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=structtype t = string moduleMap=structincludeMap.Make(String)let of_list bindings =List.fold_left (fun acc (k,v) -> add k v acc) empty bindingsendendmoduleTypeExpr=structtype t =IntType|UnitType|ArrType of t * t let rec eq t1 t2 =match t1, t2 with|UnitType,UnitType->true|IntType,IntType->true|ArrType(l1, r1),ArrType(l2, r2) -> eq l1 l2 && eq r1 r2| _ ->falseendmoduleExpression=structtype t =|VarExpr of Variable.t|AbsExpr of Variable.t * t |AppExpr of t * t |AnnExpr of t *TypeExpr.t|IntExpr of int|UnitExprlet rec synthesize expression ~env = match expression with | UnitExpr -> Some(TypeExpr.UnitType) | IntExpr _ -> Some(TypeExpr.IntType) | VarExpr var -> Variable.Map.find_opt var env | AnnExpr(e, ty) -> let env' =Variable.Map.empty in check e ~env:env' ~against:ty | AppExpr(e1, e2) -> (match synthesize e1 ~env with | Some TypeExpr.ArrType(a, b) -> (match check e2 ~env ~against:a with | Some _ -> Some b | _ -> None) | _ -> None) | AbsExpr(x, e) -> let env' = Variable.Map.add x (TypeExpr.UnitType) env in (match synthesize e ~env:env' with | Some ty -> Some (TypeExpr.ArrType(TypeExpr.UnitType, ty)) | _ -> None) and check expression ~env ~against = match expression, against with | e, ty -> (match synthesize e ~env with | Some ty' when TypeExpr.eq ty' against -> Some ty | _ -> None)endtype vertex = inttype hyperedge = { vertices: vertex list; weight: Z.t }type hypergraph = hyperedge listlet encrypt m (n, g) = let lower = Z.one in let upper = Z.pred n in let diff = Z.sub upper lower in let r = Z.add lower (Z.rem (Z.of_int (Random.State.bits (Random.get_state ()))) diff) in let gm = Z.powm g m (Z.mul n n) in let rn = Z.powm r n (Z.mul n n) in Z.rem (Z.mul gm rn) (Z.mul n n)let decrypt c (n, _) (lambda, mu) = let cl = Z.powm c lambda (Z.mul n n) in let l x = Z.div (Z.sub x Z.one) n in Z.rem (Z.mul (l cl) mu) nlet encrypted_add c1 c2 n = Z.rem (Z.mul c1 c2) (Z.mul n n)let generate_keys () = let p = Z.of_int 61 in let q = Z.of_int 53 in let n = Z.mul p q in let lambda = Z.lcm (Z.pred p) (Z.pred q) in let g = Z.succ n in let mu = Z.invert lambda n in ((n, g), (lambda, mu))let create_hyperedge m public_key = [encrypt m public_key]let add_hyperedges he1 he2 public_key = if List.length he1 <> 1 || List.length he2 <> 1 then failwith "Hyperedges must contain only one encrypted number each" else let c1 = List.hd he1 in let c2 = List.hd he2 in let n = fst public_key in [encrypted_add c1 c2 n]let analyze_hypergraph_operation a_hyperedge b_hyperedge = let open Expression in let env = Variable.Map.of_list [("encrypted_add", TypeExpr.ArrType(TypeExpr.IntType, TypeExpr.ArrType(TypeExpr.IntType, TypeExpr.IntType)))] in
let add_expr = VarExpr "encrypted_add" in let a_expr = IntExpr 0 in let b_expr = IntExpr 1 in let operation_expr = AppExpr(AppExpr(add_expr, a_expr), b_expr) in match synthesize operation_expr ~env with | Some ty -> 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 (); let public_key, private_key = generate_keys () in let a = Z.of_int 111 in let b = Z.of_int 222 in let a_hyperedge = create_hyperedge a public_key in let b_hyperedge = create_hyperedge b public_key in let sum_hyperedge = add_hyperedges a_hyperedge b_hyperedge public_key in let decrypted_sum_hyperedge = List.map (fun c -> decrypt c public_key private_key) sum_hyperedge in Printf.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: