Sharing

Sharing is a Ruby gem with implmementations of secret sharing schemes with homomorphic properties. Although secret sharing schemes and multiparty computation protocols are distinct notions, multiparty computation protocols are typically enabled by secret sharing schemes. In this setting, security comes from the use of multiple parties. If they collude, all security is lost, but satisfactory levels of security can be established by trusting a subset of them will not to collude. In many settings where corrupting security requires corrupting all the parties, and considering you are one of the computing parties, security is guaranteed if you are one of the parties.

Computing linear functions is trivial. Each non-linear operation however requires interaction between the parties/extra steps (for most secret sharing schemes).

Installation

Add this line to your application's Gemfile:

gem 'sharing'

And then execute:

$ bundle install

Or install it yourself as:

$ gem install sharing

Supported Secret Sharing Schemes

Secret Sharing currently supports two schemes:

• A first version of the Shamir's secret sharing
• The second of two modified versions of the CRT-based Asmuth-Bloom scheme proposed by Ersoy et al.

Usage

In the examples below, there are two main levels of execution:

• Computations performed by the owner of the secrets (those are computations using instance methods)
• Computations performed over the secret shares (those are computations using class methods)

This distiction is important since we are showing everything at once here, for completeness and for clarity. However it is important to keep in mind that after the secret shares are generated, the computations over the shares are intended to be computed independetly by each participant (party), each one with their corresponding shares.

Shamir's Secret Sharing V1

The Shamir's secret sharing v1 scheme is based on the work of Adi Shamir in How to Share a Secret.

n-out-of-n Shamir Secret Sharing

Let's consider the followin setup:

secret1 = 22
secret2 = 36
scalar = 2
params = {total_shares: 5, threshold: 5, lambda_: 16}
sss = Sharing::Polynomial::Shamir::V1.new params
# => #<Sharing::Polynomial::Shamir::V1:0x0000000114090618 @lambda_=16, @p=63719, @threshold=3, @total_shares=5>

We generate shares as follows:

shares1 = sss.create_shares(secret1)
# => [[1, 17038], [2, 51463], [3, 24539], [4, 33770], [5, 34327]]
shares2 = sss.create_shares(secret2)
# => [[1, 26584], [2, 37554], [3, 53948], [4, 45589], [5, 58559]]
scalar = 2

We reconstruct the secrets as follows:

reconstructed_secret1 = sss.reconstruct_secret(shares1)
# => 22
reconstructed_secret2 = sss.reconstruct_secret(shares2)
# => 36

We can compute linear functions without requiring communication between the share holders:

shares1_add_shares2 = Sharing::Polynomial::Shamir::V1.add(shares1, shares2, sss.p)
# => [[1, 43622], [2, 25298], [3, 14768], [4, 15640], [5, 29167]]
shares2_sub_shares1 = Sharing::Polynomial::Shamir::V1.sub(shares2, shares1, sss.p)
# => [[1, 9546], [2, 49810], [3, 29409], [4, 11819], [5, 24232]]
shares1_smul_scalar = Sharing::Polynomial::Shamir::V1.smul(shares1, scalar, sss.p)
# => [[1, 34076], [2, 39207], [3, 49078], [4, 3821], [5, 4935]]
shares1_sdiv_scalar = Sharing::Polynomial::Shamir::V1.sdiv(shares1, scalar, sss.p)
# => [[1, 8519], [2, 57591], [3, 44129], [4, 16885], [5, 49023]]

and we can check that:

sss.reconstruct_secret(shares1_add_shares2)
# => 58
sss.reconstruct_secret(shares2_sub_shares1)
# => 14
sss.reconstruct_secret(shares1_smul_scalar)
# => 44
sss.reconstruct_secret(shares1_sdiv_scalar)
# => 11

Using Hensel Codes

The gem Secret Sharing takes advantage of the gem Hensel Code for homomorphically encoding rational numbers as integers in order to compute over the integers and yet obtain results over the rationals.

