Multivariate public key cryptosystem (MPKC) is one of the major post quantum cryptosystems (PQC), and the National Institute of Standards and Technology (NIST) recently selected four MPKCs as candidates of their PQC. The security of MPKC depends on the hardness of solving systems of algebraic equations over finite fields. In particular, the multivariate quadratic (MQ) problem is that of solving such a system consisting of quadratic polynomials and is regarded as an important research subject in cryptography. In the Fukuoka MQ challenge project, the hardness of the MQ problem is discussed, and algorithms for solving the MQ problem and the computational results obtained by these algorithms are reported. Algorithms for computing Gröbner basis are used as the main tools for solving the MQ problem. For example, the F4 algorithm and M4GB algorithm have succeeded in solving many instances of the MQ problem provided by the project. In this paper, based on the F4-style algorithm, we present an efficient algorithm to solve the MQ problems with dense polynomials generated in the Fukuoka MQ challenge project. We experimentally show that our algorithm requires less computational time and memory for these MQ problems than the F4 algorithm and M4GB algorithm. We succeeded in solving Type II and III problems of Fukuoka MQ challenge using our algorithm when the number of variables was 37 in both problems.

In PQCrypto 2013, Yasuda, Takagi and Sakurai proposed a new signature scheme as one of multivariate public key cryptosystems (MPKCs). This scheme (called YTS) is based on the fact that there are two isometry classes of non-degenerate quadratic forms on a vector space with a prescribed dimension. The advantage of YTS is its efficiency. In fact, its signature generation is eight or nine times faster than Rainbow of similar size. For the security, it is known that the direct attack, the IP attack and the min-rank attack are applicable on YTS, and the running times are exponential time for the first and the second attacks and sub-exponential time for the third attack. In the present paper, we give a new attack on YTS whose approach is to use the diagonalization of matrices. Our attack works in polynomial time and it actually recovers equivalent secret keys of YTS having 140-bits security against min-rank attack in around fifteen seconds.

It is well known that solving randomly chosen Multivariate Quadratic equations over a finite field (MQ-Problem) is NP-hard, and the security of Multivariate Public Key Cryptosystems (MPKCs) is based on the MQ-Problem. However, this problem can be solved efficiently when the number of unknowns n is sufficiently greater than that of equations m (This is called “Underdefined”). Indeed, the algorithm by Kipnis et al. (Eurocrypt'99) can solve the MQ-Problem over a finite field of even characteristic in a polynomial-time of n when n ≥ m(m+1). Therefore, it is important to estimate the hardness of the MQ-Problem to evaluate the security of Multivariate Public Key Cryptosystems. We propose an algorithm in this paper that can solve the MQ-Problem in a polynomial-time of n when n ≥ m(m+3)/2, which has a wider applicable range than that by Kipnis et al. We will also compare our proposed algorithm with other known algorithms. Moreover, we implemented this algorithm with Magma and solved the MQ-Problem of m=28 and n=504, and it takes 78.7 seconds on a common PC.

NTRU is a public key cryptosystem based on hard problems over lattices. In this paper, we present efficient methods for convolution product computation which is a dominant operation of NTRU. The new methods are based on the observation that repeating patterns in coefficients of an NTRU polynomial can be used for the construction of look-up tables, which is a similar approach to the sliding window methods for exponentiation. We provide efficient convolution algorithms to implement this idea, and we make a comprehensive analysis of the complexity of the new algorithms. We also give software implementations over a Pentium IV CPU, a MICAz mote, and a CUDA-based GPGPU platform. According to our analyses and experimental results, the new algorithms speed up the NTRU encryption and decryption operations by up to 41%.

Shi-Hui et al. proposed a new public key cryptosystem using ergodic binary matrices. The security of the system is derived from some assumed hard problem based on ergodic matrices over GF(2). In this note, we show that breaking this system, with a security parameter n (public key of length 4n2 bits, secret key of length 2n bits and block length of length n2 bits), is equivalent to solving a set of n4 linear equations over GF(2) which renders this system insecure for practical choices of n.

The encryption and the decryption of the product-sum type public key cryptosystems can be performed extremely fast. However, when the density is low, the cryptosystem should be broken by the low-density attack. In this paper, we propose a new class of the product-sum type public key cryptosystems based on the reduced bases, which is invulnerable to the low-density attack.