Here's a simple example showcasing the use of the hypergraph approach for mathematical operations with prime numbers.

This code exemplifies a fundamental mathematical operation involving two numerical variables, denoted as a & b. Subsequently, it undertakes a homomorphic operation leveraging hypergraphs as a computational framework.

Additionally, we perform type analysis on a hypothetical hyperedge operation and determine its type.

It's important to understand that this is not production code, it is designed for the sole purpose of demonstrating the concept and approach to performing homomorphic computations using hypergraphs.

For clarity, all examples have been significantly simplified and are not directly used in the Octra production code.

Modular Arithmetic Operations

Addition

$\ (a + b) \mod n$

Where a and b are integers, and n is the modulus.

Remainder Calculation

$x \mod n$

Where x is an integer, and n is the modulus.

Homomorphic Addition of Encrypted Numbers

Encrypted Addition

$(c1, c2, n) = (c1 \cdot c2) \mod (n \cdot n)$

Where c1 and c2 are encrypted numbers, and n is the modulus.

Encryption and Decryption Operations

Encryption

$(m, (n, g)) = (g^m \cdot r^n) \mod (n \cdot n)$

Where m is the message, (n, g) is the public key, and r is a random value.

Where c is the encrypted message, (n, _) is the public key, and (λ, μ) is the private key.

Type Analysis for Hyperedge Operation

This is useful for ensuring the correctness of operations on hypergraphs and preventing type-related errors in the code.

Determine the type of a hyperedge operation, which can be int, uint or func based on the analysis performed.

openOctra.ZmoduleVariable=structtypet= string moduleMap=structincludeMap.Make(String)letof_listbindings=List.fold_left (funacc (k,v) -> add k v acc) empty bindingsendendmoduleTypeExpr=structtypet=IntType|UnitType|ArrType of t * t letreceqt1t2=match t1, t2 with|UnitType,UnitType->true|IntType,IntType->true|ArrType(l1,r1),ArrType(l2,r2) -> eq l1 l2 && eq r1 r2|_->falseendmoduleExpression=structtypet=|VarExpr of Variable.t|AbsExpr of Variable.t * t |AppExpr of t * t |AnnExpr of t *TypeExpr.t|IntExpr of int|UnitExprletrecsynthesizeexpression~env=match expression with|UnitExpr->Some(TypeExpr.UnitType)|IntExpr_->Some(TypeExpr.IntType)|VarExprvar->Variable.Map.find_opt var env|AnnExpr(e,ty) ->letenv'=Variable.Map.empty in check e ~env:env' ~against:ty|AppExpr(e1,e2) -> (match synthesize e1 ~envwith|SomeTypeExpr.ArrType(a,b) -> (match check e2 ~env~against:a with|Some_->Some b|_->None)|_->None)|AbsExpr(x,e) ->letenv'=Variable.Map.add x (TypeExpr.UnitType) env in (match synthesize e ~env:env' with|Somety->Some (TypeExpr.ArrType(TypeExpr.UnitType, ty))|_->None)andcheckexpression~env~against=match expression, against with|e,ty-> (match synthesize e ~envwith|Somety'whenTypeExpr.eq ty' against ->Some ty|_->None)endtypevertex= inttypehyperedge={vertices: vertex list;weight:Z.t }typehypergraph= hyperedge listletencryptm (n,g) =letlower=Z.one inletupper=Z.pred n inletdiff=Z.sub upper lower inletr=Z.add lower (Z.rem (Z.of_int (Random.State.bits (Random.get_state ()))) diff) inletgm=Z.powm g m (Z.mul n n) inletrn=Z.powm r n (Z.mul n n) inZ.rem (Z.mul gm rn) (Z.mul n n)letdecryptc (n,_) (lambda,mu) =letcl=Z.powm c lambda (Z.mul n n) inletlx=Z.div (Z.sub x Z.one) n inZ.rem (Z.mul (l cl) mu) nletencrypted_addc1c2n=Z.rem (Z.mul c1 c2) (Z.mul n n)letgenerate_keys()=letp=Z.of_int 61inletq=Z.of_int 53inletn=Z.mul p q inletlambda=Z.lcm (Z.pred p) (Z.pred q) inletg=Z.succ n inletmu=Z.invert lambda n in ((n, g), (lambda, mu))letcreate_hyperedgempublic_key=[encrypt m public_key]letadd_hyperedgeshe1he2public_key=ifList.length he1 <>1||List.length he2 <>1then failwith "Hyperedges must contain only one encrypted number each"elseletc1=List.hd he1 inletc2=List.hd he2 inletn= fst public_key in[encrypted_add c1 c2 n]letanalyze_hypergraph_operationa_hyperedgeb_hyperedge=letopenExpressionin let env = Variable.Map.of_list [("encrypted_add", TypeExpr.ArrType(TypeExpr.IntType, TypeExpr.ArrType(TypeExpr.IntType, TypeExpr.IntType)))] in
letadd_expr=VarExpr"encrypted_add"inleta_expr=IntExpr0inletb_expr=IntExpr1inletoperation_expr=AppExpr(AppExpr(add_expr, a_expr), b_expr) inmatch synthesize operation_expr ~envwith|Somety->Printf.printf "Type of hyperedge operation: %s\n" (match ty with|TypeExpr.IntType->"Integer"|TypeExpr.UnitType->"Unit"|TypeExpr.ArrType(_,_) ->"Function")|None->Printf.printf "Type of hyperedge operation could not be determined\n"let()=Random.self_init ();letpublic_key,private_key= generate_keys ()inleta=Z.of_int 111inletb=Z.of_int 222inleta_hyperedge= create_hyperedge a public_key inletb_hyperedge= create_hyperedge b public_key inletsum_hyperedge= add_hyperedges a_hyperedge b_hyperedge public_key inletdecrypted_sum_hyperedge=List.map (func-> decrypt c public_key private_key) sum_hyperedge inPrintf.printf "Encrypted Hyperedge A: %s\n" (Z.to_string (List.hd a_hyperedge));Printf.printf "Encrypted Hyperedge B: %s\n" (Z.to_string (List.hd b_hyperedge));Printf.printf "Encrypted Sum Hyperedge: %s\n" (Z.to_string (List.hd sum_hyperedge));Printf.printf "Decrypted Sum Hyperedge: %s\n" (Z.to_string (List.hd decrypted_sum_hyperedge)); analyze_hypergraph_operation [List.hd a_hyperedge][List.hd b_hyperedge];

The compilation of the code takes place as follows: