Clifford GA Ruby

Ruby library for data representation and data concealment with Ruby.

Requirements

This code requires Ruby installed on your system. There are several options for downloading and installing Ruby.

This project uses only Ruby standard libraries, so once you have Ruby installed (version 2.6.3 and greater), you have everything required to run the code. We tested our implementation on Mac OSX version 10.13.6 with ruby 2.6.3p62 (2019-04-16 revision 67580) [x86_64-darwin17].

Usage

Once you have the project in your working environment, at the root directory of the project, run:

gem install bundler

if you don't have Bundler installed yet, and then run

bundle install

Running tests

Once Ruby is installed on your machine, from the command line and in the root directory of the project, run the tests to check the code with the following command:

bundle exec rake

You should get a result similiar to the following:

Run options: --seed 52080

# Running:

................................

Finished in 0.009734s, 3287.4460 runs/s, 5033.9017 assertions/s.

32 runs, 49 assertions, 0 failures, 0 errors, 0 skips

Ruby Interactive Shell

You can also run code from the Ruby Interactive Shell (IRB). From the project's root directory, execute the following command on the terminal:

$ irb

You will see the IRB's prompt. Next, command snippets for specific cases that can be executed on IRB.

Working with multivectors

Require the file the will boot the entire project on IRB:

> require './boot'

Creating a multivector:

> m = Clifford::Multivector3D.new [2,3,4,5,6,7,8,9]

which returns

=> 2e0 + 3e1 + 4e2 + 5e3 + 6e12 + 7e13 + 8e23 + 9e123

Clifford conjugation:

> m.clifford_conjugation or > m.cc

=> 2e0 + -3e1 + -4e2 + -5e3 + -6e12 + -7e13 + -8e23 + 9e123

Reverse:

> m.reverse

=> 2e0 + 3e1 + 4e2 + 5e3 + -6e12 + -7e13 + -8e23 + -9e123

Amplitude squared:

> m.amplitude_squared

=> 22e0 + 0e1 + 0e2 + 0e3 + 0e12 + 0e13 + 0e23 + -16e123

Rationalize:

> m.rationalize

=> 740e0 + 0e1 + 0e2 + 0e3 + 0e12 + 0e13 + 0e23 + 0e123

Inverse:

> m.inverse

=> -5/37e0 + 31/370e1 + -10/37e2 + -7/370e3 + -53/185e12 + -9/74e13 + -56/185e23 + 23/74e123

Geometric product:

> m.geometric_product(m.inverse) or >> m.gp(m.inverse)

=> 1e0 + 0e1 + 0e2 + 0e3 + 0e12 + 0e13 + 0e23 + 0e123

> m.gp(m)

=> -176e0 + -132e1 + 142e2 + -88e3 + 114e12 + -44e13 + 86e23 + 88e123

Addition:

> m.plus(m)

=> 4e0 + 6e1 + 8e2 + 10e3 + 12e12 + 14e13 + 16e23 + 18e123

Subtraction:

> m.minus(m)

=> 0e0 + 0e1 + 0e2 + 0e3 + 0e12 + 0e13 + 0e23 + 0e123

Scalar division:

> m.scalar_div(2)

=> 1e0 + 3/2e1 + 2e2 + 5/2e3 + 3e12 + 7/2e13 + 4e23 + 9/2e123

Scalar multiplication:

> m.scalar_mul(2)

=> 4e0 + 6e1 + 8e2 + 10e3 + 12e12 + 14e13 + 16e23 + 18e123

All multivectors M in Cl(3,0) can be decomposed as in M = Z + F. Obtaining Z:

> m.z

=> 2e0 + 0e1 + 0e2 + 0e3 + 0e12 + 0e13 + 0e23 + 9e123

Obtaining F:

> m.f

=> 0e0 + 3e1 + 4e2 + 5e3 + 6e12 + 7e13 + 8e23 + 0e123

Obtaining F squared:

> m.f2

=> -99e0 + 0e1 + 0e2 + 0e3 + 0e12 + 0e13 + 0e23 + 52e123

Obtaining the eigenvalues a multivector:

