated and enciphered by using composite number modulus. In the proposed scheme it is proved that if there exists the PPT algorithm that generates the cipher text of the plaintext -p from the cipher text of any plaintext p, there exists the PPT algorithm that factors the given composite number modulus. That is, we include the random parameter in the plaintext so that if the random parameter and the plaintext are separated, then the composite number to be the modulus is factored. Since 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 Gröbner basis attack, the differential attack, rank attack and so on

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 “p and -p attack”. 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 Gröbner basis attack. The key size of this scheme and complexity for enciphering /deciphering become to be small enough to handle

Key Agreement Protocols Based on Multivariate Algebraic Equations on Quaternion Ring

In this paper we propose new key agreement protocols based on multivariate algebraic equations. We choose the multivariate function F(X) of high degree on non-commutative quaternion ring H over finite field Fq. Common keys are generated by using the public-key F(X). Our system is immune from the Gröbner bases attacks because obtaining parameters of F(X) to be secret keys arrives at solving the multivariate algebraic equations that is one of NP complete problems .Our protocols are also thought to be immune from the differential attacks and the rank attacks