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The Case for Elliptic Curve Cryptography


Over the past 30 years, public key cryptography has become a mainstay for secure communications over the Internet and throughout many other forms of communications. It provides the foundation for both key management and digital signatures. In key management, public key cryptography is used to distribute the secret keys used in other cryptographic algorithms (e.g. DES). For digital signatures, public key cryptography is used to authenticate the origin of data and protect the integrity of that data. For the past 20 years, Internet communications have been secured by the first generation of public key cryptographic algorithms developed in the mid-1970's. Notably, they form the basis for key management and authentication for IP encryption (IKE/IPSEC), web traffic (SSL/TLS) and secure electronic mail.

In their day these public key techniques revolutionized cryptography. Over the last twenty years however, new techniques have been developed which offer both better performance and higher security than these first generation public key techniques. The best assured group of new public key techniques is built on the arithmetic of elliptic curves. This paper will outline a case for moving to elliptic curves as a foundation for future Internet security. This case will be based on both the relative security offered by elliptic curves and first generation public key systems and the relative performance of these algorithms. While at current security levels elliptic curves do not offer significant benefits over existing public key algorithms, as one scales security upwards over time to meet the evolving threat posed by eavesdroppers and hackers with access to greater computing resources, elliptic curves begin to offer dramatic savings over the old, first generation techniques.

The two noteworthy first generation public key algorithms used to secure the Internet today are known as RSA and Diffie-Hellman (DH). The security of the first is based on the difficulty of factoring the product of two large primes. The second is related to a problem known as the discrete logarithm problem for finite groups. Both are based on the use of elementary number theory. Interestingly, the security of the two schemes, though formulated differently, is closely related.

Both algorithms have been subject to intense scrutiny since their invention around 1975. In the years immediately following their invention, there were a number of attacks based on a variety of degenerate ways to generate the prime numbers and such that define the system. Such parameters were quickly excluded from specifications. In the public domain, more general theoretic attacks on the fundamental problems of factoring and discrete logs made steady progress until the early 1990's. Since that time, no dramatic improvements in these attack algorithms have been published. However, there have been several efforts aimed at designing theoretical special purpose computers that would implement the existing attack algorithms far faster than general computing resources.

Since their use in cryptography was discovered in 1985, elliptic curve cryptography has also been an active area of study in academia. Similar to both RSA and Diffie-Hellman, the first years of analysis yielded some degenerate cases for elliptic curve parameters that one should avoid. However, unlike the RSA and Diffie-Hellman cryptosystems that slowly succumbed to increasingly strong attack algorithms, elliptic curve cryptography has remained at its full strength since it was first presented in 1985.

Elliptic Curve Security and Efficiency:

The majority of public key systems in use today use 1024-bit parameters for RSA and Diffie-Hellman. The US National Institute for Standards and Technology has recommended that these 1024-bit systems are sufficient for use until 2010. After that, NIST recommends that they be upgraded to something providing more security. The question is what should these systems be changed to? One option is to simply increase the public key parameter size to a level appropriate for another decade of use. Another option is to take advantage of the past 30 years of public key research and analysis and move from first generation public key algorithms and on to elliptic curves.

One way judgments are made about the correct key size for a public key system is to look at the strength of the conventional (symmetric) encryption algorithms that the public key algorithm will be used to key or authenticate. Examples of these conventional algorithms are the Data Encryption Standard (DES) created in 1975 and the Advanced Encryption Standard (AES) now a new standard. The length of a key, in bits, for a conventional encryption algorithm is a common measure of security. To attack an algorithm with a k-bit key it will generally require roughly 2k-1 operations. Hence, to secure a public key system one would generally want to use parameters that require at least 2k-1 operations to attack. The following table gives the key sizes recommended by the National Institute of Standards and Technology to protect keys used in conventional encryption algorithms like the (DES) and (AES) together with the key sizes for RSA, Diffie-Hellman and elliptic curves that are needed to provide equivalent security.

Symmetric Key Size (bits) RSA and Diffie-Hellman Key Size (bits) Elliptic Curve Key Size (bits)
80 1024 160
112 2048 224
128 3072 256
192 7680 384
256 15360 521
Table 1: NIST Recommended Key Sizes

To use RSA or Diffie-Hellman to protect 128-bit AES keys one should use 3072-bit parameters: three times the size in use throughout the Internet today. The equivalent key size for elliptic curves is only 256 bits. One can see that as symmetric key sizes increase the required key sizes for RSA and Diffie-Hellman increase at a much faster rate than the required key sizes for elliptic curve cryptosystems. Hence, elliptic curve systems offer more security per bit increase in key size than either RSA or Diffie-Hellman public key systems.

