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Approach to public key cryptography

Elliptic curve cryptography (ECC) is an approach to public key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys compared to non EC cryptography (based on plain Galois fields) to provide equivalent security. and Victor S. Miller in 1985. Elliptic curve cryptography algorithms entered wide use in 2004 to 2005. In 1999, NIST recommended fifteen elliptic curves. Specifically, FIPS 186 4 has ten recommended finite fields:
Five prime fields for certain primes p of sizes 192, 224, 256, 384, and bits. For each of the prime fields, one elliptic curve is recommended. Five binary fields for m equal 163, 233, 283, 409, and 571. For each of the binary fields, one elliptic curve and one Koblitz curve was selected. The NIST recommendation thus contains a total of five prime curves and ten binary curves. The curves were chosen for optimal security and implementation efficiency. At the RSA Conference 2005, the National Security Agency (NSA) announced Suite B, which exclusively uses ECC for digital signature generation and key exchange. The suite is intended to protect both classified and unclassified national security systems and information. National Institute of Standards and Technology (NIST) has endorsed elliptic curve cryptography in its Suite B set of recommended algorithms, specifically elliptic curve Diffie–Hellman (ECDH) for key exchange and Elliptic Curve Digital Signature Algorithm (ECDSA) for digital signature. The NSA allows their use for protecting information classified up to top secret with 384 bit keys. Recently, a large number of cryptographic primitives based on bilinear mappings on various elliptic curve groups, such as the Weil and Tate pairings, have been introduced. Schemes based on these primitives provide efficient identity based encryption as well as pairing based signatures, signcryption, key agreement, and proxy re encryption. Elliptic curve cryptography is used successfully in numerous popular protocols, such as Transport Layer Security and Bitcoin.

In 2013, The New York Times stated that Dual Elliptic Curve Deterministic Random Bit Generation (or Dual EC DRBG) had been included as a NIST national standard due to the influence of NSA, which had included a deliberate weakness in the algorithm and the recommended elliptic curve. RSA Security in September 2013 issued an advisory recommending that its customers discontinue using any software based on Dual EC DRBG. In the wake of the exposure of Dual EC DRBG as "an NSA undercover operation", cryptography experts have also expressed concern over the security of the NIST recommended elliptic curves, suggesting a return to encryption based on non elliptic curve groups. Additionally, in August 2015, the NSA announced that it plans to replace Suite B with a new cipher suite due to concerns about quantum computing attacks on ECC.

While the RSA patent expired in 2000, there may be patents in force covering certain aspects of ECC technology, including at least one ECC scheme (ECMQV). However, RSA Laboratories and Daniel J. Bernstein have argued that the US government elliptic curve digital signature standard (ECDSA; NIST FIPS 186 3) and certain practical ECC based key exchange schemes (including ECDH) can be implemented without infringing those patents.

Because all the fastest known algorithms that allow one to solve the ECDLP (baby step giant step, Pollard's rho, etc. ), need steps, it follows that the size of the underlying field should be roughly twice the security parameter. For example, for 128 bit security one needs a curve over , where This can be contrasted with finite field cryptography (e. g., DSA) which requires 3072 bit public keys and 256 bit private keys, and integer factorization cryptography (e. g., RSA) which requires a 3072 bit value of n, where the private key should be just as large. However, the public key may be smaller to accommodate efficient encryption, especially when processing power is limited. The hardest ECC scheme (publicly) broken to date had a 112 bit key for the prime field case and a 109 bit key for the binary field case. For the prime field case, this was broken in July 2009 using a cluster of over 200 PlayStation 3 game consoles and could have been finished in 3.5 months using this cluster when running continuously. The binary field case was broken in April 2004 using 2600 computers over 17 months. A current project is aiming at breaking the ECC2K 130 challenge by Certicom, by using a wide range of different hardware: CPUs, GPUs, FPGA.

A close examination of the addition rules shows that in order to add two points, one needs not only several additions and multiplications in but also an inversion operation. The inversion (for given find such that ) is one to two orders of magnitude slower than multiplication. However, points on a curve can be represented in different coordinate systems which do not require an inversion operation to add two points. Several such systems were proposed: in the projective system each point is represented by three coordinates using the following relation: , ; in the Jacobian system a point is also represented with three coordinates , but a different relation is used: , ; in the López–Dahab system the relation is , ; in the modified Jacobian system the same relations are used but four coordinates are stored and used for calculations ; and in the Chudnovsky Jacobian system five coordinates are used Note that there may be different naming conventions, for example, IEEE P1363 2000 standard uses "projective coordinates" to refer to what is commonly called Jacobian coordinates. An additional speed up is possible if mixed coordinates are used.

Reduction modulo p (which is needed for addition and multiplication) can be executed much faster if the prime p is a pseudo Mersenne prime, that is ; for example, or Compared to Barrett reduction, there can be an order of magnitude speed up. The speed up here is a practical rather than theoretical one, and derives from the fact that the moduli of numbers against numbers near powers of two can be performed efficiently by computers operating on binary numbers with bitwise operations. The curves over with pseudo Mersenne p are recommended by NIST. Yet another advantage of the NIST curves is that they use a = −3, which improves addition in Jacobian coordinates. According to Bernstein and Lange, many of the efficiency related decisions in NIST FIPS 186 2 are suboptimal. Other curves are more secure and run just as fast.

Unlike most other DLP systems (where it is possible to use the same procedure for squaring and multiplication), the EC addition is significantly different for doubling (P = Q) and general addition (P ≠ Q) depending on the coordinate system used. Consequently, it is important to counteract side channel attacks (e. g., timing or simple/differential power analysis attacks) using, for example, fixed pattern window (a. k. a. comb) methods (note that this does not increase computation time). Alternatively one can use an Edwards curve; this is a special family of elliptic curves for which doubling and addition can be done with the same operation. Another concern for ECC systems is the danger of fault attacks, especially when running on smart cards.

Cryptographic experts have expressed concerns that the National Security Agency has inserted a kleptographic backdoor into at least one elliptic curve based pseudo random generator. Internal memos leaked by former NSA contractor Edward Snowden suggest that the NSA put a backdoor in the Dual EC DRBG standard. One analysis of the possible backdoor concluded that an adversary in possession of the algorithm's secret key could obtain encryption keys given only 32 bytes of PRNG output. The SafeCurves project has been launched in order to catalog curves that are easy to implement securely and are designed in a fully publicly verifiable way to minimize the chance of a backdoor.

Shor's algorithm can be used to break elliptic curve cryptography by computing discrete logarithms on a hypothetical quantum computer. The latest quantum resource estimates for breaking a curve with a 256 bit modulus (128 bit security level) are 2330 qubits and 126 billion Toffoli gates. For the binary elliptic curve case, 906 qubits are necessary (to break 128 bits of security). In comparison, using Shor's algorithm to break the RSA algorithm requires 4098 qubits and 5.2 trillion Toffoli gates for a 2048 bit RSA key, suggesting that ECC is an easier target for quantum computers than RSA. All of these figures vastly exceed any quantum computer that has ever been built, and estimates place the creation of such computers at a decade or more away. Supersingular Isogeny Diffie–Hellman Key Exchange claimed to provide a post quantum secure form of elliptic curve cryptography by using isogenies to implement Diffie–Hellman key exchanges. This key exchange uses much of the same field arithmetic as existing elliptic curve cryptography and requires computational and transmission overhead similar to many currently used public key systems. However, new classical attacks undermined the security of this protocol. In August 2015, the NSA announced that it planned to transition "in the not distant future" to a new cipher suite that is resistant to quantum attacks. "Unfortunately, the growth of elliptic curve use has bumped up against the fact of continued progress in the research on quantum computing, necessitating a re evaluation of our cryptographic strategy."

When ECC is used in virtual machines, an attacker may use an invalid curve to get a complete PDH private key.