RNS Montgomery reduction algorithms using quadratic residuosity

The residue number system (RNS) is a method for representing an integer as an n-tuple of its residues with respect to a given base. Since RNS has inherent parallelism, it is actively researched to implement a faster processing system for public-key cryptography. This paper proposes new RNS Montgomery reduction algorithms, Q-RNSs, the main part of which is twice a matrix multiplication. Letting n be the size of a base set, the number of unit modular multiplications in the proposed algorithms is evaluated as $$(2n^2+n)$$(2n2+n). This is achieved by posing a new restriction on the RNS base, namely, that its elements should have a certain quadratic residuosity. This makes it possible to remove some multiplication steps from conventional algorithms, and thus the new algorithms are simpler and have higher regularity compared with conventional ones. From our experiments, it is confirmed that there are sufficient candidates for RNS bases meeting the quadratic residuosity requirements.

[1]  P. L. Montgomery.  Modular multiplication without trial division , 1985 .

[2]  Ingrid Verbauwhede,et al.  FPGA Implementation of Pairings Using Residue Number System and Lazy Reduction , 2011, CHES.

[3]  Julien Eynard,et al.  Montgomery reduction within the context of residue number system arithmetic , 2017, Journal of Cryptographic Engineering.

[4]  Laurent Imbert,et al.  a full RNS implementation of RSA , 2004, IEEE Transactions on Computers.

[5]  Arnaud Tisserand,et al.  Improving Modular Inversion in RNS using the Plus-Minus Method , 2013, IACR Cryptol. ePrint Arch..

[6]  Leonel Sousa,et al.  RNS-Based Elliptic Curve Point Multiplication for Massive Parallel Architectures , 2012, Comput. J..

[7]  Thomas Plantard,et al.  RNS Arithmetic Approach in Lattice-Based Cryptography: Accelerating the "Rounding-off" Core Procedure , 2015, 2015 IEEE 22nd Symposium on Computer Arithmetic.

[8]  Thanos Stouraitis,et al.  Multifunction Residue Architectures for Cryptography , 2014, IEEE Transactions on Circuits and Systems I: Regular Papers.

[9]  Thanos Stouraitis,et al.  An RNS Implementation of an $F_{p}$ Elliptic Curve Point Multiplier , 2009, IEEE Transactions on Circuits and Systems I: Regular Papers.

[10]  Arnaud Tisserand,et al.  RNS modular multiplication through reduced base extensions , 2014, 2014 IEEE 25th International Conference on Application-Specific Systems, Architectures and Processors.

[11]  Atsushi Shimbo,et al.  Cox-Rower Architecture for Fast Parallel Montgomery Multiplication , 2000, EUROCRYPT.

[12]  Nicolas Guillermin.  A High Speed Coprocessor for Elliptic Curve Scalar Multiplications over \mathbbFp\mathbb{F}_p , 2010, CHES.

[13]  Arnaud Tisserand,et al.  Single Base Modular Multiplication for Efficient Hardware RNS Implementations of ECC , 2015, CHES.

[14]  T. Stouraitis,et al.  A new approach to elliptic curve cryptography: an RNS architecture , 2006, MELECON 2006 - 2006 IEEE Mediterranean Electrotechnical Conference.

[15]  Reinhard Posch,et al.  RNS-modulo reduction upon a restricted base value set and its applicability to RSA cryptography , 1998, Comput. Secur..

[16]  P. V. Ananda Mohan.  Specialized Residue Number Systems , 2016 .

[17]  Ingrid Verbauwhede,et al.  Faster Pairing Coprocessor Architecture , 2012, Pairing.

[18]  Filippo Gandino,et al.  An Algorithmic and Architectural Study on Montgomery Exponentiation in RNS , 2012, IEEE Transactions on Computers.

[19]  Nabil Merkiche,et al.  Contributions to the Design of Residue Number System Architectures , 2015, 2015 IEEE 22nd Symposium on Computer Arithmetic.

[20]  Daniel J. Bernstein,et al.  Curve25519: New Diffie-Hellman Speed Records , 2006, Public Key Cryptography.

[21]  Jean-Claude Bajard,et al.  Double Level Montgomery Cox-Rower Architecture, New Bounds , 2014, CARDIS.

[22]  Yuichi Komano,et al.  Exact Error Bound of Cox-Rower Architecture for RNS Arithmetic , 2016, IACR Cryptol. ePrint Arch..

[23]  Reinhard Posch,et al.  Modulo Reduction in Residue Number Systems , 1995, IEEE Trans. Parallel Distributed Syst..

[24]  Atsushi Shimbo,et al.  Implementation of RSA Algorithm Based on RNS Montgomery Multiplication , 2001, CHES.