Elliptic Curve Cryptography And Digital Rights Management.pdf

Why Elliptic Curve Cryptography?
• As you saw in Section 12.12 of Lecture 12, the computational overhead of the RSA-based approach to public-key cryptography increases with the size of the keys. As algorithms for integer factorization have become more and more efficient, the RSA-based methods have had to resort to longer and longer keys.
• Elliptic curve cryptography (ECC) can provide the same level and type of security as RSA (or Diffie-Hellman as used in the manner described in Section 13.5 of Lecture 13) but with much shorter keys.
• Table 1 compares the best current estimates of the key sizes for three different approaches to encryption for comparable levels of security against brute-force attacks. What makes this table all the more significant is that for comparable key lengths the computational burdens of RSA and ECC are comparable. What that implies is that, with ECC, it takes one-sixth the computational effort to provide the same level of cryptographic security that you get with 1024-bit RSA.

Symmetric Encryption
RSA and Diffie-Hellman
ECC
Key Size
"Key" size
"Key" Size
in bits
in bits
in bits
80
1024
160
112
2048
224
128
3072
256
192
7680
384
256
15360
512

• The computational overhead of both RSA and ECC grows as O(N^3) where N is the key length in bits. Nonetheless, despite this parity in the dependence of the computational effort on key size, it takes far less computational overhead to use ECC on account of the fact that you can get away with much shorter keys.

• Because of the much smaller key sizes involved, ECC algorithms can be implemented on smartcards without mathematical coprocessors. Contactless smart cards work only with ECC because other systems require too much induction energy. Since shorter key lengths translate into faster handshaking protocols, ECC is also becoming increasingly important for wireless communications.

• For the same reasons as listed above, we can also expect ECC to become important for wireless sensor networks.

• If you want to combine forward secrecy, in the sense defined in Section 12.6 of Lecture 12, with authentication, a commonly used algorithm today is ECDHE-RSA.

In ECDHE-RSA, RSA is used for certificate-based authentication using the TLS/SSL protocol and ECDHE used for creating a one-time session key using the method described in Section 14.12.