Monero Inflation Checker

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Anatomy of a real FCMP++ proof

We are going to analyse the transaction below.

Testnet transaction 1 input / 2 outputs

{
  "version": 2, 
  "unlock_time": 0, 
  "vin": [ {
      "key": {
        "amount": 0, 
        "key_offsets": [ ], 
        "k_image": "80e251c490c65c224a758c2caab46291ffe5d994232079ea37e3806ef2876584"
      }
    }], 
  "vout": [ {
      "amount": 0, 
      "target": {
        "carrot_v1": {
          "key": "0792db64bace803ba7ce997fbe8ba6d935ca5bfbd527e44f7ca3dcaef17794a1", 
          "view_tag": "728ef0", 
          "encrypted_janus_anchor": "729fccfbb96c2be286c2519e9fb4c003"
        }
      }
    }, {
      "amount": 0, 
      "target": {
        "carrot_v1": {
          "key": "e44e0f9ef408366bbc4a11f8ae8f33e2a3e08157ed0ee5e241c8ef63462fcb5a", 
          "view_tag": "69dece", 
          "encrypted_janus_anchor": "5a84eed999a291d15ec3787523b00098"
        }
      }
    }
  ], 
  "extra": [ 4, 2, 139, 23, 249, 69, 114, 105, 158, 208, 233, 120, 62, 167, 42, 193, 110, 24, 56, 78, 19, 12, 172, 199, 163, 228, 32, 157, 245, 129, 121, 180, 82, 78, 243, 134, 81, 139, 165, 169, 85, 240, 44, 55, 189, 172, 253, 173, 108, 184, 77, 229, 175, 58, 38, 154, 133, 222, 44, 0, 23, 115, 231, 164, 221, 91
  ], 
  "rct_signatures": {
    "type": 7, 
    "txnFee": 600000000, 
    "ecdhInfo": [ {
        "amount": "86a9b4d43efe4e09"
      }, {
        "amount": "fda219b8f3030566"
      }], 
    "outPk": [ "99110cd5fd06ec29a4a32423bf04967194b3ad38ff45c99736a688689de29bc6", "9e5ebc7e7a3890fbfc51614379d52c0dee4ff93b8256fd05731b4b2fa3a18656"]
  }, 
  "rctsig_prunable": {
    "nbp": 1, 
    "bpp": [ {
        "A": "d4eabc649f89889e34f5da107b9d087339bbde961317a75b9ad665cbf6afedc4", 
        "A1": "788ed6be888fc0e6daa40b42d6b1abb6c40ec0606ab923f523a395d9f4d8203d", 
        "B": "1a264c7e8861bbcda8d5d33949564159a0e4bc1140c6f47aba1f6c2dd4251d02", 
        "r1": "c43f430e8a594b210918e6c25af54d434ce94883dd4b3bfed2ad19708ff7ad08", 
        "s1": "bb15a2a0a5d04b695e39cbf4d7185ffea40750e87c536cc24a73c7b36bfba20d", 
        "d1": "8ad7aa46c44628773995a31eb74d9129131e5cc5b7e4c528ee3b88fa4cd8d50e", 
        "L": [ "cc989f321bd35fc86abc538043cbf99e72c33a72a592c4d11854bc721cc84364", "4816b9b6ba9d87076751ffa887b48721dc3536445b70167ef4f8eae33a7e092d", "46f455ad786a659b076346582026f97dc76e16c99b60909ee7a91c13c125c7cc", "5b73a57806bd64db0364e6ac99dfe8c3a0647509c58077f5ef868f790ea22fb6", "9b242d174578aa1cc733299229b57afa0a6067ae2ae85436fe548eb8c4e9e361", "e45fdb1dd726a5d2cf6b1f69a79e76477eda93d56157ebc4efd0223ac0eb0f1c", "bfd099a99d0dc6bc165ee78ba86081668a26899f9256ed86ff6194c4cdb772bd"
        ], 
