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Tree Key Derivation Function

The tree KDF serves two purposes: building a tree of related keys, and providing a providing a post-quantum backup signature scheme in the event that BLS12-381 is no longer deemed secure. The tree of keys is useful for building out families of keys from a single source of entropy allowing keys to be derived from one another in a useful pattern while allowing keys to be separated by purpose. The post-quantum backup mechanism takes the form of a Lamport signature and is designed to be a bridge to a new signature scheme in the event that BLS12-381 is deemed insecure.

The tree key derivation function describes how to take a parent key and a child-leaf index and arrive at a child key. This is very useful as one can start with a single source of entropy and from there build out a practically unlimited number of keys. The specification can be broken into two sub-components: generating the master key, and constructing a child key from its parent. The master key is used as the root of the tree and then the tree is built in layers from this root.

The Tree Structure

The key tree is defined purely through the relationship between a child-node and its ancestors. Starting with the root of the tree, the master key, a child node can be derived by knowing the parent's private key and the index of the child. The tree is broken up into depths which are indicated by / and the master node is described as m. The first child of the master node is therefore described as m / 0.

      [m / 0] - [m / 0 / 0]
     /        \
    /           [m / 0 / 1]
[m] - [m / 1]
    \
     ...
      [m / i]

Specification

Every key generated via the key derivation process derives a child key via a set of intermediate Lamport keys. The idea behind the Lamport keys is to provide a quantum secure backup incase BLS12-381 is no longer deemed secure. At a high level, the key derivation process works by using the parent node's privkey as an entropy source for the Lamport private keys which are then hashed together into a compressed Lamport public key, this public key is then hashed into BLS12-381's private key group.

Helper functions

IKM_to_lamport_SK

Inputs:

  • IKM, a secret octet string
  • salt, an octet string

Outputs:

  • lamport_SK, an array of 255 32-octet stings

Definitions:

  • HKDF-Extract is as defined in RFC5869, instantiated with SHA256
  • HKDF-Expand is as defined in RFC5869, instantiated with SHA256
  • K = 32 is the digest size (in octets) of the hash function (SHA256)
  • L = K * 255 is the HKDF output size (in octets)
  • "" is the empty string
  • bytes_split is a function takes in an octet string and splits it into K-byte chunks which are returned as an array

Procedure:

0. PRK = HKDF-Extract(salt, IKM)
1. OKM = HKDF-Expand(PRK, "" , L)
2. lamport_SK = bytes_split(OKM, K)
3. return lamport_SK

parent_SK_to_lamport_PK

Inputs:

  • parent_SK, the BLS Secret Key of the parent node
  • index, the index of the desired child node, an integer 0 <= index < 2^256

Outputs:

  • lamport_PK, the compressed lamport PK, a 32 octet string

Definitions:

  • I2OSP is as defined in RFC3447 (Big endian decoding)
  • flip_bits is a function that returns the bitwise negation of its input
  • "" is the empty string
  • a | b is the concatenation of a with b

Procedure:

0. salt = I2OSP(index, 32)
1. IKM = I2OSP(parent_SK, 32)
2. lamport_0 = IKM_to_lamport_SK(IKM, salt)
3. not_IKM = flip_bits(IKM)
4. lamport_1 = IKM_to_lamport_SK(not_IKM, salt)
5. lamport_PK = ""
6. for i = 0 to 255
       lamport_PK = lamport_PK | SHA256(lamport_0[i])
7. for i = 0 to 255
       lamport_PK = lamport_PK | SHA256(lamport_1[i])
8. compressed_lamport_PK = SHA256(lamport_PK)
9. return compressed_lamport_PK

HKDF_mod_r

Inputs:

  • IKM, a secret octet string.

Outputs:

  • SK, the corresponding secret key, an integer 0 <= SK < r.

Definitions:

  • HKDF-Extract is as defined in RFC5869, instantiated with hash H.
  • HKDF-Expand is as defined in RFC5869, instantiated with hash H.
  • L is the integer given by ceil((1.5 * ceil(log2(r))) / 8).
  • "BLS-SIG-KEYGEN-SALT-" is an ASCII string comprising 20 octets.
  • "" is the empty string.
  • OS2IP is as defined in RFC3447 (Big endian encoding)
  • r is the order of the BLS 12-381 curve defined in the draft IETF BLS signature scheme standard r=52435875175126190479447740508185965837690552500527637822603658699938581184513

Procedure:

1. PRK = HKDF-Extract("BLS-SIG-KEYGEN-SALT-", IKM)
2. OKM = HKDF-Expand(PRK, "", L)
3. SK = OS2IP(OKM) mod r
4. return SK

derive_child_SK

The child key derivation function takes in the parent's private key and the index of the child and returns the child private key.

Inputs:

  • parent_SK, the secret key of the parent node, a big endian encoded integer
  • index, the index of the desired child node, an integer 0 <= index < 2^256

Outputs:

  • child_SK, the secret key of the child node, a big endian encoded integer

Procedure:

0. compressed_lamport_PK = parent_SK_to_lamport_PK(parent_SK, index)
1. SK = HKDF_mod_r(compressed_lamport_PK)
2. return SK

derive_master_SK

Inputs:

  • seed, the source entropy for the entire tree, a octet string

Outputs:

  • SK, the secret key of master node within the tree, a big endian encoded integer

Procedure:

0. SK = HKDF_mod_r(seed)
1. return SK

Appendix A. Test Vectors

Test Case 0

seed = 0xc55257c360c07c72029aebc1b53c05ed0362ada38ead3e3e9efa3708e53495531f09a6987599d18264c1e1c92f2cf141630c7a3c4ab7c81b2f001698e7463b04 master_SK = 12513733877922233913083619867448865075222526338446857121953625441395088009793 child_index = 0 child_SK = 7419543105316279183937430842449358701327973165530407166294956473095303972104

Test Case 1

seed = 0x3141592653589793238462643383279502884197169399375105820974944592 master_SK = 46029459550803682895343812821003080589696405386150182061394330539196052371668 child_index = 3141592653589793238462643383279502884197169399375105820974944592 child_SK = 52355059779601818323170390700812190085791545700943775185630512585202016942671

Test Case 2

seed = 0x0099FF991111002299DD7744EE3355BBDD8844115566CC55663355668888CC00 master_SK = 45379166311535261329029945990467475187325618028073620882733843918126031931161 child_index = 115792089237316195423570985008687907853269984665640564039457584007913129639935 child_SK = 3001977934078166987926353732839098506754809480904566732795462937312900783942

Test Case 3

seed = 0x000000000019d6689c085ae165831e934ff763ae46a2a6c172b3f1b60a8ce26f master_SK = 8591296517642752610571443601667923790682754368613740552668934360711284428110 child_index = 96295644508963302359223866841007920022480890644992946816264522587871600414627 child_SK = 18992511018606881439236209510845138630250367286140373438339392563268870207242

Appendix B. Test Vector with Intermediate values

seed = 0xc55257c360c07c72029aebc1b53c05ed0362ada38ead3e3e9efa3708e53495531f09a6987599d18264c1e1c92f2cf141630c7a3c4ab7c81b2f001698e7463b04 master_SK = 12513733877922233913083619867448865075222526338446857121953625441395088009793 child_index = 0 lamport_0 = [0x7b4a587eac94d7f56843e718a04965d4832ef826419b4001a3ad0ba77eb44a3b, 0x90f45a712112122429412921ece5c30eb2a6daf739dc9034fc79424daeb5eff6, 0xd061c2799de00b2be90eb1cc295f4c31e22d4b45c59a9b9b2554379bea7783cb, 0x3ad17e4cda2913b5180557fbe7db04b5ba440ce8bb035ae27878d66fbfa50d2c, 0xf5b954490933ad47f8bf612d4a4f329b3aa8914b1b83d59e15e271e2a087e002, 0x95d68d505bf4ff3e5149bc5499cf4b2f00686c674a29a8d903f70e569557d867, 0x1b59c76d9bb2170b220a87833582ede5970d4a336d91c99a812825afe963e056, 0x4310ff73cfbbf7b81c39ecbf1412da33e9388c1a95d71a75e51fe12256551ceb, 0xee696343f823e5716e16747f3bbae2fc6de233fe10eea8e45b4579018da0874f, 0xae12a437aaa7ae59f7d8328944b6a2b973a43565c55d5807dc2faf223a33aa73, 0x2a3ae0b47f145bab629452661ff7741f111272e33ec571030d0eb222e1ed1390, 0x1a3ea396e8cbd1d97733ef4753d6840b42c0795d2d693f18e6f0e7b3fff2beb2, 0x472429d0643c888bfdfe6e6ccfdeee6d345d60c6710859ac29fc289fd3656347, 0xa32d4d955949b8bed0eb20f586d8fd516d6ddec84fbbc36998d692633c349822, 0xe5ac8ac5ee1d40e53a7abf36e8269d5d5fce450a87feae8e59f432a44bcc7666, 0xddf9e497ed78032fbd72d9b8abd5204d81c3475f29afa44cdf1ded8ea72dd1dc, 0x945c62e88fb1e5f3c15ff57cd5eb1586ee93ec5ec80154c5a9c50241c5adae0a, 0xc8868b50fc8423c96b7efa1ede4d3203a6b835dbeb6b2ababc58397e6b31d9dd, 0x66de9bd86b50e2b6a755310520af655759c1753bff34b79a5cd63d6811fc8c65, 0x5b13786c6068df7735343e5591393bea8aee92ac5826d6132bf4f5ebf1098776, 0xa2038fc7d8e3cb2eda2bd303cfa76a9e5d8b88293918bec8b2fc03be75684f14, 0x47a13f6b2308a50eded830fdee7c504bf49d1fe6a95e337b0825d0d77a520129, 0xb534cdddcf1aa1c6b4cbba46d1db31b766d958e0a0306450bc031d1e3ed79d97, 0x54aa051b754c31658377f7bff00b7deaa861e74cb12e1eb84216666e19b23d69, 0x0220d57f63435948818eb376367b113c188e37451c216380f65d1ad55f73f527, 0xf9dd2e391565534a4db84980433bf5a56250f45fe294fce2679bcf115522c081, 0x1166591ee2ca59b9f4e525900f085141be8879c66ef18529968babeb87c44814, 0xf4fa2e8de39bdbeb29b64d8b440d3a6c9a6ca5bdce543877eaee93c11bd70ab8, 0x07f466d73b93db283b3f7bfaf9c39ae296adc376ab307ef12312631d0926790e, 0xb2ecff93acb4fa44c1dbf8464b81734a863b6d7142b02f5c008907ea4dc9aaa1, 0xa1d9c342f6c293ac6ef8b5013cba82c4bad6ed7024d782948cb23cd490039ba1, 0xc7d04a639ba00517ece4dbc5ef4aaf20e0ccde6e4a24c28936fabe93dec594db, 0xe3cbb9810472d9dd1cdb5eed2f74b67ea60e973d2d2e897bd64728c9b1aa0679, 0xe36884703413958ff2aba7a1f138a26d0ac0a371270f0169219beb00a5add5f0, 0xe5ea300a09895b3f98de5232d92a36d5611cbcf9aaf9e7bb20cf6d1696ad1cb4, 0xc136cda884e18175ab45148ed4f9d0d1a3c5e11ad0275058e61ae48eb151a81f, 0x3ee1101e944c040021187e93b6e0beb1048c75fb74f3fdd67756b1c8517a311f, 0x016964fd6fc32b9ad07a630949596715dee84d78230640368ff0929a280cf3a2, 0xe33865fc03120b94333bb754fd097dc0f90e69ff6fd221d6aae59fcf2d762d76, 0xe80bb3515a09ac6ecb4ec59de22701cdf954b1ae8a677fd85508c5b041f28058, 0x3889af7cd325141ec288021ede136652a0411d20364005b9d3ca9102cb368f57, 0x18dad0bc975cf8800addd54c7867389d3f7fe1b97d348bd8412a6cbfb75c520a, 0x09035218686061ee91bd2ad57dc6fb6da7243b8177a153484524b2b228da5314, 0x688fd7a97551c64eae33f91abb073a46eafbbacd5595c6bac2e57dd536acdfe2, 0x1fc164dce565a1d0da59cc8048b334cc5eb84bf04de2399ddb847c22a7e32ab7, 0xa2a340ba05c8a30dd1cab886a926b761758eba0e41b5c4c5dfd4a42f249655c1, 