Computable Clause Samples
The "Computable" clause establishes that certain terms, values, or obligations within a contract can be determined or calculated using a defined method or formula. In practice, this means that the contract specifies how to derive figures such as payments, interest, or performance metrics, often referencing external data sources or algorithms. By ensuring that these elements are objectively and transparently calculable, the clause reduces ambiguity and disputes over interpretation, thereby promoting clarity and efficiency in contract administration.
Computable. There exists an efficient algorithm to compute e(P, Q) for all P, Q ∈ G1. From the literature [1], we note that such a bilinear pairing may be realized using the modified Weil pairing associated with supersingular elliptic curves.
Computable. Given P, Q ∈ G1, there exists an efficient algorithm to compute e(P, Q).
Computable. For e(g, p) . g , p ∈ G1 , there exists an efficient algorithm to compute Typically, the map e will be derived from either the Weil or ▇▇▇▇ pairing on an elliptic curve over a finite field. Pairings and other parameters should be selected in proactive for efficiency and security.
Computable eˆ(P, Q) for any P, Q ∈ G1 is polynomial-time computable. In the literature the security of many the identity-based schemes are based on the following assumptions.
Assumption 1 ( Bilinear ▇▇▇▇▇▇-▇▇▇▇▇▇▇ (BDH) [4]) For x, y, z ∈R Z∗, P ∈ G∗, eˆ : G × G → G , given (P, xP, yP, zP ), computing eˆ(P, P )xyz
Computable. The map e is efficiently computable. The Weil [21] and modified T▇▇▇ [22] pairings on elliptic curves can be used to construct such bilinear maps.
Computable eˆ(P, Q) for any P, Q ∈ G1 is polynomial-time computable. In the literature the security of many the identity-based schemes are based on the following assumptions.
Assumption 1 ( Bilinear ▇▇▇▇▇▇-▇▇▇▇▇▇▇ (BDH) [4]) For x, y, z ∈R Z∗q , P ∈ G1∗, eˆ : G1 × G1 → G2, given (P, xP, yP, zP), computing eˆ(P, P )xyz is hard.
Assumption 2 ( Decisional Bilinear ▇▇▇▇▇▇-▇▇▇▇▇▇▇ (DBDH)) For x, y, z, r ∈R Z∗q , P ∈ G1∗, eˆ : G1 × G1 → G2, distinguishing between the distributions (P, xP , yP, zP, eˆ(P, P )xyz ) and (P, xP , yP, zP, eˆ(P, P )r) is hard.
Computable. There exists an efficient algorithm to compute e(P, Q) for all P, Q ∈ G1. CDH Problem : A Computational ▇▇▇▇▇▇-▇▇▇▇▇▇▇ (CDH) parameter generator CDH is a PPT algo- rithm takes a security parameter 1k and outputs addi- tive group G1 with an order q. When an algorithm solves CDH problem with an advantage s, the advantage is s = Pr[A(G, P, aP, bP ) = abP ], where P ∈ G1 and a, b ∈ Zq∗. - Send(U, i, M ) : Send message M to instance Πi and parameter generator IG is a PPT algorithm takes outputs the reply generated by this instance. - Execute(U1, ..., Un) : Execute the protocol between the players U1, ..., Un and outputs the transcript of ex- ecution. - Reveal(U, i) : Output the session key ski . - Corrupt(U ) : Output the long-term secret key Si. - T est(U, i) : asks any of the above queries, and then asks T est query only once. This query outputs a ran- dom bit b; if b = 1 the adversary can access ski , and a security parameter 1k and outputs G1 and G2 and bilinear map e. When an algorithm solves Decisional BDH (DBDH) problem with an advantage s, the advantage is |Pr[A(P, aP, bP, cP, e(P, P )abc) = 1] - Pr[A(P, aP, bP, cP, e(P, P )d) = 1]| ≤ s, where P ∈ G1 and a, b, c, d ∈ Zq∗.
Computable. There is an efficient algorithm to compute for all . .
Computable. An efficient algorithm to compute eˆ(P, Q) exists for any P, Q ∈ G1 where ri is a random number chosen by Ui.
Computable. There is an efficient algorithm to compute eˆ(P, Q) for any P, Q ∈ G1. By using the pairing computation and a ▇▇▇▇▇▇-▇▇▇▇▇▇▇ type scheme, the protocol2 requires each party to transmit only a single broadcast message to establish an agreed session key among three parties. After
1. A → B, C : aP
2. B → A, C : bP 3. C → A, B : cP
Fig. 1. Joux’s One-round Tripartite Key Agreement