Elliptic-curve cryptography

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 ${\displaystyle nG={\mathcal {O}}}$ (the point at infinity of the curve, and the identity element), is normally prime. Since n is the size of a subgroup of ${\displaystyle E(\mathbb {F} _{p})}$ it follows from Lagrange's theorem that the number ${\displaystyle h={\frac {1}{n}}|E(\mathbb {F} _{p})|}$ is an integer. In cryptographic applications this number h, called the cofactor, must be small (${\displaystyle h\leq 4}$) and, preferably, ${\displaystyle h=1}$. To summarize: in the prime case, the domain parameters are ${\displaystyle (p,a,b,G,n,h)}$; in the binary case, they are ${\displaystyle (m,f,a,b,G,n,h)}$.

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:

• NIST, Recommended Elliptic Curves for Government Use
• SECG, SEC 2: Recommended Elliptic Curve Domain Parameters
• ECC Brainpool (RFC 5639), ECC Brainpool Standard Curves and Curve Generation

SECG test vectors are also available.[9] 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:

• Select a random curve and use a general point-counting algorithm, for example, Schoof's algorithm or Schoof–Elkies–Atkin algorithm,
• Select a random curve from a family which allows easy calculation of the number of points (e.g., Koblitz curves), or
• Select the number of points and generate a curve with this number of points using complex multiplication technique.[10]

Several classes of curves are weak and should be avoided:

• Curves over ${\displaystyle \mathbb {F} _{2^{m}}}$ with non-prime m are vulnerable to Weil descent attacks.[11][12]
• Curves such that n divides ${\displaystyle p^{B}-1}$ (where p is the characteristic of the field: q for a prime field, or ${\displaystyle 2}$ for a binary field) for sufficiently small B are vulnerable to Menezes–Okamoto–Vanstone (MOV) attack[13][14] which applies usual discrete logarithm problem (DLP) in a small-degree extension field of ${\displaystyle \mathbb {F} _{p}}$ to solve ECDLP. The bound B should be chosen so that discrete logarithms in the field ${\displaystyle \mathbb {F} _{p^{B}}}$ are at least as difficult to compute as discrete logs on the elliptic curve ${\displaystyle E(\mathbb {F} _{q})}$.[15]
• Curves such that ${\displaystyle |E(\mathbb {F} _{q})|=q}$ are vulnerable to the attack that maps the points on the curve to the additive group of ${\displaystyle \mathbb {F} _{q}}$.[16][17][18]

Because all the fastest known algorithms that allow one to solve the ECDLP (baby-step giant-step, Pollard's rho, etc.), need ${\displaystyle O({\sqrt {n}})}$ 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 ${\displaystyle \mathbb {F} _{q}}$, where ${\displaystyle q\approx 2^{256}}$. This can be contrasted with finite-field cryptography (e.g., DSA) which requires[19] 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.[20] The binary field case was broken in April 2004 using 2600 computers over 17 months.[21]

A current project is aiming at breaking the ECC2K-130 challenge by Certicom, by using a wide range of different hardware: CPUs, GPUs, FPGA.[22]

A close examination of the addition rules shows that in order to add two points, one needs not only several additions and multiplications in ${\displaystyle \mathbb {F} _{q}}$ but also an inversion operation. The inversion (for given ${\displaystyle x\in \mathbb {F} _{q}}$ find ${\displaystyle y\in \mathbb {F} _{q}}$ such that ${\displaystyle xy=1}$) is one to two orders of magnitude slower[23] 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 ${\displaystyle (X,Y,Z)}$ using the following relation: ${\displaystyle x={\frac {X}{Z}}}$, ${\displaystyle y={\frac {Y}{Z}}}$; in the Jacobian system a point is also represented with three coordinates ${\displaystyle (X,Y,Z)}$, but a different relation is used: ${\displaystyle x={\frac {X}{Z^{2}}}}$, ${\displaystyle y={\frac {Y}{Z^{3}}}}$; in the López–Dahab system the relation is ${\displaystyle x={\frac {X}{Z}}}$, ${\displaystyle y={\frac {Y}{Z^{2}}}}$; in the modified Jacobian system the same relations are used but four coordinates are stored and used for calculations ${\displaystyle (X,Y,Z,aZ^{4})}$; and in the Chudnovsky Jacobian system five coordinates are used ${\displaystyle (X,Y,Z,Z^{2},Z^{3})}$. 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.[24]

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 ${\displaystyle p\approx 2^{d}}$; for example, ${\displaystyle p=2^{521}-1}$ or ${\displaystyle p=2^{256}-2^{32}-2^{9}-2^{8}-2^{7}-2^{6}-2^{4}-1.}$ Compared to Barrett reduction, there can be an order of magnitude speed-up.[25] 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 ${\displaystyle \mathbb {F} _{p}}$ 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 sub-optimal. Other curves are more secure and run just as fast.[26]

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[36] (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.[37] Another concern for ECC-systems is the danger of fault attacks, especially when running on smart cards.[38]

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.[39] Internal memos leaked by former NSA contractor, Edward Snowden, suggest that the NSA put a backdoor in the Dual EC DRBG standard.[40] 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.[41]

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.[42]

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.[43] 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 as a decade or more away.

Supersingular Isogeny Diffie–Hellman Key Exchange provides 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.[44]

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."[3]

When ECC is used in virtual machines, an attacker may use an invalid curve to get a complete PDH private key.[45]