As most (if not all) of secret sharing schemes over finite fields F_p for p > 2, the secret inputs are naturally required to be positive integers in F_p. In this way, if we compute subtraction and we end up with a result that is negative, the reconstruction will fail (provided we don't have any econding in place). Same will occur if we compute a scalar division involving a scalar that is not a divisor of the corresponding secret. For addressing this and many other arithmetic problems, we can use Hensel codes to allow secret inputs to be positive and negative rational numbers.

rational_secret1 = Rational(2,3)
# => 2/3
rational_secret2 = Rational(-5,7)
# => -5/7
scalar = 5
# => 5
params = {total_shares: 5, threshold: 5, lambda_: 32}
# => {:total_shares=>5, :threshold=>5, :lambda_=>32}
sss = Sharing::Polynomial::Shamir::V1.new params
# => #<Sharing::Polynomial::Shamir::V1:0x0000000103065cd0 @lambda_=32, @total_shares=5, @threshold=5, @p=4151995223>

We compute the Hensel codes for the secrets:

secret1 = HenselCode::TruncatedFinitePadicExpansion.new(sss.p, 1, rational_secret1).hensel_code
# => 2767996816
secret2 = HenselCode::TruncatedFinitePadicExpansion.new(sss.p, 1, rational_secret2).hensel_code
# => 593142174

Then, we create the shares:

shares1 = sss.create_shares(secret1)
# => [[1, 1788895381], [2, 1795799163], [3, 3852643947], [4, 58410522], [5, 2611091242]]
shares2 = sss.create_shares(secret2)
# => [[1, 2523224758], [2, 2966680092], [3, 3722500411], [4, 3217222534], [5, 656923087]]

Now we can compute all the available linear computations as before:

shares1_add_shares2 = Sharing::Polynomial::Shamir::V1.add(shares1, shares2, sss.p)
# => [[1, 160124916], [2, 610484032], [3, 3423149135], [4, 3275633056], [5, 3268014329]]
shares1_sub_shares2 = Sharing::Polynomial::Shamir::V1.sub(shares1, shares2, sss.p)
# => [[1, 3417665846], [2, 2981114294], [3, 130143536], [4, 993183211], [5, 1954168155]]
shares1_smul_scalar = Sharing::Polynomial::Shamir::V1.smul(shares1, scalar, sss.p)
# => [[1, 640486459], [2, 675005369], [3, 2655238843], [4, 292052610], [5, 599470541]]
shares1_sdiv_scalar = Sharing::Polynomial::Shamir::V1.sdiv(shares1, scalar, sss.p)
# => [[1, 2848976210], [2, 3680756011], [3, 1600927834], [4, 842081149], [5, 1352617293]]

We reconstruct the secrets:

reconstruct_secret1_add_secret2 = sss.reconstruct_secret(shares1_add_shares2)
# => 3361138990
reconstruct_secret1_sub_secret2 = sss.reconstruct_secret(shares1_sub_shares2)
# => 2174854642
reconstruct_shares1_smul_scalar = sss.reconstruct_secret(shares1_smul_scalar)
# => 1383998411
reconstruct_shares1_sdiv_scalar = sss.reconstruct_secret(shares1_sdiv_scalar)
# => 3044796497

and we can check that:

HenselCode::TruncatedFinitePadicExpansion.new(sss.p, 1, reconstructed_secret1_add_secret2).to_r
# => -1/21
HenselCode::TruncatedFinitePadicExpansion.new(sss.p, 1, reconstructed_secret1_sub_secret2).to_r
# => 29/21
HenselCode::TruncatedFinitePadicExpansion.new(sss.p, 1, reconstructed_shares1_smul_scalar).to_r
# => 10/3
HenselCode::TruncatedFinitePadicExpansion.new(sss.p, 1, reconstructed_shares1_sdiv_scalar).to_r
# => 2/15

Multiplication

As we previously saw, linear functions are easy to compute with shares created by an instance of Shamir's secret sharing scheme. Non-linear functions need some strategy that require extra steps in other to successfuly achieve the desired results. We implement multiplication in the context of Shamir's secret sharing scheme following the approach discussed by Dan Bognadov in Foundations and properties of Shamir's secret sharing scheme - Research Seminar in Cryptography.

We define an instance of Shamir's secret sharing scheme with the following parameters:

params = { lambda_: 16, total_shares: 6, threshold: 3 }
# => {:lambda_=>16, :total_shares=>6, :threshold=>3}
sss = Sharing::Polynomial::Shamir::V1.new params
# => #<Sharing::Polynomial::Shamir::V1:0x0000000105423640 @lambda_=16, @p=49367, @threshold=3, @total_shares=6>

As before, we define the secrets and create shares for them:

secret1 = 13
secret2 = 28
shares1 = sss.create_shares(secret1)
# => [[1, 43064], [2, 20333], [3, 30554], [4, 24360], [5, 1751], [6, 12094]]
shares2 = sss.create_shares(secret2)
# => [[1, 7983], [2, 18517], [3, 31630], [4, 47322], [5, 16226], [6, 37076]]

