Number
146
Author
str4d
Created
Thread
#ls2
Last updated
Status
Open

Overview

This proposal outlines a re-randomizable signature scheme suitable for creating Destinations that can be blinded. It additionally can be used to blind existing Ed25519 Destinations, with a slight reduction in efficacy.

Motivation

[Prop123] defines an encrypted LeaseSet2 format that embodies the Principle of Least Authority: each network participant is given only the information necessary for their role. In particular, an encrypted LeaseSet2 published to a floodfill does not reveal the Destination it is for, and the Leases can only be viewed by someone with prior knowledge of the Destination. However, floodfills still need to be able to authenticate the encrypted LeaseSet2s when they are published, and clients need to additionally ensure that the authentication was enforced by the Destination itself.

Prop123 achieves this by blinding the signing keys of Destinations. The blinded keys can be used to create signatures that are verifiable by floodfills, and clients can be certain that only the Destination could have created the signatures. It is therefore necessary to specify a signature scheme that can be used for blinding.

Design

Core signature scheme

The signature scheme described in this proposal, Red25519, is an instantiation of [RedDSA], a Schnorr-based signature scheme that supports key re-randomization. It has the following functions:

GENERATE_PRIVATE()
Returns a uniformly-random private key.
DERIVE_PUBLIC(sk)
Returns the public key corresponding to the given private key.
GENERATE_RANDOM()
Returns a random scalar suitable for re-randomizing a keypair.
RANDOMIZE_PRIVATE(sk, alpha)
Re-randomizes a private key, using a secret scalar alpha.
RANDOMIZE_PUBLIC(vk, alpha)
Re-randomizes a public key, using a secret scalar alpha.
SIGN(sk, m)
Returns a signature by the private key sk over the given message m.
VERIFY(vk, m, sig)
Verifies the signature sig against the public key vk and message m. Returns true if the signature is valid, false otherwise.

For a given keypair (sk, vk) the following relationship holds:

RANDOMIZE_PUBLIC(vk, alpha) == DERIVE_PUBLIC(RANDOMIZE_PRIVATE(sk, alpha))

Converting Ed25519 keys to Red25519

Ed25519 keys MAY be transiently one-way converted to Red25519 keys, in order to support re-randomization of existing Ed25519 Destinations. Other sigtypes are not compatible.

We define the following conversion functions:

CONVERT_ED25519_PRIVATE(privkey)
Returns the Red25519 private key corresponding to the given Ed25519 private key.
CONVERT_ED25519_PUBLIC(pubkey)
Returns the Red25519 public key corresponding to the given Ed25519 public key.

For a given Ed25519 keypair (privkey, pubkey) the following relationship holds:

CONVERT_ED25519_PUBLIC(pubkey) == DERIVE_PUBLIC(CONVERT_ED25519_PRIVATE(privkey))

Specification

Definitions

B
The Ed25519 basepoint as in [RFC-8032].
L
The Ed25519 order 2^252 + 27742317777372353535851937790883648493 as in [RFC-8032].
[s] B
Fixed-base scalar multiplication of the basepoint by s.
[s] A
Variable-base scalar multiplication of A by s.
x || y
Concatenate two byte arrays x and y.

Red25519

The scheme Red25519 specializes RedDSA with:

  • G := the group of points on the Edwards form of Curve25519. In particular, this means that Red25519 uses the prime-order subgroup of order L, and the cofactor h_G is 8.
  • P_G := the Ed25519 basepoint B.
  • l_H := 512
  • H(x) := SHA-512("I2P_Red25519H(x)" || x)

RedDSA assumes that H(x) is instantiated with a cryptographic hash function that is secure against length extension attacks. SHA-512 does not satisfy this on its own. To remedy this, we require that messages are prefixed with a prefix-free encoding of their length:

len_u16(M) || M

where len_u16(M) is the 2-byte representation of the length of M, in little-endian (to be consistent with the little-endian encoding of scalars and points).

Messages must be no longer than 65534 bytes. A length of 65535 is reserved for possible future extensions.

Encoding and decoding

Red25519 private keys are scalars mod L, encoded in little-endian representation. We define the functions DECODE_SCALAR and ENCODE_SCALAR for swapping between the byte array and integer forms of a scalar.

Red25519 public keys are points on the Edwards form of Curve25519. They are encoded as the 255-bit little-endian representation of the y-coordinate, followed by a single bit indicating the sign of the x-coordinate. This is the same encoding as for Ed25519. We define the functions DECODE_POINT and ENCODE_POINT for swapping between the bye array and coordinate forms of a point.

