Cyber Defense Laboratory

TinyECC: Elliptic Curve Cryptography for Sensor Networks
(Version 0.2)

Released on 09/29/06.

Introduction

TinyECC is a software package providing Elliptic Curve Cryptography (ECC) operations on TinyOS. It supports all elliptic curve operations over F_p, including point addition, point doubling and scalar point multiplication, as well as ECDSA operations over F_p (signature generation and verification). We plan to include other ECC schemes (e.g. ECDH, ECAES) in this package in the future.

The current version of TinyECC supports MICAz and TelosB motes. It supports SECG recommended 128-bit, 160-bit and 192-bit elliptic curve domain parameters.

For questions please contact An Liu at aliu3 (at) ncsu.edu.

What's new in version 0.2

• Support both MICAz and TelosB
• Support all SECG recommended 128-bit, 160-bit and 192-bit elliptic curve domain parameters
• Hybrid multiplication in assembly code for MICAz
• Shamir's trick to reduce the signature verification time
• More compact SHA1 implementation based on C code in RFC 3174

For questions about this implementation, please contact An Liu at aliu3 (at) ncsu.edu.

People

• Dr. Peng Ning
• An Liu (main contributor)

Platform

• MICAz motes and TelosB motes running TinyOS

Related Software

• TinyECC Version 0.1
• TinyKeyMan

Related Document

• TinyECC is a byproduct of our research on how to use public key cryptograpy in sensor networks safely. One result can be found in the following paper:
  • Peng Ning, An Liu, Wenliang Du, "Mitigating DoS Attacks against Broadcast Authentication in Wireless Sensor Network," Technical Report, North Carolina State University, Department of Computer Science, Revised September 2006.

Download

• Click here to download.

How to Use

• Please check README for details.
• If you need to cite TinyECC, please use the following:

  An Liu, Peng Ning, "TinyECC: Elliptic Curve Cryptography for Sensor Networks (Version 0.2)", http://discovery.csc.ncsu.edu/software/TinyECC/, September 2006.

Performance Results

The following performance results were measured on MICAz and TelosB running TinyOS. Secp128r1, secp128r2, secp160k1, secp160r1, secp160r2, secp192k1, secp192r1 are elliptic curve domain parameters over F_p[7] recommended by Standards for Efficient Cryptography Group (SECG).

• Code size

We used the sliding window method to speed up scalar point multiplication operation. Table 1 and Table 2 show the code size for different curves and window sizes. Note that the code size changes with different parameters due to the changes in the ECC parameters and some internal state. Because of the small SHA1 code size, the total code size has been reduced a lot compared with TinyECC 0.1. The code size of SHA1 is only 3,680 bytes for MICAz, and 2,442 bytes for TelosB.

• Computational cost

Since we use the sliding window method and Shamir's trick, ECDSA module needs to pre-compute intermediate points before doing any signature generation and verification. Table 3 shows the  initialization time of ECDSA module on MICAz and TelosB. Although ECDSA initialization is expensive, it is performed just one time.

Table 3: ECDSA initialization time (window size w = 4)

Parameters	ECDSA.init() on MICAz (seconds)	ECDSA.init() on TelosB (seconds)
secp128r1	2.522	3.861
secp128r2	2.518	3.847
secp160k1	3.553	5.208
secp160r1	3.548	5.225
secp160r2	3.543	5.197
secp192k1	4.992	7.190
secp192r1	4.992	7.204

We measure the time for signature generation and verification for both MICAz and TelosB. Table 4 shows the timing results. Since we haven't implemented hybrid multiplication for TelosB, TelosB is much slower than MICAz.

Table 4: Time for signature generation and verification (window size w = 4)

Parameters	MICAz		TelosB
	sign (seconds)	verify (seconds)	sign (seconds)	verify (seconds)
secp128r1	1.923	2.418	4.059	5.056
secp128r2	2.069	2.674	4.325	5.618
secp160k1	2.059	2.441	4.433	5.209
secp160r1	1.925	2.433	4.361	5.448
secp160r2	2.066	2.615	4.457	5.609
secp192k1	3.070	3.612	6.695	7.840
secp192r1	2.991	3.776	6.651	8.331

Techniques Adopted in TinyECC (Version 0.2)

We implemented several well known optimizations for TinyECC, which are listed below. However, this implementation does not include all known optimizations, such as Non-Adjacent Forms.

• Projective Coordinate Systems

There is no division instruction for ATmega128, so the inverse operation is significantly more expensive than multiplication. It is efficient to implement elliptic curve operations in projective coordinates instead of affine coordinates. We used weighted projective representation (Jacobian representation) [1] in TinyECC to speed up point addition, point doubling and scalar point multiplication.