        "R": [ "2223f9a581336d01bb3699f931ee7d5155b7f53d998803aba76b30a7f90040f7", "8606934a39f9ee01dd515614e94c44cfb3e0de13cb021d997206451d850bd4df", "9d84c0c37dbce3b1b15154c6fdc09ebc37cc882b9a0f175cd584950d990a9df1", "b3e81fd5ee6910a3446055058025a666adfb936c08775542b742a4108f51ebcd", "b76ee97ffb6d6bb557447e2ef306fafe71bca21be780efa52301cb90e9bb92be", "6ef16fc8648e68ac3de3dd215a1afccca8c83dc12c3b45ae640dd4e684c9d806", "d8753041dc3f7f7d25b41e86f04d17c2e0238c173c0413770fb8216c51bd4d99"
        ]
      }
    ], 
    "reference_block": 3073976, 
    "n_tree_layers": 6, 
    "fcmp_pp": "f71b2aa43485e2f1311d9ca29fda2af06d17f0d2dee91ea1e924f2f174f5fe5051448a0ccf621d52939d34b74e6b4c0f4ac3edad30645eba0323f4533fb0b65035c61ff53e702b2be7caed908739db92d13da85e87e1acfddf4ddc622d3104761924f4e4cfc5533081bdfc1d8eee24a62f40e61ccbed2705700d2ce3abf1ac300ac37f5eb3d5f60616e286316160f50f477d48f715cc7e5f6bad3219945537c8dd1286b09cbf6e88077112f04d3e74741c3270089887b21c931544ddfc118c307b9f1acbc068ea4cd40f61c0632137968c066054080888e310b7c5a8f39c24a79a2871b0b1e37221ac5fe4503e68742bec42d3b3808299f86d7f18432f41d7fc6e930a718752e86f80c52e9282ab782b8e5c1ab51b6e6cc0a280fdf3a2607ddd5d6fc8d36a1832c4b2affc4fd420ca25361f346fff18549e6364448a6338b50ca54b6199c6634f5b78d55ec101d6f2e59168d08d4f275d70205a4fc4938e1e01b015b0d4667cb20dfe63032644a40957ae4ee675c9240be10c579416ee10af015b930f6189a8f28b7941cdce67e4009865e869ebe68da7d863970e3805a6880bc40b21d9bcc406fa5f4422aac31f0254205500c214ed5fab3d55b491dcd46b0b1c76474565672b5464eb971e5d5b751eb0895cd69d8a86a8ad1a4b24c1199c0b9be834bc026c16cd42d81eb73cfcfe0fb5fe3d6e710e072e330d7b5b8d357f594b2c2c109c0f98472bb88b8b9b665d27249033815da3dd850fad29ef761569a09e3e52d050d6bd7869ffa209a248140bec1281924ae6c24355e4d710d84e4d671872afe8f097ec297b635d294562b5c95882326ea84ee96c3f4325282dcfc15084abce2a9704379d9639f82add2145a4ab7788b57f5ce578f54cca07fdf2346e0bdf9f2c0917e6ed2d660c0954c9e50cc50279814d0e962e63f38d8550299620e20b08f53834f024c3f242ffee939a7834a6527f396438c145c592ec7b1bce4e18127abf5a1a237c3b6325e6af7d6e83f7f5ccfc66f47eebca6ce6ec8b9f141ec939be976a4073ee90efab41defb272adc6ea89748a41e644566158f9fe587784d2eedec6e084845b95b09bbaa9dfcf0a68f259d4417554b133c5c2db6c24dc4bea608e804076e9535491a65eacf0d06d972928ff398a7614b059bf1b0ad52326d97f8a528841416dd8c60a37b942ecebf9cc6f49c9878721889a12e88d0cb8de6f89a7192b308cae8b67c045142ab332d87a5c29fd008e2bf99d72785a7a8ec240a3660f4fcaeee722c466ca85f6452eed28c6416d6254fb36a4452f21b4c43351bdcc04eceb739aa018427c49d518897b50969bfb7d9b98891d9402224b9e21b2491d1076ea4386316159d8e1bda57f840e4faffedb8dd66e1133191f453e8c98b1633d153abdecee2c19d55309a75ebcfe610643935545564ceadff98af4ed708ab3de741e084e696ccb0d6a474462bdef46bfde7562fdb8829e68cc00a167ed1879cbe0b9919260a1cdc06c3a47c4dabf08b1d8457c3f35c11e01bc8bb4b1ed2b0a98eb8e6fba86155369d3bfe3271087777b1bc664eda4513ebcd2128b45b77613b90f2b90fb4079949c48faf0f4259038bca8570bdeea0e8580f3e7626fec7cdd4b430d469a5aa393e35316e