RedDSA functions

For ease of implementation, we explicitly write out below the RedDSA functions (as well as several helper functions) already specialized for Red25519. Implementors should refer to section 5.4.6 of [RedDSA] for the general specification of the RedDSA functions.

HStar(prefix1, prefix2, m) :=
    h = SHA-512()
    h.input("I2P_Red25519H(x)")
    h.input(prefix1)
    h.input(prefix2)

    h.input(len(m) & 0xff)
    h.input((len(m) >> 8) & 0xff)
    h.input(m)

    s = h.digest()
    return s mod L

GENERATE_PRIVATE :=
    s = 64 random bytes
    return s mod L

DERIVE_PUBLIC(sk) := [sk] B

GENERATE_RANDOM :=
    s = 64 random bytes
    return s mod L

RANDOMIZE_PRIVATE(sk, alpha) := (sk + alpha) mod L

RANDOMIZE_PUBLIC(vk, alpha) := vk + [alpha] B

SIGN(sk, m) :=
    T = 80 random bytes
    vkBytes = ENCODE_POINT(DERIVE_PUBLIC(sk))
    r = HStar(T, vkBytes, m)

    R = [r] B
    Rbytes = ENCODE_POINT(R)

    c = HStar(Rbytes, vkBytes, m)
    S = (r + (c * sk)) mod L

    return Rbytes || ENCODE_SCALAR(S)

VERIFY(vk, m, sig) :=
    Rbytes = sig[0..32]
    Sbytes = sig[32..64]

    R = DECODE_POINT(Rbytes)
    if R is invalid:
        return false

    S = DECODE_SCALAR(Sbytes)
    if S >= L:
        return false

    vkBytes = ENCODE_POINT(vk)
    c = HStar(Rbytes, vkBytes, m)
    return ((-[S] B) + R + ([c] vk)).multiplyByCofactor().isIdentity()

Conversion functions

CONVERT_ED25519_PRIVATE(privkey) :=
    s = SHA-512(privkey)[0..32]
    s[0] = s[0] & 248
    s[31] = (s[31] & 63) | 64
    return s

CONVERT_ED25519_PUBLIC(pubkey) := pubkey

Note that the implementation of CONVERT_ED25519_PRIVATE is equivalent to the computation of the secret scalar s when deriving an Ed25519 public key from an Ed25519 private key, as specified in in steps 1-3 from section 5.1.5 of [RFC-8032].

Security implications

Re-randomizing a Red25519 Destination and then creating signatures with it does not leak any information about the Destination, because the distribution of Red25519 private keys generated via RANDOMIZE_PRIVATE is identical to the distribution of private keys generated via GENERATE_PRIVATE, and DERIVE_PUBLIC is deterministic.

Converting Ed25519 private keys to Red25519 via CONVERT_ED25519_PRIVATE does not result in the same distribution. However, we consider the reduction in security acceptable for the following reasons:

  • The space of Ed25519 scalars is roughly half the size of the space of Red25519 scalars (there are 2^251 possible Ed25519 scalars, and L ~= 2^252 possible Red25519 scalars). Thus the security loss is at most a factor of roughly 2, or roughly 1 bit (because we could have coincidentally chosen a Red25519 scalar that is also a valid Ed25519 scalar).
  • Existing Ed25519 Destinations have already been historically exposed on the network, and it should be assumed that malicious floodfills have already enumerated them.

Users who are concerned about this security reduction should use Red25519 as the sigtype for their Destinations instead of Ed25519.

Note that the above argument does not apply to the re-randomization scalar alpha; information about the key is leaked each time a biased alpha is chosen, because additive re-randomization behaves like a one-time pad.

Compatibility

I2P versions that support Red25519 will be able to verify network datastructures that are signed with it. I2P versions that do not support Red25519 will treat it as an unknown signature, and MAY drop the datastructures. Users should expect the reliability of Red25519-signed datastructures to be poor until sufficient deployment of Red25519 has been reached.

