Shamir’s Secret SharingA Simple Applications of Polynomial Ring to Protect Your Secret

Motivation It is not a good idea to keep the whole secret alone Especially if it is highly sensitive and highly important.

The way to solve this problem is to divide the original secret into parts.

Actually, there is a rules to divide the original secret and to reconstruct the original secret. Mathematics is important here.

Then, how we divide the secret so that it will increase the security of the original secret?

Secret Sharing Informally, Secret sharing is any method for distributing a secret amongst a group of individuals (shareholders) each of which is allocated some information (share) related to the secret.

(Adi Shamir & George Blakley, 1979)

The secret can only be reconstructed when the shares are combined together.

Individual shares are of no use on their own.

Before we talk about Shamir’s secret sharing, lets see about this scheme first.

Threshold Scheme The purpose of this scheme is to divide a secret into shares such that:

1. The reconstruction of secret requires a knowledge of or more shares.

2. A knowledge of or less shares leaves the secret completely undetermined.

This scheme is called the -threshold scheme and is the threshold value

Shamir’s Secret Sharing The main idea is: two 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, points are sufficient to define a polynomial of degree .

Shamir’s secret sharing scheme is a -threshold scheme based from polynomial interpolation.

Instead of sharing the random numbers, Shamir’s secret sharing scheme generate a polynomial

from random numbers which is an elements of finite field of size and again where is prime numbers.

Usually the finite field is used.

How to Share Recall the polynomial over

Suppose that the secret to be divided into shares. Then, compute

for So we have an ordered pairs of points , which is the shares, to be distributed to participants or shareholders.

How to Reconstruct the Secret Let contain exactly elements Formalizing the Lagrange interpolation over a finite field. Let for

Thus

is the original secret.

Example Let and the threshold value be Choose at random and in . For example and Now we have over Then generate as many share as we wish. For example if we have

Suppose we have shares then we can reconstruct the secret from

over . Hence,

Observation Properties:

1. Information theoretically secure2. Make us of Lagrange interpolation3. Space efficient

Advantages:1. Keeping fixed, shares can be easily added or removed without affecting other share2. It is easy to change the shares3. It is possible to provide more than one share per individual: hierarchy

Problems1. If the participants cheat in reconstruction of secret, the secret

cannot be recovered. That is, every persons/parties should tell the truth or the secret can not be reconstructed.

2. The scheme is one-time.3. The scheme only allows revealing a secret, not computing with it.

Another Scheme Verifiable Secret Sharing (VSS) could fix the first problem above. Proactive Secret Sharing: periodically renew the shares (from Shamir’s scheme) without changing the secret S.

Reference Munir,Rinaldi, Baratha,Addie, “Studi Dan Implementasi Clustering Penerima Kunci Dengan Metode Shamir Secret Sharing Advanced”

http://informatika.stei.itb.ac.id/~rinaldi.munir/TA/Makalah_TA%20Addie%20Barata.pdf . Tanggal akses: 26 Mei 2015

Zanin,Giorgio, “Secret Sharing Schemes and their Applications”http://wwwusers.di.uniroma1.it/smart/ppt/zanin.pdf . Tanggal akses: 21 Mei 2015

http://www.cs.berkeley.edu/~daw/teaching/cs276-s04/22.pdf . Tanggal akses: 29 Mei 2015http://scholarworks.uno.edu/cgi/viewcontent.cgi?article=2314&context=td . Tanggal akses: 30 Mei 2015