Abstract
Recently, the Residue Number System and the Cox-Rower architecture have been used to compute efficiently Elliptic Curve Cryptography over FPGA. In this paper, we are rewriting the conditions of Kawamura’s theorem for the base extension without error in order to define the maximal range of the set from which the moduli can be chosen to build a base. At the same time, we give a procedure to compute correctly the truncation function of the Cox module. We also present a modified ALU of the Rower architecture using a second level of Montgomery Representation. Such architecture allows us to select the moduli with the new upper bound defined with the condition. This modification makes the Cox-Rower architecture suitable to compute 521 bits ECC with radix downto 16 bits compared to 18 with the classical Cox-Rower architecture. We validate our results through FPGA implementation of a scalar multiplication at classical cryptography security levels (NIST curves). Our implementation uses 35 % less LUTs compared to the state of the art generic implementation of ECC using RNS for the same performance [5]. We also slightly improve the computation time (latency) and our implementation shows best ratio throughput/area for RNS computation supporting any curve independently of the chosen base.
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Notes
- 1.
Barrett reduction is also possible, but we would need larger multipliers for the same results.
- 2.
For \( m_i = 2^r \) (only one even number can be selected), we use a classical multiplier and gather the \( r \) least significant bits of the multiplier.
- 3.
It is well known that the Montgomery representation is stable for addition and product using Algorithm 3.
- 4.
The slices is the cells counting system on Xilinx FPGA. A slice on a Kintex-7 includes 4 LUTs with 6 inputs and 8 registers.
- 5.
An ALM, in the Stratix-2 family, contains 2 LUTs with 5 inputs and 2 registers, and equivalent to the Xilinx Virtex-4 slice.
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A Algorithm to Compute the Montgomery Reduction over RNS and Implementation Details
A Algorithm to Compute the Montgomery Reduction over RNS and Implementation Details
Let \( \varvec{\mathfrak {B}} \) and \( \varvec{\mathfrak {B}}' \) be 2 RNS bases such that \( \varvec{\mathfrak {B}} = \lbrace m_i \rbrace \) and \( \varvec{\mathfrak {B}}' = \lbrace {m'}_i \rbrace \) with \( M = \prod _{i=1}^{n} m_i \), \( M' = \prod _{i=1}^{n} {m'}_i \), \( \gcd (p, M) = 1 \) and \( \gcd (M, M') = 1 \). Algorithm 4 recalls the Montgomery reduction over RNS, when using the classical ALU. Precomputed values are in bald.
Algorithm 5 is the algorithm for the Montgomery reduction over RNS, when using our ALU. Operation \( \otimes \) will denote the inner Montgomery multiplication and reduction (Algorithm 3) such that \( a \otimes b \mod m = a b 2^{-r} \mod m \).
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Bajard, JC., Merkiche, N. (2015). Double Level Montgomery Cox-Rower Architecture, New Bounds. In: Joye, M., Moradi, A. (eds) Smart Card Research and Advanced Applications. CARDIS 2014. Lecture Notes in Computer Science(), vol 8968. Springer, Cham. https://doi.org/10.1007/978-3-319-16763-3_9
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