# Sharing ![GitHub Workflow Status](https://github.com/davidwilliam/sharing/actions/workflows/main.yml/badge.svg) [![Ruby Style Guide](https://img.shields.io/badge/code_style-rubocop-brightgreen.svg)](https://github.com/rubocop/rubocop) ![GitHub](https://img.shields.io/github/license/davidwilliam/sharing) ![Gem](https://img.shields.io/gem/v/sharing) ![GitHub release (latest by date)](https://img.shields.io/github/v/release/davidwilliam/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: ```ruby 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](https://web.mit.edu/6.857/OldStuff/Fall03/ref/Shamir-HowToShareASecret.pdf). ### n-out-of-n Shamir Secret Sharing Let's consider the followin setup: ```ruby secret1 = 22 secret2 = 36 scalar = 2 params = {total_shares: 5, threshold: 5, lambda_: 16} sss = Sharing::Polynomial::Shamir::V1.new params # => # ``` We generate shares as follows: ```ruby 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: ```ruby 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: ```ruby 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: ```ruby 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](https://github.com/davidwilliam/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. ```ruby 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 # => # ``` We compute the Hensel codes for the secrets: ```ruby 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: ```ruby 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: ```ruby 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: ```ruby 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: ```ruby 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](https://uuslepo.it.da.ut.ee/~peeter_l/teaching/seminar07k/bogdanov.pdf). We define an instance of Shamir's secret sharing scheme with the following parameters: ```ruby params = { lambda_: 16, total_shares: 6, threshold: 3 } # => {:lambda_=>16, :total_shares=>6, :threshold=>3} sss = Sharing::Polynomial::Shamir::V1.new params # => # ``` As before, we define the secrets and create shares for them: ```ruby 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: ```ruby 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: ```ruby 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: ```ruby 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: ```ruby 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: ```ruby 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 ```ruby secret1 * secret2 # => 364 ``` ### t-out-of-n Secret Sharing Now we defined a threshold value that is less than the total number of shares: ```ruby params = {total_shares: 5, threshold: 3, lambda_: 16} # => {:total_shares=>5, :threshold=>3, :lambda_=>16} sss = Sharing::Polynomial::Shamir::V1.new params # => # 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: ```ruby {:lambda_=>16, :total_shares=>6, :threshold=>3} sss = Sharing::Polynomial::Shamir::V1.new(params) ``` and the rational numbers ```ruby rat1 = Rational(2, 3) rat2 = Rational(-5, 7) ``` We compute their Hensel code so we define our secrets: ```ruby 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: ```ruby 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: ```ruby 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: ```ruby 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: ```ruby 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: ```ruby sss.reconstruct_division(cs, ds, r3) # => (-14/15) ``` We can also use only a subset of the shares: ```ruby 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 ```ruby 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](https://www.sciencedirect.com/science/article/pii/S0166218X20303012)). The reference is a CRT-based secret sharing scheme introduced by Asmuth-Bloom in [A modular approach to key safeguarding](https://ieeexplore.ieee.org/abstract/document/1056651). 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. ```ruby 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: ```ruby 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: ```ruby 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: ```ruby 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](https://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](https://opensource.org/licenses/MIT).