> m.eigenvalues

=> [(4.5323662175072155+19.26707741568027i), (-0.5323662175072155-1.2670774156802675i)]

Packing multivectors

Clifford Eigenvalue Packing Scheme

> a = Clifford::Packing.cep_forward(23)

181/2e0 + -35507199447/428048393e1 + -68935712985/856096786e2 + 144683344314/428048393e3 + -29616707322/428048393e12 + 145732761090/428048393e13 + 20785971411/428048393e23 + 0e123

> b = Clifford::Packing.cep_forward(18)

=> 111e0 + -244605151746/2140241965e1 + -47489046723/428048393e2 + 996707483052/2140241965e3 + -204026205996/2140241965e12 + 200787359724/428048393e13 + 143192247498/2140241965e23 + 0e123

Recovering the packed message:

> Clifford::Packing.cep_backward(a)

=> 23

> Clifford::Packing.cep_backward(b)

=> 18

Addition:

a_plus_b = a.plus(b)

=> 403/2e0 + -422141148981/2140241965e1 + -163913806431/856096786e2 + 1720124204622/2140241965e3 + -352109742606/2140241965e12 + 346520120814/428048393e13 + 247122104553/2140241965e23 + 0e123

> Clifford::Packing.cep_backward(a_plus_b)

=> 41

Multiplication:

> a_times_b = a.gp(b)

=> 16323e0 + -41843261926098/2140241965e1 + -8123690799099/428048393e2 + 170501283310476/2140241965e3 + -34901644206348/2140241965e12 + 34347592536012/428048393e13 + 24495112531674/2140241965e23 + 0e123

> Clifford::Packing.cep_backward(a_times_b)

=> 414

Complex Magnitude Squared Packing Scheme

> a = Clifford::Packing.cmsp_forward(26)

=> 6.471338332083668+31.438854634950513ie0 + 4.906228887222669+41.4679929927055ie1 + 22e2 + 29e3 + 19e12 + 22e13 + 18e23 + 24e123

> b = Clifford::Packing.cmsp_forward(41)

=> -103.99587604803894-2.337670283500854ie0 + -94.53195854374974-2.5717024463389007ie1 + 52e2 + 49e3 + 38e12 + 36e13 + 56e23 + 51e123

Multiplication:

> a_times_b = a.gp(b)

=> -2137.6509095698448-7217.307040332529ie0 + -3020.185983839764-7312.5882643378845ie1 + 1275.1442302572004+3208.0377749836557ie2 + -2027.457499218519+3022.136640447904ie3 + 2675.8162018045127+3363.173830181745ie12 + -1063.1090798591918+3186.881048207597ie13 + -3806.080154069653+3771.64457837006ie23 + -3495.689206319693+3823.1944631358633ie123

> Clifford::Packing.cmsp_backward(a_times_b)

=> 1066

Concealing multivectors

Clifford Sylvester's Equation Concelament Scheme

First, pack two multivectors A and B:

> a = Clifford::Packing.cep_forward(12)

=> 74e0 + -163070101164/2140241965e1 + -31659364482/428048393e2 + 664471655368/2140241965e3 + -136017470664/2140241965e12 + 133858239816/428048393e13 + 95461498332/2140241965e23 + 0e123

> b = Clifford::Packing.cep_forward(19)

=> 177/2e0 + -182796323079/2140241965e1 + -70978252629/856096786e2 + 744851291098/2140241965e3 + -152471196954/2140241965e12 + 150050768826/428048393e13 + 107009260227/2140241965e23 + 0e123

Generate a tuple of secret keys:

k = Clifford::Tools.generate_regular_keys

=> [189e0 + 216e1 + 202e2 + 222e3 + 152e12 + 172e13 + 202e23 + 187e123, 184e0 + 235e1 + 176e2 + 225e3 + 148e12 + 177e13 + 144e23 + 145e123]

Conceal:

> ca = Clifford::Concealment.csec_forward(k,a)

