A DNA Computing System of Modular-Multiplication over Finite Field GF(2n)

The enormous parallel computing ability and high memory density of DNA computing bring potential challenges and opportunities to traditional cryptography. Finite field GF(2n) is one of the most commonly used mathematic sets for cryptography. It is an open problem that how to implement the arithmetic operations over finite field GF(2n) based on DNA computing. Existing research has the problem that the lengths of parameters in the DNA tile assembly process could not match each other strictly. This paper proposes a parallel molecular computing system to compute the modular-multiplication, an operation combining multiplication and reduction over finite field GF(2n). The multiplication and the reduction are executed simultaneously in this system. One concrete example of $1100 \cdot 1001 \ mod \ 10011$ is proposed to show the details of our tile assembly system. The time complexity of this system is Θ(n) and the space complexity is Θ(n 2). This system requires 210 types of computation tiles and 17 types of boundary tiles.

[1]  Hao Wang.  Proving theorems by pattern recognition — II , 1961 .

[2]  Paul W. K. Rothemund,et al.  Design of DNA origami , 2005, ICCAD-2005. IEEE/ACM International Conference on Computer-Aided Design, 2005..

[3]  M. Sahani,et al.  Algorithmic Self-Assembly of DNA , 2006 .

[4]  Mohammed Benaissa,et al.  A scalable hardware/software co-design for elliptic curve cryptography on PicoBlaze microcontroller , 2010, Proceedings of 2010 IEEE International Symposium on Circuits and Systems.

[5]  P. Rothemund.  Folding DNA to create nanoscale shapes and patterns , 2006, Nature.

[6]  Mohammed Benaissa,et al.  Flexible hardware/software co-design for scalable elliptic curve cryptography for low-resource applications , 2010, ASAP 2010 - 21st IEEE International Conference on Application-specific Systems, Architectures and Processors.

[7]  Victor S. Miller,et al.  Use of Elliptic Curves in Cryptography , 1985, CRYPTO.

[8]  Erik Winfree,et al.  The program-size complexity of self-assembled squares (extended abstract) , 2000, STOC '00.

[9]  N. Koblitz.  Elliptic curve cryptosystems , 1987 .

[10]  Zhen Cheng.  Computation of Multiplicative Inversion and Division in GF(2 n ) by Self-Assembly of DNA Tiles , 2012 .

[11]  Zhigang Chen,et al.  A Public-Key Cryptosystem Scheme on Conic Curves Over Zn , 2007, 2007 International Conference on Machine Learning and Cybernetics.

[12]  Limin Xiao,et al.  Parallelization of Two Arithmetic Operations over Finite Field GF(2 n ) , 2012 .

[13]  Erik Winfree,et al.  Programmable Control of Nucleation for Algorithmic Self-Assembly , 2009, SIAM J. Comput..

[14]  Keshab K. Parhi,et al.  Fast Reconfigurable Elliptic Curve Cryptography Acceleration for GF(2m) on 32 bit Processors , 2010, J. Signal Process. Syst..

[15]  R J Lipton,et al.  DNA solution of hard computational problems. , 1995, Science.

[16]  Sun-Yuan Hsieh,et al.  A DNA-based solution to the graph isomorphism problem using Adleman-Lipton model with stickers , 2008, Appl. Math. Comput..

[17]  Limin Xiao,et al.  Parallel molecular computation of modular-multiplication based on tile assembly model , 2013, ICPADS 2013.

[18]  Yuriy Brun.  Arithmetic computation in the tile assembly model: Addition and multiplication , 2007, Theor. Comput. Sci..

[19]  Seokhie Hong,et al.  TinyECCK: Efficient Elliptic Curve Cryptography Implementation over GF(2m) on 8-Bit Micaz Mote , 2008, IEICE Trans. Inf. Syst..

[20]  David I. Lewin,et al.  DNA computing , 2002, Comput. Sci. Eng..

[21]  Weng-Long Chang.  Fast Parallel DNA-Based Algorithms for Molecular Computation: The Set-Partition Problem , 2007, IEEE Transactions on NanoBioscience.

[22]  R. Blahut.  Theory and practice of error control codes , 1983 .

[23]  Majid Darehmiraki,et al.  Molecular solution to the 0-1 knapsack problem based on DNA computing , 2007, Appl. Math. Comput..

[24]  Limin Xiao,et al.  Arithmetic Computation Using Self-Assembly of DNA Tiles: Modular-Square over Finite Field GF(2n) , 2013, 2013 IEEE 10th International Conference on High Performance Computing and Communications & 2013 IEEE International Conference on Embedded and Ubiquitous Computing.

[25]  L M Adleman,et al.  Molecular computation of solutions to combinatorial problems. , 1994, Science.

[26]  Shantanu Das,et al.  Finite field arithmetic using self-assembly of DNA tilings , 2003, The 2003 Congress on Evolutionary Computation, 2003. CEC '03..

[27]  Seokhie Hong,et al.  TinyECCK16: An Efficient Field Multiplication Algorithm on 16-bit Environment and Its Application to Tmote Sky Sensor Motes , 2009, IEICE Trans. Inf. Syst..

[28]  Paul W. K. Rothemund,et al.  Beyond Watson and Crick: Programming DNA self-assembly for nanofabrication , 2012, 2012 7th IEEE International Conference on Nano/Micro Engineered and Molecular Systems (NEMS).

[29]  Aggelos Kiayias,et al.  BiTR: Built-in Tamper Resilience , 2011, IACR Cryptol. ePrint Arch..

[30]  Yuriy Brun.  Nondeterministic polynomial time factoring in the tile assembly model , 2008, Theor. Comput. Sci..

[31]  P D Kaplan,et al.  DNA solution of the maximal clique problem. , 1997, Science.