Package net.i2p.crypto.eddsa.math.ed25519

Class Ed25519LittleEndianEncoding

  • All Implemented Interfaces:
    Serializable
    public class Ed25519LittleEndianEncoding
extends Encoding
    Helper class for encoding/decoding from/to the 32 byte representation.

    Reviewed/commented by Bloody Rookie (nemproject@gmx.de)

    See Also:
    Serialized Form

    • Constructor Detail

      • Ed25519LittleEndianEncoding

        public Ed25519LittleEndianEncoding()

    • Method Detail

      • encode

        public byte[] encode​(FieldElement x)
        Encodes a given field element in its 32 byte representation. This is done in two steps:
        1. Reduce the value of the field element modulo $p$.
        2. Convert the field element to the 32 byte representation.

        The idea for the modulo $p$ reduction algorithm is as follows:

        Assumption:

        • $p = 2^{255} - 19$
        • $h = h_0 + 2^{25} * h_1 + 2^{(26+25)} * h_2 + \dots + 2^{230} * h_9$ where $0 \le |h_i| \lt 2^{27}$ for all $i=0,\dots,9$.
        • $h \cong r \mod p$, i.e. $h = r + q * p$ for some suitable $0 \le r \lt p$ and an integer $q$.

        Then $q = [2^{-255} * (h + 19 * 2^{-25} * h_9 + 1/2)]$ where $[x] = floor(x)$.

        Proof:

        We begin with some very raw estimation for the bounds of some expressions:

        $$ \begin{equation} |h| \lt 2^{230} * 2^{30} = 2^{260} \Rightarrow |r + q * p| \lt 2^{260} \Rightarrow |q| \lt 2^{10}. \\ \Rightarrow -1/4 \le a := 19^2 * 2^{-255} * q \lt 1/4. \\ |h - 2^{230} * h_9| = |h_0 + \dots + 2^{204} * h_8| \lt 2^{204} * 2^{30} = 2^{234}. \\ \Rightarrow -1/4 \le b := 19 * 2^{-255} * (h - 2^{230} * h_9) \lt 1/4 \end{equation} $$

        Therefore $0 \lt 1/2 - a - b \lt 1$.

        Set $x := r + 19 * 2^{-255} * r + 1/2 - a - b$. Then:

        $$ 0 \le x \lt 255 - 20 + 19 + 1 = 2^{255} \\ \Rightarrow 0 \le 2^{-255} * x \lt 1. $$

        Since $q$ is an integer we have

        $$ [q + 2^{-255} * x] = q \quad (1) $$

        Have a closer look at $x$:

        $$ \begin{align} x &= h - q * (2^{255} - 19) + 19 * 2^{-255} * (h - q * (2^{255} - 19)) + 1/2 - 19^2 * 2^{-255} * q - 19 * 2^{-255} * (h - 2^{230} * h_9) \\ &= h - q * 2^{255} + 19 * q + 19 * 2^{-255} * h - 19 * q + 19^2 * 2^{-255} * q + 1/2 - 19^2 * 2^{-255} * q - 19 * 2^{-255} * h + 19 * 2^{-25} * h_9 \\ &= h + 19 * 2^{-25} * h_9 + 1/2 - q^{255}. \end{align} $$

        Inserting the expression for $x$ into $(1)$ we get the desired expression for $q$.

        Specified by:
        encode in class Encoding
        Parameters:
        x - the FieldElement to encode
        Returns:
        the $(b-1)$-bit encoding of this FieldElement.

      • load_3

        static int load_3​(byte[] in,
                  int offset)

      • load_4

        static long load_4​(byte[] in,
                   int offset)

      • decode

        public FieldElement decode​(byte[] in)
        Decodes a given field element in its 10 byte $2^{25.5}$ representation.
        Specified by:
        decode in class Encoding
        Parameters:
        in - The 32 byte representation.
        Returns:
        The field element in its $2^{25.5}$ bit representation.

      • isNegative

        public boolean isNegative​(FieldElement x)
        Is the FieldElement negative in this encoding?

        Return true if $x$ is in $\{1,3,5,\dots,q-2\}$
        Return false if $x$ is in $\{0,2,4,\dots,q-1\}$

        Preconditions:

        • $|x|$ bounded by $1.1*2^{26},1.1*2^{25},1.1*2^{26},1.1*2^{25}$, etc.
        Specified by:
        isNegative in class Encoding
        Parameters:
        x - the FieldElement to check
        Returns:
        true if $x$ is in $\{1,3,5,\dots,q-2\}$, false otherwise.