Doppler spectroscopy

Shamir's Secret Sharing is an algorithm in cryptography created by Adi Shamir. It is a form of secret sharing, where a secret is divided into parts, giving each participant its own unique part, where some of the parts or all of them are needed in order to reconstruct the secret.

Counting on all participants to combine together the secret might be impractical, and therefore sometimes the threshold scheme is used where any $k$ of the parts are sufficient to reconstruct the original secret.

Mathematical definition

The goal is to divide data $D$ (e.g., a safe combination) into $n$ pieces $D_{1},\ldots ,D_{n}$ in such a way that:

1. Knowledge of any $k$ or more $D_{i}$ pieces makes $D$ easily computable.
2. Knowledge of any $k-1$ or fewer $D_{i}$ pieces leaves $D$ completely undetermined (in the sense that all its possible values are equally likely).

This scheme is called $\left(k,n\right)$ threshold scheme. If $k=n$ then all participants are required to reconstruct the secret.

Shamir's secret-sharing scheme

One can draw an infinite number of polynomials of degree 2 through 2 points. 3 points are required to define a unique polynomial of degree 2. This image is for illustration purposes only — Shamir's scheme uses polynomials over a finite field, not representable on a 2-dimensional plane.

The essential idea of Adi Shamir's threshold scheme is that 2 points are sufficient to define a line, 3 points are sufficient to define a parabola, 4 points to define a cubic curve and so forth. That is, it takes $k\,\!$ points to define a polynomial of degree $k-1\,\!$.

Suppose we want to use a $\left(k,n\right)\,\!$ threshold scheme to share our secret $S\,\!$, without loss of generality assumed to be an element in a finite field $F$ of size $0<k\leq n<P$ where $P$ is a prime number.

Choose at random $k-1\,\!$ coefficients $a_{1},\cdots ,a_{k-1}\,\!$ in $F$, and let $a_{0}=S\,\!$. Build the polynomial $f\left(x\right)=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots +a_{k-1}x^{k-1}\,\!$. Let us construct any $n\,\!$ points out of it, for instance set $i=1,\cdots ,n\,\!$ to retrieve $\left(i,f\left(i\right)\right)\,\!$. Every participant is given a point (an integer input to the polynomial, and the corresponding integer output). Given any subset of $k\,\!$ of these pairs, we can find the coefficients of the polynomial using interpolation. The secret is the constant term $a_{0}\,\!$.

Usage

Example

The following example illustrates the basic idea. Note, however, that calculations in the example are done using integer arithmetic rather than using finite field arithmetic. Therefore the example below does not provide perfect secrecy,Template:Clarify and is not a true example of Shamir's scheme.

Preparation

Suppose that our secret is 1234 $(S=1234)\,\!$.

We wish to divide the secret into 6 parts $(n=6)\,\!$, where any subset of 3 parts $(k=3)\,\!$ is sufficient to reconstruct the secret. At random we obtain two ($k-1$) numbers: 166 and 94.

$(a_{1}=166;a_{2}=94)\,\!$

Our polynomial to produce secret shares (points) is therefore:

$f\left(x\right)=1234+166x+94x^{2}\,\!$

We construct 6 points from the polynomial:

$\left(1,1494\right);\left(2,1942\right);\left(3,2578\right);\left(4,3402\right);\left(5,4414\right);\left(6,5614\right)\,\!$

We give each participant a different single point (both $x\,\!$ and $f\left(x\right)\,\!$).

Reconstruction

In order to reconstruct the secret any 3 points will be enough.

Let us consider $\left(x_{0},y_{0}\right)=\left(2,1942\right);\left(x_{1},y_{1}\right)=\left(4,3402\right);\left(x_{2},y_{2}\right)=\left(5,4414\right)\,\!$.

We will compute Lagrange basis polynomials:

$\ell _{0}={\frac {x-x_{1}}{x_{0}-x_{1}}}\cdot {\frac {x-x_{2}}{x_{0}-x_{2}}}={\frac {x-4}{2-4}}\cdot {\frac {x-5}{2-5}}={\frac {1}{6}}x^{2}-{\frac {3}{2}}x+{\frac {10}{3}}\,\!$

$\ell _{1}={\frac {x-x_{0}}{x_{1}-x_{0}}}\cdot {\frac {x-x_{2}}{x_{1}-x_{2}}}={\frac {x-2}{4-2}}\cdot {\frac {x-5}{4-5}}=-{\frac {1}{2}}x^{2}+{\frac {7}{2}}x-5\,\!$

$\ell _{2}={\frac {x-x_{0}}{x_{2}-x_{0}}}\cdot {\frac {x-x_{1}}{x_{2}-x_{1}}}={\frac {x-2}{5-2}}\cdot {\frac {x-4}{5-4}}={\frac {1}{3}}x^{2}-2x+{\frac {8}{3}}\,\!$

Therefore

$f(x)=\sum _{j=0}^{2}y_{j}\cdot \ell _{j}(x)\,\!$

$=1234+166x+94x^{2}\,\!$

Recall that the secret is the free coefficient, which means that $S=1234\,\!$, and we are done.

Javascript example

var prime = 257;

/*
 * Split number into the shares
 */
function split(number, available, needed)
{
 var coef = [number, 166, 94], x, exp, c, accum, shares = [];
 /*
  * Normally, we use the line:
  * for(c = 1, coef[0] = number; c < needed; c++) coef[c] = Math.floor(Math.random() * (prime  - 1));
  * where (prime - 1) is the maximum allowable value.
  * However, to follow this example, we hardcode the values:
  * coef = [number, 166, 94];
  * For production, replace the hardcoded value with the random loop
  *
  * For each share that is requested to be available, run through the formula plugging the corresponding coefficient
  * The result is f(x), where x is the byte we are sharing (in the example, 1234)
  */
 for(x = 1; x <= available; x++)
 {
  /*
   * coef = [1234, 166, 94] which is 1234x^0 + 166x^1 + 94x^2
   */
  for(exp = 1, accum = coef[0]; exp < needed; exp++)
   accum = (accum + (coef[exp] * (Math.pow(x, exp) % prime) % prime)) % prime; // Modular math
  /*
   * Store values as (1, 1494), (2, 1942), (3, 2578), (4, 3402), (5, 4414) (6, 5614)
   */
  shares[x - 1] = [x, accum];
 }
 return shares;
}

/*
 * Gives the decomposition of the gcd of a and b.
 * Returns [x,y,z] such that x = gcd(a,b) and y*a + z*b = x
 */
function gcdD(a,b) { 
  if (b == 0) return [a, 1, 0]; 
  else { 
    var n = Math.floor(a/b), c = a % b, r = gcdD(b,c); 
    return [r[0], r[2], r[1]-r[2]*n];
  }
}

/*
 * Gives the multiplicative inverse of k mod prime.
 * In other words (k * modInverse(k)) % prime = 1 for all 1 <= k < prime
 */
function modInverse(k) { 
  k = k % prime;
  var r = (k < 0) ? -gcdD(prime,-k)[2] : gcdD(prime,k)[2];
  return (prime + r) % prime;
}
/*
 * Join the shares into a number
 */
function join(shares)
{
 var accum, count, formula, startposition, nextposition, value, numerator, denominator;
 for(formula = accum = 0; formula < shares.length; formula++)
 {
  /*
   * Multiply the numerator across the top and denominators across the bottom to do Lagrange's interpolation
   * Result is x0(2), x1(4), x2(5) -> -4*-5 and (2-4=-2)(2-5=-3), etc for l0, l1, l2...
   */
  for(count = 0, numerator = denominator = 1; count < shares.length; count++)
  {
   if(formula == count) continue; // If not the same value
   startposition = shares[formula][0];
   nextposition = shares[count][0];
   numerator = (numerator * -nextposition) % prime;
   denominator = (denominator * (startposition - nextposition)) % prime;
  }
  value = shares[formula][1];
  accum = (prime + accum + (value * numerator * modInverse(denominator))) % prime;
 }
 return accum;
}

var sh = split(129, 6, 3) /* split the secret value 129 into 6 components - at least 3 of which will be needed to figure out the secret value */
var newshares = [sh[1], sh[3], sh[4]]; /* pick any any selection of 3 shared keys from sh */

alert(join(newshares));

Properties

Some of the useful properties of Shamir's $\left(k,n\right)\,\!$ threshold scheme are:

1. Secure: Information theoretic security.
2. Minimal: The size of each piece does not exceed the size of the original data.
3. Extensible: When $k\,\!$ is kept fixed, $D_{i}\,\!$ pieces can be dynamically added or deleted without affecting the other pieces.
4. Dynamic: Security can be easily enhanced without changing the secret, but by changing the polynomial occasionally (keeping the same free term) and constructing new shares to the participants.
5. Flexible: In organizations where hierarchy is important, we can supply each participant different number of pieces according to their importance inside the organization. For instance, the president can unlock the safe alone, whereas 3 secretaries are required together to unlock it.

See also

• Secret sharing
• Lagrange polynomial
• Homomorphic secret sharing - A simplistic decentralized voting protocol.
• Two-man rule

References

• 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.

• 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.

• 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.

• 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

• A proper Javascript implementation of Shamir's secret sharing scheme with open source (MIT) license
• ssss: An open source (GPL) implementation of Shamir's Scheme with online demo
• An open source (GPL) perl implementation of Shamir's Secret Sharing
• Secret Sharp: An open source (GPL) implementation of Shamir's Scheme for windows
• Christophe David's web based implementation of Shamir's scheme 'How to share a Secret'
• Shamir's Secret Sharing in Java : An open source (LGPL) implementation of Shamir's scheme in Java
• An open source implementation of the Shamir's Secret Sharing as open Web application, augmented by additional security features
• libgfshare: a secret sharing library in GF(2**8), opensource (MIT)
• Web implementation of Shamir's method
• Java library implementation of multiple secret sharing methods, opensource(LGPLv2)