=> -618818306222/428048393e0 + -19501147572462/2140241965e1 + 265089164434222/2140241965e2 + 361931982799054/2140241965e3 + 263927103941858/2140241965e12 + 352630738821406/2140241965e13 + 51070915692586/2140241965e23 + -3998489933290/428048393e123

> cb = Clifford::Concealment.csec_forward(k,b)

=> 384383003877/856096786e0 + -33009163124779/4280483930e1 + 603290221657139/4280483930e2 + 822044307451883/4280483930e3 + 598832474024101/4280483930e12 + 798864039917347/4280483930e13 + 122715122589577/4280483930e23 + -7387370951481/856096786e123

Addition over concealed data:

> ca_plus_cb = ca.plus(cb)

=> -853253608567/856096786e0 + -72011458269703/4280483930e1 + 1133468550525583/4280483930e2 + 1545908273049991/4280483930e3 + 1126686681907817/4280483930e12 + 1504125517560159/4280483930e13 + 224856953974749/4280483930e23 + -15384350818061/856096786e123

> a_plus_b = Clifford::Concealment.csec_backward(k,ca_plus_cb)

=> 325/2e0 + -345866424243/2140241965e1 + -134296981593/856096786e2 + 1409322946466/2140241965e3 + -288488667618/2140241965e12 + 283909008642/428048393e13 + 202470758559/2140241965e23 + 0e123

> Clifford::Packing.cep_backward(a_plus_b)

=> 31

Modular Concelament Scheme

First, generate modular secret keys:

> k = Clifford::Tools.generate_modular_keys

=> [449/2e0 + 9205570227/2140241965e1 + 3574444377/856096786e2 + -37510496674/2140241965e3 + 7678405602/2140241965e12 + -7556513538/428048393e13 + -5388955551/2140241965e23 + 0e123, 351/2e0 + 74959643277/2140241965e1 + 29106189927/856096786e2 + -305442615774/2140241965e3 + 62524159902/2140241965e12 + -61531610238/428048393e13 + -43881495201/2140241965e23 + 0e123]

Create two multivectors A and B:

> a = Clifford::Packing.cep_forward(22)

=> 159/2e0 + -30246873603/428048393e1 + -58723014765/856096786e2 + 123248774786/428048393e3 + -25229046978/428048393e12 + 124142722410/428048393e13 + 17706568239/428048393e23 + 0e123

> b = Clifford::Packing.cep_forward(25)

=> 129e0 + -273536943888/2140241965e1 + -53106030744/428048393e2 + 1114597615456/2140241965e3 + -228158337888/2140241965e12 + 224536402272/428048393e13 + 160128964944/2140241965e23 + 0e123

Conceal:

> ca = Clifford::Concealment.mc_forward(k,a)

=> 14908785/2e0 + 947656906366827/428048393e1 + 1839835456554885/856096786e2 + -3861475210965874/428048393e3 + 790444689377202/428048393e12 + -3889483250769690/428048393e13 + -554759870391351/428048393e23 + 0e123

> cb = Clifford::Concealment.mc_forward(k,b)

=> 16241565/2e0 + 3841917117527769/2140241965e1 + 1491783637407819/856096786e2 + -15654893255403878/2140241965e3 + 3204559542777894/2140241965e12 + -3153688255543686/428048393e13 + -2249064432343197/2140241965e23 + 0e123

> ca_times_cb = ca.gp(cb)

=> 63167002896385e0 + 13423547250531991237194/428048393e1 + 13030622168392992896235/428048393e2 + -54697744091672128203228/428048393e3 + 11196638325009710340444/428048393e12 + -55094477596356546423180/428048393e13 + -7858166054471528380722/428048393e23 + 0e123

> a_times_b = Clifford::Concealment.mc_backward(k,ca_times_cb)

=> 29521e0 + -76198450013262/2140241965e1 + -14793604006581/428048393e2 + 310490456897844/2140241965e3 + -63557453913012/2140241965e12 + 62548501059828/428048393e13 + 44606694648006/2140241965e23 + 0e123

> Clifford::Packing.cep_backward(a_times_b)

=> 550