Test vectors

Legend:
  edsk:  Ed25519 private key (random)
  edpk:  Ed25519 public key corresponding to edsk
  sk:    CONVERT_ED25519_PRIVATE(edsk)
  vk:    CONVERT_ED25519_PUBLIC(edpk)
  msg:
  sig:   SIGN(sk, msg)
  alpha: GENERATE_RANDOM()
  rsk:   RANDOMIZE_PRIVATE(sk, alpha)
  rvk:   RANDOMIZE_PUBLIC(vk, alpha)
  rsig:  SIGN(rsk, msg)

Test vector 1
  edsk:  0101010101010101010101010101010101010101010101010101010101010101
  edpk:  8a88e3dd7409f195fd52db2d3cba5d72ca6709bf1d94121bf3748801b40f6f5c
  sk:    58e86efb75fa4e2c410f46e16de9f6acae1a1703528651b69bc176c088bef36e
  vk:    8a88e3dd7409f195fd52db2d3cba5d72ca6709bf1d94121bf3748801b40f6f5c
  msg:   0202020202020202020202020202020202020202020202020202020202020202
  sig:   61f5527f4d3b46de4b2c234390370bf715ae9098907a0d191ba1b44b23a8ac1a
         6a40437a5294e9503faaf9bd2b7f2fe7ba44dec487b3185aba7ff7d7a17cd40f
  alpha: ae9ba9cbbc047c442448fca7c9f4e288a202ed520bfad0c784b792b7773cee08
  rsk:   8bb85f3c7a494a08890d7d142109c1a3501d04565d80227e2079097800fbe107
  rvk:   6fe128737b8e76fa66698a748b0dc0a89168dd8a0601c2b1c0b26835d323e9b3
  rsig:  533053074d3b44f08723aab988ede9880a001b7a684d4a98f2d1b88fabee07a5
         b5c9430c69a690321e0cb8365d7aeb6688bcbad2c0780e0c69e8a1b4a45f3001

Test vector 2
  edsk:  0202020202020202020202020202020202020202020202020202020202020202
  edpk:  8139770ea87d175f56a35466c34c7ecccb8d8a91b4ee37a25df60f5b8fc9b394
  sk:    a83c626bc9c38c8c201878ebb1d5b0b50ac40e8986c78793db1d4ef369fca14e
  vk:    8139770ea87d175f56a35466c34c7ecccb8d8a91b4ee37a25df60f5b8fc9b394
  msg:   0303030303030303030303030303030303030303030303030303030303030303
  sig:   0829e58eb5399870f009bd1f0270264e556424bda7a93fbcec99f6d9d75db46d
         5c3cb546d9947ca7c1200876c8775a90c357a2aef3d2f16388242ee1914b1a0a
  alpha: 98b615d9027e996cc2796c019d9c8beb46aa7d2b6eea2e5d98eb29eb1584c203
  rsk:   9fcfaa734852ca40b3810ebef590e138516e8cb4f4b1b6f0730978de7f806402
  rvk:   527e121090158419609e4a0d8de6f7d3271b353a8cd0b8172fe41468ea1e9177
  rsig:  9a6961f35ed264a946cd6214b2326a6e6caa426c2a61bc14367fd278e0b5fb51
         3ac065a69210a457f17d12ba8a496cfd835002691affa8efcdecae48135c090f

Test vector 3
  edsk:  0303030303030303030303030303030303030303030303030303030303030303
  edpk:  ed4928c628d1c2c6eae90338905995612959273a5c63f93636c14614ac8737d1
  sk:    98aebbb178a551876bfaf8e1e530dac6aaf6c2ea1c8f8406a3ab37dfb40fbc65
  vk:    ed4928c628d1c2c6eae90338905995612959273a5c63f93636c14614ac8737d1
  msg:   0404040404040404040404040404040404040404040404040404040404040404
  sig:   ef5fd1488048fb0247e5883bd90f7b2ce1ffe9b143a5bf6156b76ac2c39d8fdb
         d0730d7111d9cec69a808f3d18268a91f035b41b82c1fe06f394a615f93a8709
  alpha: ba17f5110fcea8a12e0bd3677e4088b874332c4e3e6c9911d9ec3fe0233d3e0a
  rsk:   c4ceed95e9208c189458fe772c9628021f2aef385bfb1d187c9877bfd84cfa0f
  rvk:   6e2b9b129bbe00fa964c996d40307dd01aff120e94fd15f17119341ecda3d7a0
  rsig:  900ecc6306f895a8ccde97d3624799fd939062a87b69e09351903ba83ceeab2b
         ef6e3c15e5d8400ed9151f7dce14bf4cfc7ce3f4399e22455fc18a68ed931c03