• Curve-Specific Optimizations

For all NIST and most SECG curves, the underlying field primes p were chosen as pseudo-Mersenne primes to allow optimized modular reduction [2]. We implemented this optimized modular reduction algorithm to speed up modular multiplication and modular square.

• Sliding Window Method (The m-ary Method)

We implemented the sliding window method to speed up scalar point multiplication. The traditional method to do scalar point multiplication is binary method. Binary method scans bits of scalar n from left to right, one bit at a time. A point doubling is performed at each step. Depending on the scanned bit value, a subsequent point addition is performed. Sliding window method [3] scans k bits at a time. Point doubling is performed k times at each step, depending on the scanned k bits value a subsequent point addition is performed. We have to pre-compute all the added points, which is the result of possible k bits value multiply the base point. The sliding window method can speed up scalar point multiplication by reducing the total number of point additions, but extra memory is required.

• Inline Assembly and Hybrid Multiplication

The natural number operations in TinyECC are based on the implementation in RSAREF2.0 [6]. The natural number operations in RSAREF 2.0 are platform independent, but they are not efficient. We used inline assembly code [4] to speed up critical operations such as multiplication and squaring for MICAz motes. In particular, hybrid multiplication in [2] is implemented in inline assembly, which can speed up TinyECC effectively. Note that this cannot be used with sensor nodes that do not use ATmega128 (e.g. TelosB).

• Optimization for Modular Addition and Modular Subtraction

We implemented algorithms 2.7 and 2.8 in [5] to speed up modular addition and modular subtraction.

• Shamir's Trick

We applied algorithm 3.48 in [5] to reduce the signature verification time.

Copyright and Disclaimer

All new code in this distribution is Copyright 2005 by North Carolina State University. All rights reserved. Redistribution and use in source and binary forms are permitted provided that this entire copyright notice is duplicated in all such copies, and that any documentation, announcements, and other materials related to such distribution and use acknowledge that the software was developed at North Carolina State University, Raleigh, NC. No charge may be made for copies, derivations, or distributions of this material without the express written consent of the copyright holder. Neither the name of the University nor the name of the author may be used to endorse or promote products derived from this material without specific prior written permission.

IN NO EVENT SHALL THE NORTH CAROLINA STATE UNIVERSITY BE LIABLE TO ANY PARTY FOR DIRECT, INDIRECT, SPECIAL, INCIDENTAL, OR CONSEQUENTIAL DAMAGES ARISING OUT OF THE USE OF THIS SOFTWARE AND ITS DOCUMENTATION, EVEN IF THE NORTH CAROLINA STATE UNIVERSITY HAS BEEN ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. THE SOFTWARE PROVIDED HEREUNDER IS ON AN "AS IS" BASIS, AND THE NORTH CAROLINA STATE UNIVERSITY HAS NO OBLIGATION TO PROVIDE MAINTENANCE, SUPPORT, UPDATES, ENHANCEMENTS, OR MODIFICATIONS."

Acknowledgment

The natural number operations in TinyECC are based on the implementation in RSAREF 2.0 [6].

References

[1] I. Blake, G. Seroussi, and N. Smart. Elliptic Curves in Cryptography. Cambridge University Press, 1999. London Mathematical Society Lecture Note Series 265.

[2] N. Gura, A. Patel, and A. Wander. Comparing elliptic curve cryptography and RSA on 8-bit CPUs. In Proceedings of the 2004 Workshop on Cryptographic Hardware and Embedded Systems (CHES 2004), August 2004.

[3] Çetin Kaya Kac. High-Speed RSA Implementation, RSA Laboratories Technical Report TR-201, Version 2.0, November 22, 1994.

[4] 8-bit AVR Instruction Set. http://www.atmel.com/dyn/resources/prod_documents/DOC0856.PDF

[5] D. Hankerson, A. Menezens, and S. Vanstone. Guide to Elliptic Curve Cryptography. Springer, 2004.

[6] RSA Laboratories. RSAREF: A cryptographic toolkit (version 2.0), March 1994.

[7] Certicom Research. Standards for efficient cryptography - SEC2: Recommended elliptic curve domain parameters. http://www.secg.org/download/aid-386/sec2_final.pdf, September 2000.

[8] D. Eastlake, P. Jones. US Secure Hash Algorithm 1 (SHA1). RFC 3174, September 2001.

Sponsors

This project has been generously supported by

This material is based upon work supported by the National Science Foundation (NSF) under grant CAREER-0447761 and US Army Research Office (ARO) under grant W911NF-05-1-0247. Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the NSF or the ARO.