Finite Field Theory and Its applications

Many of modern ciphers are supported by discrete mathematics such as integer theory and finite field (extension field) theory, as typified by RSA cryptography and ElGamal cryptography. In particular, in order to secure cryptographic security, we utilize four arithmetic operations in a vector space with very large integers and very many elements. In order to make it safe, secure and comfortable for ubiquitous mobile terminals such as smartphones and IC cards, we aim to achieve the fastest and lightest world in the world, realize high-speed calculation processing and compact program implementation. We develop algorithms and provide theoretical support and proof for that. Based on these four arithmetic operations, public key cryptography and secret key cryptography introduced below will be calculated and implemented at a high speed and compactly.

IoT (Internet of Things) / IoE (Internet of Everything)

IoT (Internet of things) / IoE (Internet of Everything) era has come. Everything will be connected to the Internet and various types of information are transmitted between not only computers but also small devices. Of course, it includes very sensitive and private information such as ID, birthday, credit card number, and so on. PC can efficiently and securely carry out encryption and decryption for the secure transmission; however, it is very heavy for small devices such as IC card and microcontrollers, namely IoT devices. In the Internet of Things (IoT) paradigm, many of the objects that surround us will be connected to each other on the network that is why the importance of security becoming the most crucial concern. Generally, most of the people think that only the cyber attack (such as virus attacks, malware ) can happen through the internet. On the other hand, in the IoT era, we have to consider the physical attack along with the cyber attack. Our target hardwares and cryptosystems are Raspberry Pi, Arduino, FPGA, and ECC (especially Montgomery curve), AES, respectively.

Error Correcting Code

To achieve a highly reliable digital communication system over a noisy channel, an Error Correcting Code can be used to control errors on transmitted information symbols. Linear codes can be used for the purpose with effective decoding algorithms. Since a class of linear codes has special structures which can be effectively used to implement the optimum or sub-optimum soft-decision decoding algorithms, research on decoding algorithms are available. Especially, we are working on research on efficient soft-decision decoding algorithms, which achieve the optimum or sub-optimum error performance with small average computational complexity.

Elliptic Curve Discrete Logarithm Problem (ECDLP)

Currently, security is necessary for us to protect our personal information from a vicious attacker. Recently, Elliptic Curve Cryptography(ECC) is attracting attention as a stronger security than before. The security of ECC is based on the computational difficulty of the Elliptic Curve Discrete Logarithm Problem(ECDLP). It depends on the computer's performance and the number of computers that can be parallelized. As an example, 112-bit ECDLP was actually solved by a cluster of more than 200 PlayStation 3 game consoles for half a year and it is the largest size of ECDLP ever solved. The threat of parallel attacks has increased because advanced information and communication technologies facilitate parallel computing through the internet. It is crucially important to verify the practical security against parallel attacks. Therefore, We aim to evaluate the security of ECDLP by attacking to it actually. Now, we attack to the 114-bit ECDLP by about 200 computers (starBED) which provided from NICT.

Elliptic Curve Cryptography

Elliptic Curve Cryptography(ECC) is a public-key cryptography based on the algebraic structure of elliptic curves over finite fields. Elliptic curves are applicable elliptic curve Diffie-Hellman (ECDH) for key exchange, Elliptic Curve Digital Signature Algorithm (ECDSA) for digital signature and pseudo-random generators. ECC primarily focuses on the efficiency of attacks with the structure of rational points group and speeding up elliptic curve addition. We consider an efficient implementation of ECC over Barreto-Naehrig curves, Bernstein curves, and other curves.

Pairing-based Cryptography

Pairings on elliptic curves is a relatively new and active area of research in cryptography which is often known as Pairing-based Crypto (PBC). By using some certain mathematics, pairing maps a pair of points on an elliptic curve into the multiplicative group of a finite field. Such technique yields several new cryptographic protocols that had not previously been feasible.One of the widely known protocols realized by PBC is identity-based encryption (IBE), which overcomes the need of knowing the receivers public key in prior of sending the secure message. Another innovative application of PBC is functional encryption which allows anyone who possesses a particular set of attributes defined during encryption stage, can decrypt the message. It is also known as attribute-based encryption (ABE). More and more novel ideas are in the pipeline from the academia and industry who are involved in research of PBC. Therefore we can say, pairing-based cryptography is now in the frontline of next generation of security.

Lattice-based Cryptography

The time for replacing the conventional computational system with the quantum computer is coming in the near future. It is known that the Shor's algorithm with quantum computer enables us to solve the mathematical hardness such as prime factorization and Discrete Logarithm Problem. In other words, the conventional public key cryptographies, for example, RSA and Elliptic Curve Cryptography, will be broken by the quantum computer in polynomial time. In this context, a lattice-based cryptography called NTRU has been paid much attentions as post-quantum cryptography. It is constructed on the shortest vector problems and is able to encrypt data much faster than RSA. In addition, it can use for making a searchable cryptosystem which allows us to find the data without decryption. Such kinds of mathematical background and applications are investigated in our lab.