Security is not the only attractive feature of elliptic curve cryptography. Elliptic curve cryptosystems also are more computationally efficient than the first generation public key systems, RSA and Diffie-Hellman. Although elliptic curve arithmetic is slightly more complex per bit than either RSA or DH arithmetic, the added strength per bit more than makes up for any extra compute time. The following table shows the ratio of DH computation versus EC computation for each of the key sizes listed in Table 1.

Security Level
Ratio of
DH Cost : EC Cost
80 3:1
112 6:1
128 10:1
192 32:1
256 64:1
Table 2: Relative Computation Costs of Diffie-Hellman and Elliptic Curves1

Closely related to the key size of different public key systems is the channel overhead required to perform key exchanges and digital signatures on a communications link. The key sizes for public key in Table 1 (above) is also roughly the number of bits that need to be transmitted each way over a communications channel for a key exchange2. In channel-constrained environments, elliptic curves offer a much better solution than first generation public key systems like Diffie-Hellman.

In choosing an elliptic curve as the foundation of a public key system there are a variety of different choices. The National Institute of Standards and Technology (NIST) has standardized on a list of 15 elliptic curves of varying sizes. Ten of these curves are for what are known as binary fields and 5 are for prime fields. Those curves listed provide cryptography equivalent to symmetric encryption algorithms (e.g. AES, DES or SKIPJACK) with keys of length 80, 112, 128, 192, and 256 bits and beyond.

For protecting both classified and unclassified National Security information, the National Security Agency has decided to move to elliptic curve based public key cryptography. Where appropriate, NSA plans to use the elliptic curves over finite fields with large prime moduli (256, 384, and 521 bits) published by NIST.

The United States, the UK, Canada and certain other NATO nations have all adopted some form of elliptic curve cryptography for future systems to protect classified information throughout and between their governments. The Cryptographic Modernization Initiative in the US Department of Defense aims at replacing almost 1.3 million existing equipments over the next 10 years. In addition, the Department's Global Information Grid will require a vast expansion of the number of security devices in use throughout the US Military. This will necessitate change and rollover of equipment with all major US allies. Most of these needs will be satisfied with a new generation of cryptographic equipment that uses elliptic curve cryptography for key management and digital signatures.

Elliptic Curve Intellectual Property:

Despite the many advantages of elliptic curves and despite the adoption of elliptic curves by many users, many vendors and academics view the intellectual property environment surrounding elliptic curves as a major roadblock to their implementation and use. Various aspects of elliptic curve cryptography have been patented by a variety of people and companies around the world. Notably the Canadian company, Certicom Inc. holds over 130 patents related to elliptic curves and public key cryptography in general.

As a way of clearing the way for the implementation of elliptic curves to protect US and allied government information, the National Security Agency purchased from Certicom a license that covers all of their intellectual property in a restricted field of use. The license would be limited to implementations that were for national security uses and certified under FIPS 140-2 or were approved by NSA. Further, the license would be limited to only prime field curves where the prime was greater than 2255. On the NIST list of curves 3 out of the 15 fit this field of use: the prime field curves with primes of 256 bits, 384 bits and 521 bits. Certicom identified 26 patents that covered this field of use. NSA's license includes a right to sublicense these 26 patents to vendors building products within the restricted field of use. Certicom also retained a right to license vendors both within the field of use and under other terms that they may negotiate with vendors.

Commercial vendors may receive a license from NSA provided their products fit within the field of use of NSA's license. Alternatively, commercial vendors may contact Certicom for a license for the same 26 patents. Certicom is planning on developing and selling software toolkits that implement elliptic curve cryptography in the field of use. With the toolkit a vendor will also receive a license from Certicom to sell the technology licensed by NSA in the general commercial marketplace. Vendors wishing to implement elliptic curves outside the scope of the NSA license will need to work with Certicom if they wish to be licensed.


Elliptic Curve Cryptography provides greater security and more efficient performance than the first generation public key techniques (RSA and Diffie-Hellman) now in use. As vendors look to upgrade their systems they should seriously consider the elliptic curve alternative for the computational and bandwidth advantages they offer at comparable security.

1These estimates are based on the theoretic costing of an n-bit multiply modulo a large prime as costing roughly n2 operations. It is also based on an estimate that computing an inverse modulo a large prime is roughly 8 multiplies. Actual implementations could be radically different based on computer architecture.

2In the elliptic curve case, there is actually one additional bit that needs to be transmitted in each direction which allows the recovery of both the x and y coordinates of an elliptic curve point.


Date Posted: Jan 15, 2009 | Last Modified: Jan 15, 2009 | Last Reviewed: Jan 15, 2009