24b70695e6423f9cc8efb9624245d0a3f694c5c84941ba74eebe3cad71ad34cca4e5ee1d65b8413d1e7d9a40409b2bae53b0fc067f31bd92f84d5531ca88ec8814599f0e870f0333a3a9332a18993eeca12410a6cfa146f3153f8dc1c5c1ad90dc3fbfcd171c73344448b8e24dde01dd058ce8a81a1d8c83cf5424a6b51f9e528339432139154ea55280b0678691c3f672772a7f272986475a5c3a1a1385f87388a7f609f9801c1d3d3da9521e93ec38ae70748e6282662a7f214dfb7141a12c5bd7454156b8aace66df628c2f5e13ec63e35bb545f2e9cc5e34e5fdb0ebc3ff9bd4cb22d601a7f12d4b4131db6dae5fe17c3ac016959bbbdc71f9d3d30aa9c6edf43d07ff4e0469b2d0e865b792f71ab5b68e24c4f1f220485879b6469df3e857519bb8f051f3b91803e3455f17cf00b5d8ca6ec7121d2c8771785d2737f2bac7bc75780669e01017e05a02d056b0327984d7f1d2dee4226e85043bd887164acd8b7b430d54b5fb1bd5be7af8312e333d3b2410ad93d81f2a3c925cfbbc480077a6c173e230647b0ddf5fdb5d4699f40d299861cc6bff900442033e608bf8428beea737a40f856302f93e732099b50907498f67949bc8dd68d833bd0307e54626f63806a83f1fd953bd171b9cefec651ccf24a83f4039cde1a5dd9d8877444d9426328c58a6265d700074c448c4360fac88e33db27874d3b80050639f561c614ef753f1df9a3f6252578079bd68223d3c70b8d76a5754ac1e3dcd36d18197dfd592038ec03156fcb87ba220b19bb970733ecdb9419a28bcb2015198b6de0fc024dc076a32e0a2839b8a80ccfd73ea7824f16ed075a12a7bdaea9b8963826e905dcb172eac7f33ec31e43f358d9dead0828edaa0c97772cb7d72501e24d328753c9caf60e2f67f868a80cdea57a4a6232ee44f1cc8fb8f257e4fc87b93a985abdaa7743f92943b424ed59f306e1e13d222142715b2cc99e93c2e20935ac4b75b571facd4293fc68d8334d839da8ddcda7c00eb71a0837d2b1fba74cbdc776896c80b39d5d6724112cb99d79df2539cf112183e4113edd7888bc4b5a6ac7b6b9dd25084b091868da82024b5aeee7f2e151735c8de3445755d4fe5d9291e9af90d8aab053885835161d21713c611f89113da3f4ded2fce7889807d64704d2e3ff8f8542ea7e329a847ba9b636e3ec9850e885973b4cc0b8e4df0533a78bccefc7f9ea3534992c3535e00f7e7ef7bb56357ff4c45f6bfdb412c717240f931e6edd245c4af5a4c0f8b33d40d3dc8d466a9a52e21d41fb703114a516412303183df0e4d0bcc7076ed925d00f3daba27359be8b24e67244a04d4183062e5051d15635c159e3a5c59f281be6e76ef627f89ec76195b0e8d7b8b03dc73f75ee1994ea5a3868c0bb4a315f43ca4a12977af86ae9756b16cf72425712dcd60fa6711835c6a9c50145d948d84f4cdf62ddcb4d12ea31dc8aa3c4a9c564af4563aa9538f0edb433bce6cc8be9429d41a6370560bbeb2dee8ce7ef2b6f192d73ec8405c8da22405a925f2ad8e30237281ced4eb4569c4fb1a2620ea8d3ef530c274d149ab9b1801530055dd03500f043608d40a54e6cbf5a0e773f660064e0c373d79ade3abc1f9e76209037dbb15ea1322af4059ee4bb9c77d88c23b17eb58790a5e041b901513dbc595aa7bd40cdc904e728caab4282ed9f1e3576afab58d0c9884a3d2ebd31fd11860613dac33b32d51de1c065a39e2242207757995986e7124cbf00fd6e1790921db7891649d02c290ac742523efeae80be3fae9267d12eb313e54956fc571ac03c0f57d7cfc0e7a70102f1facfe6eb722c0bd83071ac9df8db1982884a66f48891833932aa9f6f9afb30a3ae0dc8ca3a7a2c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
    "pseudoOuts": [ "454e867009aef302815597139182d3b8123f57c3e09cd6ac724c4e7c9fe59d49"]
  }
}

