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Inoculating multivariate schemes against differential attacks
 In PKC, volume 3958 of LNCS
, 2006
"... Abstract. We demonstrate how to prevent differential attacks on multivariate public key cryptosystems using the Plus (+) method of external perturbation. In particular, we prescribe adding as few as 10 Plus polynomials to the Perturbed MatsumotoImai (PMI) cryptosystem when g =1andr =6,whereθ is the ..."
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Abstract. We demonstrate how to prevent differential attacks on multivariate public key cryptosystems using the Plus (+) method of external perturbation. In particular, we prescribe adding as few as 10 Plus polynomials to the Perturbed MatsumotoImai (PMI) cryptosystem when g =1andr =6,whereθ is the MatsumotoImai exponent, n is the message length, g =gcd(θ, n), and r is the internal perturbation dimension; or as few as g +10 when g � = 1. The external perturbation does not significantly decrease the efficiency of the system, and in fact has the additional benefit of resolving the problem of finding the true plaintext among several preimages of a given ciphertext. We call this new scheme the Perturbed MatsumotoImaiPlus (PMI+) cryptosystem. Keywords: multivariate, public key, cryptography, MatsumotoImai, perturbation, plus, differential.
Multivariate public key cryptography
, 2009
"... A multivariate public key cryptosystem (MPKCs for short) have a set of (usually) quadratic polynomials over a finite field as its public map. Its main security assumption is backed by the NPhardness of the problem to solve nonlinear equations over a nite eld. This family is considered as one of th ..."
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A multivariate public key cryptosystem (MPKCs for short) have a set of (usually) quadratic polynomials over a finite field as its public map. Its main security assumption is backed by the NPhardness of the problem to solve nonlinear equations over a nite eld. This family is considered as one of the major families of PKCs that could resist potentially even the powerful quantum computers of the future. There has been fast and intensive development in Multivariate Public Key Cryptography in the last two decades. Some constructions are not as secure as was claimed initially, but others are still viable. The paper gives an overview of multivariate public key cryptography and discusses the current status of the research in this area.
Cryptanalysis of HFE with Internal Perturbation
 In Public Key Cryptography
, 2007
"... Abstract. Multivariate Cryptography has been an active line of research for almost twenty years. While most multivariate cryptosystems have been under attack, variations of the basic schemes came up as potential repairs. In this paper, we study the Internal Perturbation variation of HFE recently pro ..."
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Abstract. Multivariate Cryptography has been an active line of research for almost twenty years. While most multivariate cryptosystems have been under attack, variations of the basic schemes came up as potential repairs. In this paper, we study the Internal Perturbation variation of HFE recently proposed by Ding and Schmidt. Although several results indicate that HFE is vulnerable against algebraic attacks for moderate size parameters, Ding and Schmidt claim that the cryptosystem with internal perturbation should be immune against them. However in this paper, we apply the recently discovered method of differential analysis to the Internal Perturbation of HFE and we find a subtle property which allows to disclose the kernel of the perturbation. Once this has been achieved, the public key can be inverted by attacking the underlying HFE provided the parameters were taken low enough to make the perturbed scheme of competitive performance.
ℓInvertible Cycles for Multivariate Quadratic (MQ) Public Key Cryptography
"... Abstract. We propose a new basic trapdoor ℓIC (ℓInvertible Cycles) of the mixed field type for Multivariate Quadratic public key cryptosystems. This is the first new basic trapdoor since the invention of Unbalanced Oil and Vinegar in 1997. ℓICcanbeconsideredanextended form of the wellknown Matsumo ..."
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Abstract. We propose a new basic trapdoor ℓIC (ℓInvertible Cycles) of the mixed field type for Multivariate Quadratic public key cryptosystems. This is the first new basic trapdoor since the invention of Unbalanced Oil and Vinegar in 1997. ℓICcanbeconsideredanextended form of the wellknown MatsumotoImai Scheme A (also MIA or C ∗), and share some features of stagewise triangular systems. However ℓIC has very distinctive properties of its own. In practice, ℓIC is much faster than MIA, and can even match the speed of singlefield MQ schemes.
