Counterion condensation

Proactive secret sharing is a method to update distributed keys (shares) in a secret sharing scheme periodically such that an attacker has less time to compromise shares. This contrasts to a non-proactive scheme where if the threshold number of shares are compromised during the lifetime of the secret, the secret is compromised.

Motivation

If the players (holders of the shared secret) store their shares on insecure computer servers, an attacker could crack in and steal the shares. Since it is not often practical to change the secret, the uncompromised (Shamir-style) shares should be updated in a way that they generate the same secret, yet the old shares are invalidated.

Mathematics

In order to update the shares, the dealer (i.e., the person who gives out the shares) generates a new random polynomial with constant term zero and calculates for each remaining player a new ordered pair, where the x-coordinates of the old and new pairs are the same. Each player then adds the old and new y-coordinates to each other and keeps the result as the new y-coordinate of the secret.

• The dealer constructs a random polynomial over a field of degree $k$ where $k$ is the threshold
• Each player gets the share $x_i^0=f^0(i)$ where $i\in \{1,...,n\}$, $n$ is the number of players, and $x_i^0$ is the share for player $i$ at time interval $0$
• The secret can be reconstructed via interpolation of $k$ shares
• To update the shares, all parties need to construct a random polynomial of the form ${\delta }_i(z)={\delta }_{i,1}{z}^1+{\delta }_{i,2}{z}^2+...+{\delta }_{i,k}{z}^k$
• Each player $i$ sends all other players $u_{i,j}={\delta }_i(j)$
• Each player updates their share by $x_i^{t+1}=x_i^t+u_{1,i}^t+...+u_{n,i}^t$ where $t$ is the time interval in which the shares are valid

All of the non-updated shares the attacker accumulated become useless. An attacker can only recover the secret if he can find enough other non-updated shares to reach the threshold. This situation should not happen because the players deleted their old shares. Additionally, an attacker cannot recover any information about the original secret from the update process because it only contains random information.

The dealer can change the threshold number while distributing updates, but must always remain vigilant of players keeping expired shares.

Example

The following example has 2 shares and a threshold of 2 with 2 players and 1 dealer. Since shares and polynomials are only valid for a certain time period, the time period they are valid is denoted with a superscript.

• All parties agree on a finite field: $Z_{11}$
• The dealer establishes a secret: $x=6\in Z_{11}$
• The dealer constructs a random polynomial over $Z_{11}$ of degree 2 - 1 (threshold of 2)
  • $f^0(x)=6+2\times x$
  • note $f^0(0)=x=6$
• Player 1 gets share $x_1^0=f^0(1)=6+2\times 1=8$ and player 2 gets share $x_2^0=f^0(2)=6+2\times 2=10$
• To reconstruct the secret, use $x_1^0$ and $x_2^0$
  • Since $f^0(x)$ is a line, we can use point slope form to interpolate
  • $m=(f^0(2)-f^0(1))/(2-1)=(x_2^0-x_1^0)/(2-1)=(10-8)/(2-1)=2/1=2$
  • $b=f^0(1)-m=x_1^0-2=8-2=6$
  • $f^0(x)=b+m\times x=6+2\times x$
  • $f^0(0)=6+2\times 0=6=x$
• To update the shares, all parties need to construct random polynomials of degree 1 such that the free coefficient is zero
  • Player 1 constructs ${\delta }_1^0(z)={\delta }_{1,1}^0\times z^1=2\times z^1$
  • Player 2 constructs ${\delta }_2^0(z)={\delta }_{2,1}^0\times z^1=3\times z^1$
• Each player evaluates their polynomial and shares some information with other players
  • Player 1 computes $u_{1,1}^0={\delta }_1^0(1)=2$ and $u_{1,2}^0={\delta }_1^0(2)=4$ in $Z_{11}$
  • Player 1 sends Player 2 $u_{1,2}^0$
  • Player 2 computes $u_{2,1}^0={\delta }_2^0(1)=3$ and $u_{2,2}^0={\delta }_2^0(2)=6$ in $Z_{11}$
  • Player 2 sends Player 1 $u_{2,1}^0$
• Each player updates their share by $x_i^1=x_i^0+u_{1,i}^0+u_{2,i}^0$
  • Player 1 computes $x_1^1=x_1^0+u_{1,1}^0+u_{2,1}^0=8+2+3=2\in Z_{11}$
  • Player 2 computes $x_2^1=x_2^0+u_{1,2}^0+u_{2,2}^0=10+4+6=9\in Z_{11}$
• Confirm updated shares generate same original secret
  • Use $x_1^1$ and $x_2^1$ to reconstruct the polynomial $f^1(x)$
  • Since $f^1(x)$ is a line, we can use point slope
  • $m=(f^1(2)-f^1(1))/(2-1)=(x_2^1-x_{1}1)/(2-1)=(9-2)/(2-1)=7/1=7$
  • $b=f^1(1)-m=x_1^1-7=2-7=-5=6$
  • $f^1(x)=b+m\times x=6+7\times x$
  • $f^1(0)=6+7\times 0=6=x$

See also

• Key (cryptography)
• Key generation
• Key distribution
• Key management
• Shamir's Secret Sharing

References

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  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang