Softening

In mathematics, the Weil pairing is a pairing (bilinear form, though with multiplicative notation) on the points of order dividing n of an elliptic curve E, taking values in nth roots of unity. More generally there is a similar Weil pairing between points of order n of an abelian variety and its dual. It was introduced by André Weil ([[#Template:Harvid|1940]]) for Jacobians of curves, who gave an abstract algebraic definition; the corresponding results for elliptic functions were known, and can be expressed simply by use of the Weierstrass sigma function.

Formulation

Choose an elliptic curve E defined over a field K, and an integer n > 0 (we require n to be prime to char(K) if char(K) > 0) such that K contains a primitive nth root of unity. Then the n-torsion on $E(\bar{K})$ is known to be a Cartesian product of two cyclic groups of order n. The Weil pairing produces an n-th root of unity

$w(P,Q)\in {\mu }_n$

by means of Kummer theory, for any two points $P,Q\in E(K)[n]$, where $E(K)[n]=\{T\in E(K)\mid n\cdot T=O\}$ and ${\mu }_n=\{x\in K\mid x^n=1\}$.

A down-to-earth construction of the Weil pairing is as follows. Choose a function F in the function field of E over the algebraic closure of K with divisor

$\mathrm{d}\mathrm{i}\mathrm{v}(F)=\sum _{0\le k<n}(P+k\cdot Q)-\sum _{0\le k<n}(k\cdot Q).$

So F has a simple zero at each point P + kQ, and a simple pole at each point kQ if these points are all distinct. Then F is well-defined up to multiplication by a constant. If G is the translation of F by Q, then by construction G has the same divisor, so the function G/F is constant.

Therefore if we define

$w(P,Q):=\frac{G}{F}$

we shall have an n-th root of unity (as translating n times must give 1) other than 1. With this definition it can be shown that w is antisymmetric and bilinear, giving rise to a non-degenerate pairing on the n-torsion.

The Weil pairing does not extend to a pairing on all the torsion points (the direct limit of n-torsion points) because the pairings for different n are not the same. However they do fit together to give a pairing T_ℓ(E) × T_ℓ(E) → T_ℓ(μ) on the Tate module T_ℓ(E) of the elliptic curve E (the inverse limit of the ℓⁿ-torsion points) to the Tate module T_ℓ(μ) of the multiplicative group (the inverse limit of ℓⁿ roots of unity).

Generalisation to abelian varieties

For abelian varieties over an algebraically closed field K, the Weil pairing is a nondegenerate pairing

$A[n]\times A^{\vee }[n]⟶{\mu }_n$

for all n prime to the characteristic of k. Here $A^{\vee }$ denotes the dual abelian variety of A. This is the so-called Weil pairing for higher dimensions. If A is equipped with a polarisation

$\lambda :A⟶A^{\vee }$,

then composition gives a (possibly degenerate) pairing

$A[n]\times A[n]⟶{\mu }_n.$

If C is a projective, nonsingular curve of genus ≥ 0 over k, and J its Jacobian, then the theta-divisor of J induces a principal polarisation of J, which in this particular case happens to be an isomorphism (see autoduality of Jacobians). Hence, composing the Weil pairing for J with the polarisation gives a nondegenerate pairing

$J[n]\times J[n]⟶{\mu }_n$

for all n prime to the characteristic of k.

As in the case of elliptic curves, explicit formulae for this pairing can be given in terms of divisors of C.

Applications

The Weil pairing is used in number theory and algebraic geometry, and has also been applied in elliptic curve cryptography and identity based encryption.

See also

• Tate pairing
• Pairing-based cryptography
• Boneh/Franklin scheme
• Homomorphic Signatures for Network Coding

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

External links

• The Weil pairing on elliptic curves over C (PDF)