Nonlinear Piece In Hand Matrix Method for Enhancing Security of Multivariate Public Key Cryptosystems
"... Abstract. It is widely believed to take exponential time to find a solution of a system of random multivariate polynomials because of the NPcompleteness of such a task. On the other hand, in most of multivariate public key cryptosystems proposed so far, the computational complexity of cryptanalysis ..."
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Abstract. It is widely believed to take exponential time to find a solution of a system of random multivariate polynomials because of the NPcompleteness of such a task. On the other hand, in most of multivariate public key cryptosystems proposed so far, the computational complexity of cryptanalysis is apt to be polynomial time due to the trapdoor structure. In this paper, we develop the concept, piece in hand matrix (PH matrix, for short), which aims to bring the computational complexity of cryptanalysis of multivariate public key cryptosystems close to exponential time by adding random polynomial terms to original cryptosystems. This is a general concept which can be applicable to any reasonable type of multivariate public key cryptosystems for the purpose of enhancing their security. There are two types of the PH matrices: a linear matrix whose elements are constants and a nonlinear matrix whose elements are polynomial functions of the plain text or random numbers. In the present paper, we focus our thought on the nonlinear PH matrix method and develop the framework of it. The nonlinear PH matrix method is obtained by generalizing the linear PH matrix method, and the nonlinearity may bring an additional randomization to the original linear PH matrix method. Thus, the nonlinear PH matrix method may enhance the security of the original multivariate public key cryptosystem more than the linear PH matrix method. We show, in an experimental manner, that this actually holds in the enhancement of the security of the MatsumotoImai cryptosystem and RSE(2)PKC against the Gröbner basis attack. Key words: public key cryptosystem, multivariate polynomial, multivariate public key cryptosystem, piece in hand concept, nonlinear matrix 1
Nonlinear piece in hand perturbation vector method for enhancing security of multivariate public key cryptosystems
 Proc. PQCrypto 2008, Lecture
"... Abstract. The piece in hand (PH) is a general scheme which is applicable to any reasonable type of multivariate public key cryptosystems for the purpose of enhancing their security. In this paper, we propose a new class PH method called NLPHPV (NonLinear Piece in Hand Perturbation Vector) method. Al ..."
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Abstract. The piece in hand (PH) is a general scheme which is applicable to any reasonable type of multivariate public key cryptosystems for the purpose of enhancing their security. In this paper, we propose a new class PH method called NLPHPV (NonLinear Piece in Hand Perturbation Vector) method. Although our NLPHPV uses similar perturbation vectors as is used for the previously known internal perturbation method, this new method can avoid redundant repetitions in decryption process. With properly chosen parameter sizes, NLPHPV achieves an observable gain in security from the original multivariate public key cryptosystem. We demonstrate these by both theoretical analyses and computer simulations against major known attacks and provides the concrete sizes of security parameters, with which we even expect the grater security against potential quantum attacks. Key words: public key cryptosystem, multivariate polynomial, multivariate public key cryptosystem, piece in hand concept, perturbation vector 1
Proposal of PPS Multivariate Public Key Cryptosystems
"... Abstract. In this paper we propose a new MPKC, called PPS, based on (i) the 2layer nonlinear piece in hand method, (ii) PMI, and (iii) STS. The PPS is a specific MPKC obtained by applying the 2layer nonlinear piece in hand method to STS, in the manner that the rank and randomness of the lower rank ..."
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Abstract. In this paper we propose a new MPKC, called PPS, based on (i) the 2layer nonlinear piece in hand method, (ii) PMI, and (iii) STS. The PPS is a specific MPKC obtained by applying the 2layer nonlinear piece in hand method to STS, in the manner that the rank and randomness of the lower rank steps in the original secret polynomial vector of STS are enhanced by adding a perturbation polynomial vector and moreover PMI is used in the auxiliary part. The PPS overcomes the drawbacks of the three schemes by the advantage of the three schemes themself. Thus, PPS can be thought to be immune simultaneously from the algebraic attacks, such as the Gröbner bases attacks, from the rank attacks, and from the differential attacks.