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. 2026 Jan 26;26(3):825. doi: 10.3390/s26030825
Phase 3: Partial_Private_Key_Generation—it is executed by KGC
Input: User identity IDu, spp, master public key U
Output: Partial Private Key (βu,γu)

  1. KGC chooses a random number ϒu from {1, 2, 3,…, q − 1}

  2. It defines the values βu and γu such that
    βu = ϒu.D and γu = ϒu + w . Ha(IDu||βu||δu) (6)

  3. It forwards a pair (βu,γu) to each user

  4. The condition of acceptance of the partial private key by the user whose identity is IDu is
    γu . D = ϒu. D + w . Ha(IDu||βu||δu) D = βu + Ha(IDu||βu||δu) . U (7)