TY - GEN

T1 - An estimate of the complexity of the section finding problem on algebraic surfaces

AU - Okumura, Shinya

AU - Akiyama, Koichiro

AU - Takagi, Tsuyoshi

N1 - Publisher Copyright:
© 2016 IEEE.

PY - 2017/1/13

Y1 - 2017/1/13

N2 - Researching post-quantum cryptography has been an important task in cryptography. The section finding problem on algebraic surfaces (AS-SFP) is considered to be intractable also after building quantum computers. Thus AS-SFP is used as a basis of the security of the Algebraic Surface Cryptosystem (ASC), which is a candidate of post-quantum cryptosystems, and it is important for designing parameters which make ASC secure to estimate the complexity of AS-SFP. Solving AS-SFP is reduced to solving certain multivariate equation systems (section equation systems) of high degrees, and one can solve such equation systems by using the Gröbner basis technique. Although estimating the complexity of computing a Gröbner basis associated with an equation system is difficult in general, it becomes easy if the equation system is semi-regular. In this paper, we experimentally estimate the complexity of AS-SFP. From our experimental results, although we see that section equation systems do not become semi-regular in most cases for small parameters, we can infer parameters closely related to the difficulty of computing Gröbner bases associated with section equation systems. According to our inference, we estimate the complexity of AS-SFP and parameters which make ASC 128-bit security against the attack by the Gröbner basis technique. We also consider a brute-force attack against ASSFP and conjecture that the brute-force attack is more efficient than the attack by the Gröbner basis technique. Finally, we estimate parameters and sizes of public keys such that ASC has 128-bit security against the brute-force attack. Its size (876 bits) is much smaller than sizes of public keys in other efficient candidates of PQC.

AB - Researching post-quantum cryptography has been an important task in cryptography. The section finding problem on algebraic surfaces (AS-SFP) is considered to be intractable also after building quantum computers. Thus AS-SFP is used as a basis of the security of the Algebraic Surface Cryptosystem (ASC), which is a candidate of post-quantum cryptosystems, and it is important for designing parameters which make ASC secure to estimate the complexity of AS-SFP. Solving AS-SFP is reduced to solving certain multivariate equation systems (section equation systems) of high degrees, and one can solve such equation systems by using the Gröbner basis technique. Although estimating the complexity of computing a Gröbner basis associated with an equation system is difficult in general, it becomes easy if the equation system is semi-regular. In this paper, we experimentally estimate the complexity of AS-SFP. From our experimental results, although we see that section equation systems do not become semi-regular in most cases for small parameters, we can infer parameters closely related to the difficulty of computing Gröbner bases associated with section equation systems. According to our inference, we estimate the complexity of AS-SFP and parameters which make ASC 128-bit security against the attack by the Gröbner basis technique. We also consider a brute-force attack against ASSFP and conjecture that the brute-force attack is more efficient than the attack by the Gröbner basis technique. Finally, we estimate parameters and sizes of public keys such that ASC has 128-bit security against the brute-force attack. Its size (876 bits) is much smaller than sizes of public keys in other efficient candidates of PQC.

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U2 - 10.1109/CANDAR.2016.51

DO - 10.1109/CANDAR.2016.51

M3 - Conference contribution

AN - SCOPUS:85015169906

T3 - Proceedings - 2016 4th International Symposium on Computing and Networking, CANDAR 2016

SP - 28

EP - 36

BT - Proceedings - 2016 4th International Symposium on Computing and Networking, CANDAR 2016

PB - Institute of Electrical and Electronics Engineers Inc.

T2 - 4th International Symposium on Computing and Networking, CANDAR 2016

Y2 - 22 November 2016 through 25 November 2016

ER -