Cyber Defense Laboratory 



TinyECC: Elliptic Curve Cryptography for Sensor Networks

Parameters  MICAz  TelosB  
ROM (bytes)  RAM (bytes)  ROM (bytes)  RAM (bytes)  
secp128r1  13,120  762  12,528  830 
secp128r2  13,102  762  12,494  830 
secp160k1  13,912  938  12,566  1,006 
secp160r1  13,880  938  12,560  1,006 
secp160r2  13,988  938  12,564  1,182 
secp192k1  13,510  1,114  12,628  1,182 
secp192r1  13,204  1,114  12,718  1,182 
Parameters  MICAz  TelosB  Imote2  
ROM (bytes)  RAM (bytes)  ROM (bytes)  RAM (bytes)  ROM (bytes)  RAM (bytes)  
secp128r1  13,094  1,168  12,532  1,254  26,404 
1,468 
secp128r2  13,076  1,168  12,498  1,254  26,420 
1,468 
secp160k1  13,890  1,440  12,570  1,526  26,416 
1,740 
secp160r1  13,858  1,440  12,564  1,526  26,368 
1,740 
secp160r2  13,966  1,440  12,568  1,526  26,436 
1,740 
secp192k1  13,488  1,712  12,632  1,798  26,508 
2,012 
secp192r1  13,182  1,712  12,722  1,798  26,572 
2,012 
Since we use the sliding window method and Shamir's trick, ECDSA module needs to precompute intermediate points before doing any signature generation and verification. Table 3 shows the initialization time of ECDSA module on MICAz, TelosB and Imote2. Although ECDSA initialization is expensive, it is performed just one time.
Table 3: ECDSA initialization time (window size w = 4) Parameters MICAz
(secs)TelosB
(secs)Imote2 (secs) core freq: 13MHz core freq: 104MHz secp128r1 2.522 3.861 1.164
0.145
secp128r2 2.518 3.847 1.162
0.145
secp160k1 3.553 5.208 1.313
0.164
secp160r1 3.548 5.225 1.288
0.161
secp160r2 3.543 5.197 1.311
0.163
secp192k1 4.992 7.190 1.732
0.216
secp192r1 4.992 7.204 1.736
0.217
We measure the time for signature generation and verification for both MICAz, TelosB and Imote2. Table 4 shows the timing results. Since we haven't implemented hybrid multiplication for TelosB, TelosB is much slower than MICAz. Also, Imote2 32bit processor and higher frequency improve more than 10fold the performance of the motes.
Table 4: Time for signature generation and verification (window size w = 4) Parameters MICAz TelosB Imote2 sign (secs) verify (secs) sign (secs) verify (secs) core freq: 13MHz core freq: 104MHz sign (secs) verify (secs) sign (secs) verify (secs) secp128r1 1.923 2.418 4.059 5.056 2.234
2.815
0.279
0.351
secp128r2 2.069 2.674 4.325 5.618 2.416
3.146
0.301
0.393
secp160k1 2.059 2.441 4.433 5.209 1.854
2.197
0.231
0.274
secp160r1 1.925 2.433 4.361 5.448 1.699
2.161
0.212
0.270
secp160r2 2.066 2.615 4.457 5.609 1.857
2.359
0.232
0.294
secp192k1 3.070 3.612 6.695 7.840 2,710
3,184
0.338
0.398
secp192r1 2.991 3.776 6.651 8.331 2.709
3.409
0.338
0.426
The greater RAM capabilities of Imote2 allow us to increase the window size to improve the timing results even more (Table 5).
Parameters  Imote2  
core freq: 104MHz  
sign (secs)  verify (secs)  
secp128r1  0.238 
0.352 
secp128r2  0.261 
0.393 
secp160k1  0.200 
0.274 
secp160r1  0.183 
0.269 
secp160r2  0.200 
0.294 
secp192k1  0.292 
0.397 
secp192r1  0.292 
0.425 
We used the formula E = U*I*t to estimate the energy consumption of signature generation and verification. For MICAz, when processor is in active mode, I = 8 mA. For TelosB, I = 1.8mA for active mode. Typically, U = 3.0 V if two new AA batteries are used. As for Imote2, the default mode used is deep sleep mode where the nominal current is 390μA and the core voltage is set to 850mV and 950mV for 13 MHz and 104 MHz core frequency respectively. Table 6 shows the results.
Window size (bits)  MICAz  TelosB  Imote2  
sign (mJ)  verify (mJ)  sign (mJ)  verify (mJ)  core freq: 13MHz  core freq: 104MHz  
sign (mJ)  verify (mJ)  sign (mJ)  verify (mJ)  
2  52.9  58.4  27.5  29.4  0.64 
0.71 
0.09 
0.11 
4  46.2  58.4  23.5  29.4  0.56 
0.72 
0.08 
0.10 
We have implemented several well known optimizations for TinyECC from the previous version of TinyECC. The same techniques are still supported in TinyECC0.3 and are listed below. However, this implementation does not include all known optimizations, such as NonAdjacent Forms.
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.
For all NIST and most SECG curves, the underlying field primes p were chosen as pseudoMersenne primes to allow optimized modular reduction [2]. We implemented this optimized modular reduction algorithm to speed up modular multiplication and modular square.
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 precompute 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.
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).
We implemented algorithms 2.7 and 2.8 in [5] to speed up modular addition and modular subtraction.
We applied algorithm 3.48 in [5] to reduce the signature verification time.
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."
The natural number operations in TinyECC are based on the implementation in RSAREF 2.0 [6].
[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 8bit CPUs. In Proceedings of the 2004 Workshop on Cryptographic Hardware and Embedded Systems (CHES 2004), August 2004.
[3] Çetin Kaya Kac. HighSpeed RSA Implementation, RSA Laboratories Technical Report TR201, Version 2.0, November 22, 1994.
[4] 8bit 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/aid386/sec2_final.pdf, September 2000.
[8] D. Eastlake, P. Jones. US Secure Hash Algorithm 1 (SHA1). RFC 3174, September 2001.
This project has been generously supported by
This material is based upon work supported by the National Science Foundation (NSF) under grant CAREER0447761 and US Army Research Office (ARO) under grant W911NF0510247. 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.