0xc43dffe01479db836a6a1a74564b297fad0d69c6b06cf593f6db9f26b4f307d5, 0x73cef7f3ff724a30a79e1dca74cef74954afeefa2e476c4dec65afe50c16c5c4, 0xa54002253ab7b95cc5b664b3f08976400475cc56f170b939f6792e730ff5170b, 0x9ade43053d41afebc002f09476dffd1b13ecbf67f810791540b92ca56d5e63e4, 0x234e7cbfbe45b22a871db26738fa05de09213a925439d7f3e5108132e521b280, 0x066b712417332c7cfca871fb1bb5839f0341acf9266229603a3eddbc8a93b59f, 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0x4fa2cd877aa115b476082b11053309f3537fa03d9158085f5f3f4bab6083e6da, 0xed12db4069eb9f347735816afcee3fe43d4a6999fef8240b91bf4b05447d734f, 0x3ecbe5eda469278f68548c450836a05cc500864664c7dda9b7526f084a891032, 0x690d8f928fc61949c22e18cceaa2a446f8e1b65bd2e7af9e0a8e8284134ab3d2, 0x99e09167a09f8261e7e8571d19148b7d7a75990d0702d9d582a2e4a96ac34f8e, 0x6d33931693ed7c2e1d080b6a37da52c279a06cec5f534305819f7adf7db0afe3, 0xc4b735462a9a656e28a52b1d4992ea9dea826b858971d698453a4be534d6bb70, 0xedf92b10302dc41f8d362b360f4c2ef551d50e2ded012312c964002d2afc46d7, 0x58f6691cca081ae5c3661dd171b87cc49c90359bb03cc0e57e503f7fcf14aefc, 0x5d29b8b4ee295a73c4a8618927b3d14b76c7da049133a2257192b10be8c17a6a, 0x646802fa42801e0ae24011fb4f62e87219ef1da01f7fc14bf8d6bd2d9e7c21f1, 0x23abf45eee65cc4c1e95ccab42ad280a00bb3b14d243e2021a684075f900141e, 0x2b1ae95c975bf9c387eae506fdb5e58afd2d198f00a21cd3fddb5855e8021e4d, 0x0ef9f6e1c0583493d343e75f9c0c557fa6da0dc12b17a96c5757292916b72ee3, 0x04c7fc76195c64a3285af14161077c045ff6ddbb67c0ff91b080f98eb6781e5c, 0xba12679b97027d0e7076e6d19086c07792eaa7f78350842fbef8ddf5bcd3ecc0, 0xcead458e6799df4d2f6cbf7f13cb3afec3441a354816e3071856ed49cbdbb1a7, 0xbe6c56256556bb5c6727a1d9cb641d969677f56bb5ad7f8f7a7c9cfd128427b4, 0xc80f11963ff40cb1888054b83c0463d32f737f2e7d42098e639023db0dfc84d4, 0xac80006c1296bcfde86697efebb87fb0fddfb70dd34dd2ee4c152482af4687eb, 0xbb7d13ce184249df4576fc3d13351e1683500e48726cd4198423f14f9094068b, 0x1b2d9c40c55bd7362664fa46c1268e094d56c8193e3d991c08dc7a6e4ca14fa1, 0x9bd236254d0565f5b2d24552d4b4d732de43b0adaa64ecd8be3efc6508577591, 0x38078cefccc04e8312d79e0636e0e3157434c50a2ad4e3e87cc6584c41eec8b5, 0xb5d15a8527ff3fa254ba61ffceb02d2570b53361894f351a9e839c0bb716857d, 0x6763dad684bf2e914f40ae0a7ee0cdf12c97f41fc05a485d5991b4daad21a3f8, 0xc80363c20df589333ecbe05bd5f2c19942ebc2593626dc50d00835c40fb8d005, 0x48502b56ae93acd2794f847cbe825525d5d5f59f0f75c67aff84e5338776b3af, 0xfd8e033493ba8af264a855a78ab07f37d936351d2879b95928909ed8df1b4f91, 0x11f75bee9eac7356e65ebc7f004ccdc1da80807380d69143293d1421f50b1c97, 0x903a88a3ebe84ca1c52a752b1faffa9ca1daedac9cbf1aa70942efc9beb44b79, 0x2c0dcd68837f32a69da651045ad836b8cd6b48f2c8c5d73a3bd3bba6148d345a, 0x0aa0f49b3476f3fdb6393f2ab601e0009586090b72ee54a525734f51598960d5, 0xf7a789f013f702731656c562caa15b04cb7c9957376c4d80b8839167bb7fa626, 0x4e0be1b19e305d82db3fd8affd67b0d2559da3edbfb08d19632a5cc46a90ed07, 0x3caaccfc546d84d543eaf4f4c50c9c8fd831c12a8de56fdb9dfd04cc082882fe, 0x894f6a01fd34f0642077e22981752011678548eb70eb55e8072c1caffc16fe02, 0xae7eb54adaa68679348ea3537a49be669d1d61001fbab9fac259ba727dbc9a1a, 0x291a1cbdceff957b5a65440ab67fb8672de881230fe3108a15ca487c2662c2c7, 0x891d43b867137bf8beb9df4da2d951b5984a266a8cd74ec1593801d005f83f08, 0xc558407f6491b37a10835e0ad7ce74f4e368aa49157a28873f7229310cb2d7fd, 0x9ce061b0a072e1fe645f3479dac089b5bfb78cfa6cfbe5fd603bcdb504711315, 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