Test vector 4
  edsk:  0404040404040404040404040404040404040404040404040404040404040404
  edpk:  ca93ac1705187071d67b83c7ff0efe8108e8ec4530575d7726879333dbdabe7c
  sk:    483e3c145d7e680a16676925fc045183d2f510cb2f660a1fc517c73762185d43
  vk:    ca93ac1705187071d67b83c7ff0efe8108e8ec4530575d7726879333dbdabe7c
  msg:   0505050505050505050505050505050505050505050505050505050505050505
  sig:   d76b8133e08e4ff58de6b7f2df95c84a8bb968addd1d1ff585d79a90f5cfe11f
         9aa21d0277908aecae3853ee44493f95f2445df2da712f28eea435044e6fed03
  alpha: 9a14f2755512a72a3a5a514379f3458c3f912fc5eba8711b0cf2bfda49c79104
  rsk:   2e0357164904c6d4f64ddcdcfa101bbc118740901b0f7c3ad1098712acdfee07
  rvk:   de0a291ee45634de9a051c9373b9378ffbe45a8532067a9a93a86b837c762908
  rsig:  010dcc6a44e942a6f7d52704d957ad66a5c6452ad9b9442cc8ef724e41d6c3ce
         a24eace9b22e0f9d2b9ade14c61bded33286e7e6340faaa0167a9f1f90001503

Test vector 5
  edsk:  0505050505050505050505050505050505050505050505050505050505050505
  edpk:  6e7a1cdd29b0b78fd13af4c5598feff4ef2a97166e3ca6f2e4fbfccd80505bf1
  sk:    48370d6146de919cc1ce472897775d9a6c2834c509e08e14efcb2b52188f946e
  vk:    6e7a1cdd29b0b78fd13af4c5598feff4ef2a97166e3ca6f2e4fbfccd80505bf1
  msg:   0606060606060606060606060606060606060606060606060606060606060606
  sig:   2c56c96801f99ae1f5e8d8edc87725e08aaf7fc77071f222f7c46084b41c5b41
         de1ee3df97217865633f7cceb11cedc3a637ce047d2111cb6f372882e2d6b20b
  alpha: 687944d00a53ca02a0787da2acb8f99994ea7453c8d140d93efbc2b70d852a07
  rsk:   35e598a6987bdb3685fdff552d5b3ea20013a918d2b1cfed2dc7ee092614bf05
  rvk:   9951414e4f29408031f212edc6c9cfe36550b4ce2aa968db49de6c93ca9d565b
  rsig:  4b8f3e3baa8b4fdb99b0053036d569352e49c98c61800288f676aed77b9929b3
         f3278565d824c5566666d2c9ff789207098d5f9d09dd89aa4945ca3866831e02

Test vector 6
  edsk:  0606060606060606060606060606060606060606060606060606060606060606
  edpk:  8a875fff1eb38451577acd5afee405456568dd7c89e090863a0557bc7af49f17
  sk:    a83f248f80ff04de20a82fe12bd3551887168e372d239932ce812d0992d34078
  vk:    8a875fff1eb38451577acd5afee405456568dd7c89e090863a0557bc7af49f17
  msg:   0707070707070707070707070707070707070707070707070707070707070707
  sig:   f4a00093daa26b48465e07ee5697ba44191fb5673b6ab71a31d2349a18aecbd6
         c4801be60ebf18cd7ce8ec5fe0988fc4762908095063b55068ce4c7578c91504
  alpha: 0158cda553d7e9769829a5d36d2b7ce05e9171d8d058a8630d31029001ffd409
  rsk:   41f8424d01be5b9406eb179da42fda51e5a7ff0ffe7b4196dbb22f9993d21502
  rvk:   cef5dc9b4a246025df56fb118e34c3f06d6213c4c6ab8a1d4297eb7845cb2824
  rsig:  de23eec573f35ebf7ea9539b511ca5129213821525190fdf1c186c2788c1abb3
         5bd40937defbc4112225d399a79a171cf02c3eebbd6340bbdca7383906af1902

Test vector 7
  edsk:  0707070707070707070707070707070707070707070707070707070707070707
  edpk:  ea4a6c63e29c520abef5507b132ec5f9954776aebebe7b92421eea691446d22c
  sk:    28ad39fefd7fa3e200a9c626eef599e61a2d055c48a8288a4e7e4c4bca392878
  vk:    ea4a6c63e29c520abef5507b132ec5f9954776aebebe7b92421eea691446d22c
  msg:   0808080808080808080808080808080808080808080808080808080808080808
  sig:   e78bf2d340d9ae0af5dd81e4d58801b3872189a71573a12be343ed39cebab56a
         6bc1f01872bbb1d16b2be4a943a9ba90fb7a4c97c3e5f20416890ceedf6e7e0c
  alpha: 8e16161802e3c87857ae725dfa28d6222b326907f652e6c89f806882c0fb1a00
  rsk:   3bf8968b47adebf27b0d740fd2495777455f6e633efb0e53eefeb4cd8a354308
  rvk:   755a8f05633c45d0fac471a386776f63a7d28bc8d80e326ddde5484b20565e89
  rsig:  6efdca4ba705bc05d4564f0ca626646679ac1cb2c3093618e95238ebd1c7aa09
         632ccefc324594447a01074bb473c3ce94ccaae86e18f8f43477326a12ae6207

Test vector 8
  edsk:  0808080808080808080808080808080808080808080808080808080808080808
  edpk:  1398f62c6d1a457c51ba6a4b5f3dbd2f69fca93216218dc8997e416bd17d93ca
  sk:    3826c9c31226edde9501fd2589203cb3e6fe737876a845512b53ada2fa2ace74
  vk:    1398f62c6d1a457c51ba6a4b5f3dbd2f69fca93216218dc8997e416bd17d93ca
  msg:   0909090909090909090909090909090909090909090909090909090909090909
  sig:   1ed88a926dd80999d3a40efd3b74fa731729e28bb84e0430663822a69f9f4bcc
         fd2bd0aa7325d9887eac76ddf08da65c42eedaaec244c3241307570910778f05
  alpha: c4cc56d9c3e787ca60a54dfca65b4556f2dccabcac97e7a975e4efa1acb8920e
  rsk:   945371b503f5e1e843c08d0a3bad8962d8db3e3523402dfba0379d44a7e36003
  rvk:   db1730a0730ca0746a73f1880660ea5ea42f9d931760f3cdedc9cbe1c1d1b8d9
  rsig:  db60c64b61e888696ca0a7ef7adb92b784e0e6070d0435818e99788022db8e83
         81ddcc1e27f044b8c3c75044e715d870f3273a7f9cf85f1a59f4a7c95fded408

Test vector 9
  edsk:  0909090909090909090909090909090909090909090909090909090909090909
  edpk:  fd1724385aa0c75b64fb78cd602fa1d991fdebf76b13c58ed702eac835e9f618
  sk:    388fe3ab30c0aabf54acd276f3d8bbbc2b7ca4a9495d204f255bacf578c74c46
  vk:    fd1724385aa0c75b64fb78cd602fa1d991fdebf76b13c58ed702eac835e9f618
  msg:   0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a
  sig:   dd5a8c6ed9331c074ea11f65b9290900931bdf01a47f01adc75583d2a3dcfc10
         b65c77a3e992678865e7dc713295749b4dddd33fa167b96c6d6904818e4d6806
  alpha: b851f206eba78325ed5231cad059e8bd8a1e3d7f1e391058b3d9ab08d096cb03
  rsk:   3c91fe3eb2dbe484e88b25b5494b2827b69ae128689630a7d83458fe485e180a
  rvk:   601ab762eea5cd89ff34e0f661d1ca3932ba166ca67154b2e62afb85282dda81
  rsig:  5a453378fdbb22b8f037ad61d144ce006201fea0c2c1472d463617c432786dfc
         47430d27649817a7fc26296c94bf922f3863867c648ddd6709710bfa199aee02

Test vector 10
  edsk:  0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a0a
  edpk:  43a72e714401762df66b68c26dfbdf2682aaec9f2474eca4613e424a0fbafd3c
  sk:    0099bf92c41b5d3d309c3b074756e9707e40a9bcea229857f7cf551e8bb0fd45
  vk:    43a72e714401762df66b68c26dfbdf2682aaec9f2474eca4613e424a0fbafd3c
  msg:   0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b0b
  sig:   c54d64d550f7690ffdd108efc49f1c25a54282825e10328630710924b354cb4c
         138a523b1ada66a8fdc8b7efcae939fd54b05552c30ca280d23199c391c5b707
  alpha: 5eebb60818299d581fa68f5fcae4c2bb398a7e10876e27994d93d555075e7d05
  rsk:   aa349f2773b8b035f6ceecda965330d9b7ca27cd7191bff044632b74920e7b0b
  rvk:   d0c5fe8f83fd42202265efff804a1527c0eb0e1cce9781cf14571cd506eeed36
  rsig:  28e96b6d4251b356e635e382ed89a37e7650d3035f98909e09a6cbe82c13e418
         fddb2106b7b527e198039da7221dae1a0227f0a4ab88f06567e8fd9238acc106