We combine both shares on a single array in preparation for the multiplication steps:

operands_shares = [shares1, shares2]
# => [[[1, 43064], [2, 20333], [3, 30554], [4, 24360], [5, 1751], [6, 12094]], [[1, 7983], [2, 18517], [3, 31630], [4, 47322], [5, 16226], [6, 37076]]]

Recall we are working with a t-out-of-n secret sharing scheme and this is actually required in this setting. We have a total of n = 6 shares and threshold t = 3. In order to correctly recover the result of the multiplication over shares, we need to select any combination of 2 * t - 1 shares out of the total number of shares:

selected_shares = Sharing::Polynomial::Shamir::V1.select_mul_shares(sss.total_shares, sss.threshold, operands_shares)
# => => [[[2, 20333], [1, 43064], [5, 1751], [3, 30554], [4, 24360]], [[2, 18517], [1, 7983], [5, 16226], [3, 31630], [4, 47322]]]

Now we have everything we need to compute multiplication over the secret shares, which we do in two rounds. First round:

mul_round1 = Sharing::Polynomial::Shamir::V1.mul_first_round(selected_shares, sss.total_shares, sss.threshold, sss.lambda_, sss.p)
# => [[2, [[1, 25284], [2, 5881], [3, 2537], [4, 15252], [5, 44026], [6, 39492]]], [1, [[1, 36061], [2, 17299], [3, 32435], [4, 32102], [5, 16300], [6, 34396]]], [5, [[1, 30221], [2, 32724], [3, 33210], [4, 31679], [5, 28131], [6, 22566]]], [3, [[1, 46172], [2, 33017], [3, 8081], [4, 20731], [5, 21600], [6, 10688]]], [4, [[1, 12410], [2, 39133], [3, 5920], [4, 11505], [5, 6521], [6, 40335]]]]

Then we perform the second round:

mul_round2 = Sharing::Polynomial::Shamir::V1.mul_second_round(mul_round1)
# => [[1, 150148], [2, 128054], [3, 82183], [4, 111269], [5, 116578], [6, 147477]]

Then we only need a number equal to the threshold to reconstruct the result of the multipliction over the shares:

selected_multiplication_shares = mul_round2.sample(sss.threshold)
# => [[6, 147477], [2, 128054], [1, 150148]
sss.reconstruct_secret(selected_multiplication_shares)
# => 364

and we can check that

secret1 * secret2
# => 364

t-out-of-n Secret Sharing

Now we defined a threshold value that is less than the total number of shares:

params = {total_shares: 5, threshold: 3, lambda_: 16}
# => {:total_shares=>5, :threshold=>3, :lambda_=>16}
sss = Sharing::Polynomial::Shamir::V1.new params
# => #<Sharing::Polynomial::Shamir::V1:0x000000010b046e90 @lambda_=16, @p=61343, @threshold=3, @total_shares=5>
secret = 25
# => 25
shares = sss.create_shares(secret)
# => [[1, 54707], [2, 50401], [3, 48450], [4, 48854], [5, 51613]]
selected_shares = shares.sample(3)
reconstructed_secret = sss.reconstruct_secret(selected_shares)
# => 25

Everything else works the sabe as before except the fact that only 3 shares are required to reconstruct the secret.

Division

Here we simulate a division algorithm computed over the shares. Consider the following configuration:

{:lambda_=>16, :total_shares=>6, :threshold=>3}
sss = Sharing::Polynomial::Shamir::V1.new(params)

and the rational numbers

rat1 = Rational(2, 3)
rat2 = Rational(-5, 7)

We compute their Hensel code so we define our secrets:

secret1 = HenselCode::TFPE.new(sss.p, 1, rat1).hensel_code
# => 40896
secret2 = HenselCode::TFPE.new(sss.p, 1, rat2).hensel_code
# => 52579

We then comute the shares for each secret:

shares1 = sss.create_shares(secret1)
# => [[1, 22792], [2, 38518], [3, 26731], [4, 48774], [5, 43304], [6, 10321]]
shares2 = sss.create_shares(secret2)
# => [[1, 37982], [2, 294], [3, 858], [4, 39674], [5, 55399], [6, 48033]]

We select a subset of the shares of eacg secret based on the threshold:

selected_shares1 = shares1.sample(sss.threshold)
# => [[1, 22792], [5, 43304], [6, 10321]]
selected_shares2 = shares2.sample(sss.threshold)
# => [[6, 48033], [2, 294], [1, 37982]]

A party that is not one of the selected parties (a trusted party for instance) generates three random values that will later used:

r1, r2, r3 = Sharing::Polynomial::Shamir::V1.generate_division_masking(sss.p)
=> [23078, 18925, 18812]

Each party multiply their shares of the first operand by r1 and the shares of the second operand by r2. For convenience (given our simulation), this step is here done all at once as follows:

cs, ds = Sharing::Polynomial::Shamir::V1.compute_numerator_denominator(selected_shares1, selected_shares2, r1, r2, sss.p)
=> [[[1, 38894], [5, 30899], [6, 54512]], [[6, 43951], [2, 43080], [1, 53419]]]

cs denote the shares representing the numerator of the division and ds represent the shares of the denominator of the division, as in c/d.

Finally, in the reconstruction step, c and d are reconstructed and r3 is used to invert the multiplication by r1 and r2 that was previously computed by the parties. With the correct values recovered, we compute the Hensel decoding and then we obtain the final result without revealing what the numerator and denominator were. For convenience, we execute these steps as follows:

sss.reconstruct_division(cs, ds, r3)
# => (-14/15)

We can also use only a subset of the shares:

selected_cs = cs.sample(sss.threshold)
=> [[5, 30899], [1, 38894], [6, 54512]]
selected_ds = ds.sample(sss.threshold)
# => [[2, 43080], [1, 53419], [6, 43951]]
sss.reconstruct_division(selected_cs, selected_ds, r3)
=> (-14/15)

And we can check that

rat1 / rat2
# => (-14/15)

Asmuth-Bloom V2

The Asmuth-Bloom V2 was proposed by Ersoy et al. in in Homomorphic extensions of CRT-based secret sharing). The reference is a CRT-based secret sharing scheme introduced by Asmuth-Bloom in A modular approach to key safeguarding.

We have currently the class Sharing::CRT::AsmuthBloom::V2. To initialize it, we need to pass the following parameters:

• lambda_: the bit length of the secret prime moduli.
• threshold: the recovery threshold in which it is guaranteed to recover the secret for all possible values.
• secrecy: the secrecy threshold in which it is guaranteed that no information is revealed about the secret.
• total_shares: the total number of shares we want to create for any given secret.
• k_add: the provisioned number of additions we want to compute over shares.
• k_add: the provisioned number of multiplicaitons we want to compute over shares.

params = { lambda_: 64, threshold: 10, secrecy: 3, total_shares: 13, k_add: 5000, k_mul: 2 }
crtss = Sharing::CRTAsmuthBloomV2.new params
secret1 = 5
secret2 = 8
secret3 = 9
shares1 = crtss.compute_shares(secret1)
# => [[0, 4185685952388161215], [1, 7431082072249155627], [2, 3172867207673420707], [3, 14094855932661978905], [4, 8449128283552032507], [5, 7274923078167548868], [6, 3443672123003372167], [7, 3449028625755130838], [8, 5794221801968596287], [9, 3328886357835317095], [10, 7488573194762652917], [11, 8211719263601562780], [12, 12709118192143848454]]
shares2 = crtss.compute_shares(secret2)
# => [[0, 2233143134937563130], [1, 10799196764783177850], [2, 3895399176949806798], [3, 1198864688298180029], [4, 10749884548017217271], [5, 8208674321670086887], [6, 2822739185939232463], [7, 6792525158886356123], [8, 11182441441011760155], [9, 5065252015479538675], [10, 14231070486785344674], [11, 12955329422395114581], [12, 13444354079541844356]]
shares3 = crtss.compute_shares(secret3)
# => [[0, 8784211624348791413], [1, 3173871378881529520], [2, 13248531955944997083], [3, 1782634630778360250], [4, 13054421101338573568], [5, 12464404777826322232], [6, 10309434923908541341], [7, 11837628883332260554], [8, 7022273320911219172], [9, 2554741791512322214], [10, 11331979459726025879], [11, 5610455238685743922], [12, 3003841353993251584]]

