Elliptic curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC requires smaller keys compared to non-ECC cryptography (based on plain Galois fields) to provide equivalent security.
Contents
- Rationale
- History
- Theory
- Cryptographic schemes
- Implementation
- Domain parameters
- Key sizes
- Projective coordinates
- Fast reduction NIST curves
- Applications
- Side channel attacks
- Backdoors
- Quantum computing attacks
- Patents
- Alternative representations
- References
Elliptic curves are applicable for encryption, digital signatures, pseudo-random generators and other tasks. They are also used in several integer factorization algorithms based on elliptic curves that have applications in cryptography, such as Lenstra elliptic curve factorization.
Rationale
Public-key cryptography is based on the intractability of certain mathematical problems. Early public-key systems are secure assuming that it is difficult to factor a large integer composed of two or more large prime factors. For elliptic-curve-based protocols, it is assumed that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point is infeasible: this is the "elliptic curve discrete logarithm problem" (ECDLP). The security of elliptic curve cryptography depends on the ability to compute a point multiplication and the inability to compute the multiplicand given the original and product points. The size of the elliptic curve determines the difficulty of the problem.
The primary benefit promised by elliptic curve cryptography is a smaller key size, reducing storage and transmission requirements, i.e. that an elliptic curve group could provide the same level of security afforded by an RSA-based system with a large modulus and correspondingly larger key: for example, a 256-bit elliptic curve public key should provide comparable security to a 3072-bit RSA public key.
The U.S. 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 U.S. National Security Agency (NSA) allows their use for protecting information classified up to top secret with 384-bit keys. However, 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. However some argue 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 them, including RSA Laboratories and Daniel J. Bernstein
History
The use of elliptic curves in cryptography was suggested independently by Neal Koblitz and Victor S. Miller in 1985. Elliptic curve cryptography algorithms entered wide use in 2004 to 2005.
Theory
For current cryptographic purposes, an elliptic curve is a plane curve over a finite field (rather than the real numbers) which consists of the points satisfying the equation
along with a distinguished point at infinity, denoted ∞. (The coordinates here are to be chosen from a fixed finite field of characteristic not equal to 2 or 3, or the curve equation will be somewhat more complicated.)
This set together with the group operation of elliptic curves is an Abelian group, with the point at infinity as identity element. The structure of the group is inherited from the divisor group of the underlying algebraic variety.
Cryptographic schemes
Several discrete logarithm-based protocols have been adapted to elliptic curves, replacing the group
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.
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.
Implementation
Some common implementation considerations include:
Domain parameters
To use ECC, all parties must agree on all the elements defining the elliptic curve, that is, the domain parameters of the scheme. The field is defined by p in the prime case and the pair of m and f in the binary case. The elliptic curve is defined by the constants a and b used in its defining equation. Finally, the cyclic subgroup is defined by its generator (a.k.a. base point) G. For cryptographic application the order of G, that is the smallest positive number n such that
Unless there is an assurance that domain parameters were generated by a party trusted with respect to their use, the domain parameters must be validated before use.
The generation of domain parameters is not usually done by each participant because this involves computing the number of points on a curve which is time-consuming and troublesome to implement. As a result, several standard bodies published domain parameters of elliptic curves for several common field sizes. Such domain parameters are commonly known as "standard curves" or "named curves"; a named curve can be referenced either by name or by the unique object identifier defined in the standard documents:
SECG test vectors are also available. NIST has approved many SECG curves, so there is a significant overlap between the specifications published by NIST and SECG. EC domain parameters may be either specified by value or by name.
If one (despite the above) wants to construct one's own domain parameters, one should select the underlying field and then use one of the following strategies to find a curve with appropriate (i.e., near prime) number of points using one of the following methods:
Several classes of curves are weak and should be avoided:
Key sizes
Because all the fastest known algorithms that allow one to solve the ECDLP (baby-step giant-step, Pollard's rho, etc.), need
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.
Projective coordinates
A close examination of the addition rules shows that in order to add two points, one needs not only several additions and multiplications in
Fast reduction (NIST curves)
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
The curves over
According to Bernstein and Lange, many of the efficiency-related decisions in NIST FIPS 186-2 are sub-optimal. Other curves are more secure and run just as fast.
Applications
Elliptic curves are applicable for encryption, digital signatures, pseudo-random generators and other tasks. They are also used in several integer factorization algorithms that have applications in cryptography, such as Lenstra elliptic curve factorization.
In 1999, NIST recommended 15 elliptic curves. Specifically, FIPS 186-3 has 10 recommended finite fields:
The NIST recommendation thus contains a total of 5 prime curves and 10 binary curves. The curves were ostensibly chosen for optimal security and implementation efficiency.
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.
Side-channel attacks
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.
Backdoors
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 ciphertext.
The SafeCurves project has been launched in order to catalog curves that are easy to securely implement and are designed in a fully publicly verifiable way to minimize the chance of a backdoor.
Quantum computing attacks
In contrast with its current standing over RSA, elliptic curve cryptography is expected to be more vulnerable to an attack based on Shor's algorithm. In theory, making a practical attack feasible many years before an attack on an equivalently secure RSA scheme is possible. This is because smaller elliptic curve keys are needed to match the classical security of RSA. The work of Proos and Zalka show how a quantum computer for breaking 2048-bit RSA requires roughly 4096 qubits, while a quantum computer to break the equivalently secure 224-bit Elliptic Curve Cryptography requires between 1300 and 1600 qubits.
To avoid quantum computing concerns, an elliptic curve-based alternative to Elliptic Curve Diffie Hellman which is not susceptible to Shor's attack is the Supersingular Isogeny Diffie–Hellman Key Exchange of De Feo, Jao and Plut. It uses elliptic curve isogenies to create a drop-in replacement for the quantum attackable Diffie–Hellman and Elliptic curve Diffie–Hellman key exchanges. This key exchange uses the same elliptic curve computational primitives of existing elliptic curve cryptography and requires computational and transmission overhead similar to many currently used public key systems.
In August, 2015, 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."
Patents
At least one ECC scheme (ECMQV) and some implementation techniques are covered by patents.
Alternative representations
Alternative representations of elliptic curves include: