SECRET SHARING STEP BY STEP

In this blog article I will show the different types of secret sharing methods especially the common used Shamires secret sharing method. Thereafter I will explain the mathematical background of this procedure.

What is Secret Sharing about?

Let’s start with the following situation. A big heritage should be distributed over 6 heirs. The heritage is stored in a safe with a password. To avoid that only one person opens the safe and takes all the money a so called dealer splits the password into six parts (keys) and gives every part (key) to one of the six heirs (so called players). He also gives them a set of rules on how to recombine these pieces together to reconstitute the former password so that they can open the safe.

“Secret sharing” or secret splitting describes the methodology of distributing secret information among a group of participants in a way that a subset of this group is needed to reconstruct the secret. Whereby a single piece of information cannot be used to reconstruct the secret.

Secret sharing has a wide operation range. It can be used in high security areas for validation reasons e.g. in the military or in the banking area where it might be necessary to have a 4, 6 or 8 eye validation principle. It also can be used to ensure the secure transport of information e.g. transport of a cryptographic key over several channels. Last but not least it can be used in secure access regulation where only a combination of trustworthy people might get access to a secure area.

Different types of secret sharing methods

You can distinguish between two types of secret sharing methods. The ones which needs the partial information of all players to reconstruct the secret called (n,n) schemes (pronounced N out of N scheme). The others need only a sufficient amount of partial information called (k,n) threshold scheme (pronounced K out of N threshold scheme).

(n,n) schemes

The trivial way to share some information (e.g. password) amongst a group of participants is to divide the information into pieces and give every piece to a participant. Like in the following example where each colored number is a piece of the whole.

123 456 789 = 123456789

Other (n,n) schemes include more complex processing than just appending the information pieces. E.g. the concatenation of keys with an xor operator

123 xor 456 xor 789 = 678

These approaches have the same disadvantage they each need all keys to reconstruct the secret. In case one of the keys is lost the secret is lost as well (at least theoretically; depending on the approach it may just need some computational effort to calculate the missing key[1]). Additionally (n,n) schemes are not flexible for key expansion. If further keys are needed all keys have to be recomputed.

(K ,N) threshold schemes

In contrast to the before discussed (n,n) schemes (k,n) threshold schemes are flexible in terms of participant expansion. It is easy to add a new key to the set of the existing keys without recalculating them.

The simple methods are based on reflection about geometrical primitives crossing each other in one point (the secret). E.g. (2,n) threshold scheme: at least two lines in a Euclidean two dimensional space cross each other at one specific point. It is also easy to find another line which cross the crossing point of the first two at the same point.

On the one hand it is easy to get a (k,n) threshold scheme by extending the former method in dimensionality of the space, by k dimensions e.g. (3, n) would be the computation of the intersection point between 3 planes in 3 dimensions. On the other hand the more keys that come together, the more information is known about the secret and the reconstruction of the secret in spite of missing keys becomes easier.

Shamires secret sharing

In 1979 Adi Shamires – the S in RSA – published a paper with the title “How to share a Secret”[2].

Inside this paper he presents a (K,N) threshold scheme which is easy to compute for K out of N keys but gives no information about the secret if less than K keys are known (in the sense that all possible keys are equally likely).

It is based on the fact that every polynomial (q(x) = a0+a1x1 … + ak-1xk-1) is unique determined by the degree of the polynomial and the quantity of given points (which has to be one more than the degree of the polynomial).

Some useful properties of this scheme as being secure, minimal, extensible, dynamic and flexible are good explained in [3].

Before we continue I would recommend you to try it out here[4].

How it works basic concept step by step?

1) k is the least quantity of participants you would like to be able to reconstruct the secret

2) Choose a polynomial of grade k-1

3) Define k random indices (a0…ak-1)

4) Define k points which can be calculated by the polynomial

5) Distribute k points to the participants.

6) To calculate the secret out of the keys use e.g. Lagrange interpolation [5] with x=0 .

Shamires secret sharing graphical explanation

Let’s start with a simple polynomial y = a 0+ a1x1.

a0 is our secret and since the grade of the polynomial is 1 we need at least two points to determine it (see figure 1).

figure 1: graph of a first grade polinomial. a0 is the secret e.g. password. The points are the given keys

If you have only one point given an infinite account of possible solutions can be calculated (see figure 2).

figure 2: graph of a first grade polinomial. a0 is the secret e.g. password. Since there is only one fix point the secret is not reconstructible, because an infinite amount of lines are crossing the point

If K is supposed to be bigger than 2 just take a polynomial with a higher grade e.g. y = a0-a2x2+a0x4 in this case K is equal 5 (see figure 3).

figure 3: graph of a fourth grade polynomial. a0 is the secret e.g. password. The points are the given keys.

Now if you would like to add more participants to have a key you could just create a new key by taking another point of the polynomial (see figure 6).

figure 4: graph of a fourth grade polynomial. a0 is the secret e.g. password. The points are the given keys. The x are additional keys that could be created without interfering with already existing keys.

Summary

In this blog you have learned about different types of secret sharing methods, the possible fields of their usage, the weak points of simple approaches and the benefits of conventional methods. Especially the basics of Shamires secret sharing method has been explained step by step as well as in a comprehensive graphical form.

Yet I did not discuss the possibility of a corrupt key dealer and how to handle this problem.

For detailed explanation please visit the following links or write me a mail.

[1] http://www.gm.fh-koeln.de/~konen/DisMa/secret-sharing.pdf

[2] http://groups.csail.mit.edu/cis/crypto/classes/6.857/papers/secret-shamir.pdf

[3] http://en.wikipedia.org/wiki/Shamir%27s_Secret_Sharing

[4] http://point-at-infinity.org/ssss/demo.html

[5] http://de.wikipedia.org/wiki/Polynominterpolation#Lagrangesche_Interpolationsformel

http://point-at-infinity.org/ssss/

http://de.wikipedia.org/wiki/Shamir%E2%80%99s_Secret_Sharing

SECRET SHARING STEP BY STEP相关推荐

Asp Net Core 5 REST API 使用 RefreshToken 刷新 JWT - Step by Step（三）
翻译自 Mohamad Lawand 2021年1月25日的文章 <Refresh JWT with Refresh Tokens in Asp Net Core 5 Rest API Step ...
Asp.Net Core 5 REST API 使用 JWT 身份验证 - Step by Step（二）
翻译自 Mohamad Lawand 2021年1月22日的文章 <Asp Net Core 5 Rest API Authentication with JWT Step by Step> ...
JWT 介绍 - Step by Step
翻译自 Mohamad Lawand 2021年3月11日的文章 <Intro to JWT - Step by Step> [1] 在本文中,我将向您介绍 JWT[2]. 我们今天要讲的 ...
ChatGPT初体验step by step：ChatGPT解决人类提出的数理逻辑问题，Python编程实践
ChatGPT初体验step by step:ChatGPT解决人类提出的数理逻辑问题,Python编程实践如果已有有效的open ai的api key,则跳过本文(1)(2)(3)(4),直接从( ...
【Step By Step】将Dotnet Core部署到Docker下
一.使用.Net Core构建WebAPI并访问Docker中的Mysql数据库这个的过程大概与我之前的文章<尝试.Net Core-使用.Net Core + Entity FrameWor ...
python写一个通讯录step by step V3.0
python写一个通讯录step by step V3.0 更新功能: 数据库进行数据存入和读取操作字典配合函数调用实现switch功能其他:函数.字典.模块调用注意问题: 1.更优美的格式化输 ...
C#2.0实例程序STEP BY STEP－－实例二:数据类型
C#2.0实例程序STEP BY STEP--实例二:数据类型与其他.NET语言一样,C#支持Common Type Sysem(CTS),其中的数据类型集合不仅包含我们熟悉的基本类型,例如int, ...
文本分类step by step(二)
(注:如有转载请标明作者:finallyliuyu, 和出处:博客园) <文本分类 step by step(一)> 在<文本分类step by step(一)>中,我们从处理 ...
数据库设计Step by Step (9)——ER-to-SQL转化
2019独角兽企业重金招聘Python工程师标准>>> 引言:前文(数据库设计 Step by Step (8)--视图集成)讨论了如何把局部ER图集成为全局ER图.有了全局ER图后 ...

SECRET SHARING STEP BY STEP

SECRET SHARING STEP BY STEP相关推荐

最新文章

热门文章