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Comparison analysis of Ding's RLWE-based key exchange protocol and NewHope variants

  • * Corresponding author: Xinwei Gao

    * Corresponding author: Xinwei Gao
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  • In this paper, we present a comparison study on three RLWE key exchange protocols: one from Ding et al. in 2012 (DING12) and two from Alkim et al. in 2016 (NewHope and NewHope-Simple). We compare and analyze protocol construction, notion of designing and realizing key exchange, signal computation, error reconciliation and cost of these three protocols. We show that NewHope and NewHope-Simple share very similar notion as DING12 in the sense that NewHope series also send small additional bits with small size (i.e. signal) to assist error reconciliation, where this idea was first practically proposed in DING12. We believe that DING12 is the first work that presented complete LWE & RLWE-based key exchange constructions. The idea of sending additional information in order to realize error reconciliation and key exchange in NewHope and NewHope-Simple remain the same as DING12, despite concrete approaches to compute signal and reconcile error are not the same.

    Mathematics Subject Classification: Primary: 94A60, 11T71; Secondary: 14G50.

    Citation:

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  • Figure 1.  DING12 RLWE key exchange protocol

    Figure 2.  NewHope RLWE key exchange protocol

    Figure 3.  Illustrated figure of simplified NewHope error reconciliation ($ d = 2 $)

    Figure 4.  NewHope-Simple RLWE key exchange protocol

    Figure 5.  Illustrated figure of NewHope-Simple and DING12 signal value computation

    Table 1.  Summary Chart of DING12, BCNS15, NewHope, NewHope-Simple

    DING12 BCNS15 NewHope NewHope-Simple
    Signal
    computation
    ● Divide $ \mathbb{Z}_q $ into 2 regions.
    Different region gives
    corresponding value
    ● Signal value $ \in\{0, 1\} $
    ● Function: Sig()
    ● Divide $ \mathbb{Z}_q $ into 2 regions.
    Different region gives corresponding value
    ● Signal value $ \in\{0, 1\} $
    ● Function: $ \langle\cdot\rangle_{q, 2} $
    Difference vector between coefficient vector and center of closest Voronoi cell
    ● Signal value $ \in\{0, 1, 2, 3\} $
    ● Function: HelpRec()
    ● Divide $ \mathbb{Z}_q $ into 8 regions.
    Different region gives corresponding value
    ● Signal value $ \in\{0, \cdots, 7\} $
    ● Function: NHSCompress()
    Error
    reconciliation
    ● Add then mod 2 on each coefficient
    ● Extract least significant bit
    ● Function: Mod$ _2() $
    ● Multiply, round then mod 2 on each coefficient
    ● Extract most significant bit
    ● Function: rec(), $ \lfloor\cdot\rceil_{q, 2} $
    ● Add signal vector onto coefficient vector
    ● Decide key bit according to location of the sum vector
    ● Function: Rec()
    ● Variant of RLWE decryption
    ● Recovers encapsulated key
    ● Function: NHSDecode()
    Degree $ n $ of $ R_q $ 1024 1024 1024 1024
    Signal size ● 1-bit per coefficient
    ● $ 1\cdot n = 1024 $ bits
    ● 1-bit per coefficient
    ● $ 1\cdot n = 1024 $ bits
    ● 2-bit per coefficient
    ● $ 2\cdot n = 2048 $ bits
    ● 3-bit per coefficient
    ● $ 3\cdot n = 3072 $ bits
    Key size $ n $ bits $ n $ bits 256 bits 256 bits
    Category Diffie-Hellman-like Diffie-Hellman-like Diffie-Hellman-like KEM (Encryption)
     | Show Table
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