With the shares created, we can compute basic arithmetic:

shares1_add_shares2 = Sharing::CRTAsmuthBloomV2.add(shares1, shares2)
# => [[0, 6418829087325724345], [1, 18230278837032333477], [2, 7068266384623227505], [3, 15293720620960158934], [4, 19199012831569249778], [5, 15483597399837635755], [6, 6266411308942604630], [7, 10241553784641486961], [8, 16976663242980356442], [9, 8394138373314855770], [10, 21719643681547997591], [11, 21167048685996677361], [12, 26153472271685692810]]
shares1_mul_shares2 = Sharing::CRTAsmuthBloomV2.mul(shares1, shares2)
# => [[0, 9347235849580217942880423213680002950], [1, 80249717473471354529348429396269261950], [2, 12359584309342074742148630183422566186], [3, 16897825064318556900157377557090288245], [4, 90827153579571227732364682022273828397], [5, 59717474263879064686877969113378493916], [6, 9720588245128167152782160092717057321], [7, 23427613714160960605536958120927421074], [8, 64793545996747467446843059564467544485], [9, 16861648333327680706874620252941149125], [10, 106570412980118630776546521824476514058], [11, 106385528184186069985041673188764895180], [12, 170865885013928618679442811690919225624]]
shares1_add_shares2_add_shares1_mul_shares2 = Sharing::CRTAsmuthBloomV2.add(shares1_add_shares2, shares1_mul_shares2)
# => [[0, 9347235849580217949299252301005727295], [1, 80249717473471354547578708233301595427], [2, 12359584309342074749216896568045793691], [3, 16897825064318556915451098178050447179], [4, 90827153579571227751563694853843078175], [5, 59717474263879064702361566513216129671], [6, 9720588245128167159048571401659661951], [7, 23427613714160960615778511905568908035], [8, 64793545996747467463819722807447900927], [9, 16861648333327680715268758626256004895], [10, 106570412980118630798266165506024511649], [11, 106385528184186070006208721874761572541], [12, 170865885013928618705596283962604918434]]

In order to recover the associated secrets with the shares we just generated via basic arithmetic computations, we select a number of shares for reconstruction (in any random order). It could be the total number of shares or a smaller number. We will choose the exact number of the recovery threshold:

selected_shares1_add_shares2 = shares1_add_shares2.sample(params[:threshold])
# => [[10, 21719643681547997591], [0, 6418829087325724345], [9, 8394138373314855770], [6, 6266411308942604630], [1, 18230278837032333477], [7, 10241553784641486961], [11, 21167048685996677361], [5, 15483597399837635755], [8, 16976663242980356442], [12, 26153472271685692810]]
selected_shares1_mul_shares2 = shares1_mul_shares2.sample(params[:threshold])
# => [[5, 59717474263879064686877969113378493916], [1, 80249717473471354529348429396269261950], [12, 170865885013928618679442811690919225624], [7, 23427613714160960605536958120927421074], [6, 9720588245128167152782160092717057321], [0, 9347235849580217942880423213680002950], [3, 16897825064318556900157377557090288245], [9, 16861648333327680706874620252941149125], [2, 12359584309342074742148630183422566186], [4, 90827153579571227732364682022273828397]]
selected_shares1_add_shares2_add_shares1_mul_shares2 = shares1_add_shares2_add_shares1_mul_shares2.sample(params[:threshold])
# => [[11, 106385528184186070006208721874761572541], [6, 9720588245128167159048571401659661951], [2, 12359584309342074749216896568045793691], [4, 90827153579571227751563694853843078175], [12, 170865885013928618705596283962604918434], [7, 23427613714160960615778511905568908035], [9, 16861648333327680715268758626256004895], [5, 59717474263879064702361566513216129671], [10, 106570412980118630798266165506024511649], [0, 9347235849580217949299252301005727295]]

Finally, we reconstruct the secrets:

crtss.reconstruct_secret(selected_shares1_add_shares2)
# => 13 
crtss.reconstruct_secret(selected_shares1_mul_shares2)
# => 40
crtss.reconstruct_secret(selected_shares1_add_shares2_add_shares1_mul_shares2)
# => 53

and we can check that 5 + 8 = 13, 5 * 8 = 40, and 13 + 40 = 53.

Author

David William Silva

Contributors

David William Silva (Algemetric) Marcio Junior (Algemetric)

Acknowledgements

Luke Harmon (Algemetric) and Gaetan Delavignette (Algemetric) have been instrumental by providing/conducting mathematical analyses, tests, and overall recommendations for improving the gem.

Development

After checking out the repo, run bin/setup to install dependencies. Then, run rake test to run the tests. You can also run bin/console for an interactive prompt that will allow you to experiment.

To install this gem onto your local machine, run bundle exec rake install. To release a new version, update the version number in version.rb, and then run bundle exec rake release, which will create a git tag for the version, push git commits and the created tag, and push the .gem file to rubygems.org.

Contributing

Bug reports and pull requests are welcome on GitHub at https://github.com/davidwilliam/sharing.

License

The gem is available as open source under the terms of the MIT License.