This transaction spends 1 input to 2 outputs against a tree of 6 layers, and its fcmp_pp is 6304 bytes = 197 words of 32 bytes each, where every word is either a compressed curve point or a field scalar.

in0480

C992

Selene BP2464

Helios BP2304

per-input: re-randomized tuple + SAL tree, committed (

C

) the two Bulletproofs (circuit) root anchor

The `fcmp_pp` field

Three regions, in order: one 480-byte block per input, then the membership proof, then a 64-byte anchor. For 1 input at 6 layers, $1 \cdot 480 + 5760 + 64 = 6304$ bytes.

byte range	size	region	source
0 – 479	480	input 0 · tuple + SAL	sal.py, blinds.py
480 – 6239	5760	membership proof	fcmp.py, gbp.py
6240 – 6303	64	root-blind Schnorr PoK	fcmp.py

What is not in `fcmp_pp` and where it lives

Two things the proof depends on are located in another field of the transaction: the amount pseudo-outputs $\tilde{C}$ sit in rctsig_prunable.pseudoOuts and the key images $L$ sit in each vin. For this transaction, we have:

value	lives in	32 bytes
$L = x H_{p} (O)$	vin[0].k_image	80e251c490c65c224a758c2caab46291ffe5d994232079ea37e3806ef2876584
$\tilde{C}$	pseudoOuts[0]	454e867009aef302815597139182d3b8123f57c3e09cd6ac724c4e7c9fe59d49

How the transaction scales

The table below gives the size of fcmp_pp, in bytes, for 1, 2, 4, 8, 16, 32, 64 and 128 inputs, broken down by region, so we can see both what each part contributes and how the whole thing scales:

inputs	$c_{1}$ / $c_{2}$	IPA rounds	tuple+SAL	$C$	Selene BP	Helios BP	PoK	total	Bytes / input
1	16 / 15	8 / 7	480	992	2464	2304	64	6304	6304
2	19 / 17	9 / 8	960	1152	2816	2560	64	7552	3776
4	25 / 21	10 / 9	1920	1472	3456	3008	64	9920	2480
8	37 / 29	11 / 10	3840	2112	4672	3840	64	14528	1816
16	61 / 45	12 / 11	7680	3392	7040	5440	64	23616	1476
32	109 / 77	13 / 12	15360	5952	11712	8576	64	41664	1302
64	205 / 141	14 / 13	30720	11072	20992	14784	64	77632	1213
128	397 / 269	15 / 14	61440	21312	39488	27136	64	149440	1167.5

Every 32-byte word

The whole proof, all 197 words of 32 bytes, each with its field, its defining formula, and the first and last bytes of each (full hex is in the dump above). Points and scalars are both 32 bytes.

word · bytes	field	equation	bytes
input 0 · tuple + SAL
w0 · 0–31	$\tilde{O}$ point	$\tilde{O} = x G + y T$	f71b2aa4…74f5fe50
w1 · 32–63	$\tilde{I}$ point	$\tilde{I} = I + r_{i} U$	51448a0c…3fb0b650
w2 · 64–95	$R$ point	$R = r_{i} V + r_{r_{i}} T$	35c61ff5…2d310476
w3 · 96–127	$P$ point	$P = x G + r_{i} V + x r_{i} U + r_{p} T$	1924f4e4…abf1ac30
w4 · 128–159	$A$ point	$A = α G + β V + (α r_{i} + β x) U + δ T$	0ac37f5e…945537c8
w5 · 160–191	$B$ point	$B = α β U + μ T$	dd1286b0…fc118c30
w6 · 192–223	$R_{O}$ point	$R_{O} = α G + r_{y} T$	7b9f1acb…f39c24a7
w7 · 224–255	$R_{P}$ point	$R_{P} = r_{z} U + r_{r_{p}} T$	9a2871b0…2f41d7fc
w8 · 256–287	$R_{L}$ point	$R_{L} = α \tilde{I} - r_{z} U$	6e930a71…a2607ddd
w9 · 288–319	$s_{α}$ scalar	$s_{α} = α + e x$	5d6fc8d3…6338b50c
w10 · 320–351	$s_{β}$ scalar	$s_{β} = β + e r_{i}$	a54b6199…938e1e01
w11 · 352–383	$s_{δ}$ scalar	$s_{δ} = μ + e δ + r_{p} e^{2}$	b015b0d4…ee10af01
w12 · 384–415	$s_{y}$ scalar	$s_{y} = r_{y} + e y$	5b930f61…05a6880b
w13 · 416–447	$s_{z}$ scalar	$s_{z} = r_{z} + e x r_{i}$	c40b21d9…dcd46b0b
w14 · 448–479	$s_{r_{p}}$ scalar	$s_{r_{p}} = r_{r_{p}} + e (r_{p} - y - r_{r_{i}})$	1c764745…c1199c0b
vector commitments C on Selene: C0 leaf floor, C1 floor 2, C2 floor 4, C3–C4 input 0: o_blind on T, C5–C6 input 0: i_blind on U, C7 input 0: i_blind on V (divisor only), C8–C9 input 0: i_blind_blind on T, C10–C11 input 0: c_blind on G, C12–C13 opens the blind of Helios C0, C14–C15 opens the blind of Helios C1
w15 · 480–511	$C_{0}$ point	$C_{i} = \sum_{j < n_{i}} v_{j} G_{j} - s_{i} H$	9be834bc…8d357f59
w16 · 512–543	$C_{1}$ point	$C_{i} = \sum_{j < n_{i}} v_{j} G_{j} - s_{i} H$	4b2c2c10…761569a0
w17 · 544–575	$C_{2}$ point	$C_{i} = \sum_{j < n_{i}} v_{j} G_{j} - s_{i} H$	9e3e52d0…d84e4d67
w18 · 576–607	$C_{3}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	1872afe8…2dcfc150
w19 · 608–639	$C_{4}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	84abce2a…fdf2346e
w20 · 640–671	$C_{5}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	0bdf9f2c…50299620
w21 · 672–703	$C_{6}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	e20b08f5…7b1bce4e
w22 · 704–735	$C_{7}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	18127abf…8b9f141e
w23 · 736–767	$C_{8}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	c939be97…9fe58778
w24 · 768–799	$C_{9}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	4d2eedec…b6c24dc4
w25 · 800–831	$C_{10}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	bea608e8…b0ad5232
w26 · 832–863	$C_{11}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	6d97f8a5…88d0cb8d
w27 · 864–895	$C_{12}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	e6f89a71…85a7a8ec
w28 · 896–927	$C_{13}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	240a3660…f21b4c43
w29 · 928–959	$C_{14}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	351bdcc0…2224b9e2
w30 · 960–991	$C_{15}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	1b2491d1…91f453e8
vector commitments C on Helios: C0 floor 1, C1 floor 3, C2 the root, C3–C6 opens the blind of Selene C0, C7–C10 opens the blind of Selene C1, C11–C14 opens the blind of Selene C2
w31 · 992–1023	$C_{0}$ point	$C_{i} = \sum_{j < n_{i}} v_{j} G_{j} - s_{i} H$	c98b1633…ff98af4e
w32 · 1024–1055	$C_{1}$ point	$C_{i} = \sum_{j < n_{i}} v_{j} G_{j} - s_{i} H$	d708ab3d…8cc00a16
w33 · 1056–1087	$C_{2}$ point	$C_{i} = \sum_{j < n_{i}} v_{j} G_{j} + b_{root} H$	7ed1879c…1bc8bb4b
w34 · 1088–1119	$C_{3}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	1ed2b0a9…cd2128b4
w35 · 1120–1151	$C_{4}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	5b77613b…0f3e7626
w36 · 1152–1183	$C_{5}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	fec7cdd4…45d0a3f6
w37 · 1184–1215	$C_{6}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	94c5c849…9b2bae53
w38 · 1216–1247	$C_{7}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	b0fc067f…993eeca1
w39 · 1248–1279	$C_{8}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	2410a6cf…de01dd05
w40 · 1280–1311	$C_{9}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	8ce8a81a…91c3f672
w41 · 1312–1343	$C_{10}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	772a7f27…93ec38ae
w42 · 1344–1375	$C_{11}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	70748e62…5e13ec63
w43 · 1376–1407	$C_{12}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	e35bb545…6dae5fe1
w44 · 1408–1439	$C_{13}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	7c3ac016…92f71ab5
w45 · 1440–1471	$C_{14}$ point	$C_{i} = \sum_{j} v_{j} G_{j} + b_{i} H$	b68e24c4…17cf00b5
Selene (C1) Bulletproof
w46 · 1472–1503	$A_{I}$ point	$A_{I} = ⟨ a_{L}, G ⟩ + ⟨ a_{R}, H ⟩ + α h$	d8ca6ec7…56b03279
w47 · 1504–1535	$A_{O}$ point	$A_{O} = ⟨ a_{O}, G ⟩ + β h$	84d7f1d2…312e333d
w48 · 1536–1567	$S$ point	$S = ⟨ s_{L}, G ⟩ + ⟨ s_{R}, H ⟩ + ρ h$	3b2410ad…4699f40d
w49 · 1568–1599	$T_{0}$ point	$T_{j} = t_{j} g + τ_{j} h$	299861cc…99b50907
w50 · 1600–1631	$T_{1}$ point	$T_{j} = t_{j} g + τ_{j} h$	498f6794…efec651c
w51 · 1632–1663	$T_{2}$ point	$T_{j} = t_{j} g + τ_{j} h$	cf24a83f…c4360fac
w52 · 1664–1695	$T_{3}$ point	$T_{j} = t_{j} g + τ_{j} h$	88e33db2…68223d3c
w53 · 1696–1727	$T_{4}$ point	$T_{j} = t_{j} g + τ_{j} h$	70b8d76a…9bb97073
w54 · 1728–1759	$T_{5}$ point	$T_{j} = t_{j} g + τ_{j} h$	3ecdb941…73ea7824
w55 · 1760–1791	$T_{6}$ point	$T_{j} = t_{j} g + τ_{j} h$	f16ed075…9dead082
w56 · 1792–1823	$T_{7}$ point	$T_{j} = t_{j} g + τ_{j} h$	8edaa0c9…a4a6232e
w57 · 1824–1855	$T_{8}$ point	$T_{j} = t_{j} g + τ_{j} h$	e44f1cc8…1e13d222
w58 · 1856–1887	$T_{9}$ point	$T_{j} = t_{j} g + τ_{j} h$	142715b2…8ddcda7c
w59 · 1888–1919	$T_{10}$ point	$T_{j} = t_{j} g + τ_{j} h$	00eb71a0…539cf112
w60 · 1920–1951	$T_{11}$ point	$T_{j} = t_{j} g + τ_{j} h$	183e4113…7f2e1517
w61 · 1952–1983	$T_{12}$ point	$T_{j} = t_{j} g + τ_{j} h$	35c8de34…f89113da
w62 · 1984–2015	$T_{13}$ point	$T_{j} = t_{j} g + τ_{j} h$	3f4ded2f…c9850e88
w63 · 2016–2047	$T_{14}$ point	$T_{j} = t_{j} g + τ_{j} h$	5973b4cc…b56357ff
w64 · 2048–2079	$T_{15}$ point	$T_{j} = t_{j} g + τ_{j} h$	4c45f6bf…66a9a52e
w65 · 2080–2111	$T_{16}$ point	$T_{j} = t_{j} g + τ_{j} h$	21d41fb7…359be8b2
w66 · 2112–2143	$T_{17}$ point	$T_{j} = t_{j} g + τ_{j} h$	4e67244a…89ec7619
w67 · 2144–2175	$T_{18}$ point	$T_{j} = t_{j} g + τ_{j} h$	5b0e8d7b…86ae9756
w68 · 2176–2207	$T_{19}$ point	$T_{j} = t_{j} g + τ_{j} h$	b16cf724…d12ea31d
w69 · 2208–2239	$T_{20}$ point	$T_{j} = t_{j} g + τ_{j} h$	c8aa3c4a…0bbeb2de
w70 · 2240–2271	$T_{21}$ point	$T_{j} = t_{j} g + τ_{j} h$	e8ce7ef2…4569c4fb
w71 · 2272–2303	$T_{22}$ point	$T_{j} = t_{j} g + τ_{j} h$	1a2620ea…54e6cbf5
w72 · 2304–2335	$T_{23}$ point	$T_{j} = t_{j} g + τ_{j} h$	a0e773f6…59ee4bb9
w73 · 2336–2367	$T_{24}$ point	$T_{j} = t_{j} g + τ_{j} h$	c77d88c2…aab4282e
w74 · 2368–2399	$T_{25}$ point	$T_{j} = t_{j} g + τ_{j} h$	d9f1e357…065a39e2
w75 · 2400–2431	$T_{26}$ point	$T_{j} = t_{j} g + τ_{j} h$	24220775…2523efea
w76 · 2432–2463	$T_{27}$ point	$T_{j} = t_{j} g + τ_{j} h$	e80be3fa…1facfe6e
w77 · 2464–2495	$T_{28}$ point	$T_{j} = t_{j} g + τ_{j} h$	b722c0bd…3ae0dc8c
w78 · 2496–2527	$T_{29}$ point	$T_{j} = t_{j} g + τ_{j} h$	a3a7a2cc…bc2f8fd9
w79 · 2528–2559	$T_{30}$ point	$T_{j} = t_{j} g + τ_{j} h$	37790974…4a7d203f
w80 · 2560–2591	$T_{31}$ point	$T_{j} = t_{j} g + τ_{j} h$	4cf7ec76…3868e119
w81 · 2592–2623	$T_{32}$ point	$T_{j} = t_{j} g + τ_{j} h$	e14fc975…daf06023
w82 · 2624–2655	$T_{33}$ point	$T_{j} = t_{j} g + τ_{j} h$	bc22dacd…f29464c0
w83 · 2656–2687	$T_{34}$ point	$T_{j} = t_{j} g + τ_{j} h$	8c62dbf4…11bc02a2
w84 · 2688–2719	$T_{35}$ point	$T_{j} = t_{j} g + τ_{j} h$	142f3612…005aa4c4
w85 · 2720–2751	$T_{36}$ point	$T_{j} = t_{j} g + τ_{j} h$	2fd12dd3…fa5de2ed
w86 · 2752–2783	$T_{37}$ point	$T_{j} = t_{j} g + τ_{j} h$	d8ec1b85…95be28e1
w87 · 2784–2815	$T_{38}$ point	$T_{j} = t_{j} g + τ_{j} h$	bb847113…c97faf20
w88 · 2816–2847	$T_{39}$ point	$T_{j} = t_{j} g + τ_{j} h$	5aa8a95d…f2717423
w89 · 2848–2879	$T_{40}$ point	$T_{j} = t_{j} g + τ_{j} h$	1bd86beb…0832b767
w90 · 2880–2911	$T_{41}$ point	$T_{j} = t_{j} g + τ_{j} h$	aa7a2a77…04b8438a
w91 · 2912–2943	$T_{42}$ point	$T_{j} = t_{j} g + τ_{j} h$	05659f8e…911abbdd
w92 · 2944–2975	$T_{43}$ point	$T_{j} = t_{j} g + τ_{j} h$	d800f56a…f07e035d
w93 · 2976–3007	$T_{44}$ point	$T_{j} = t_{j} g + τ_{j} h$	cbb9def8…7158aef3
w94 · 3008–3039	$T_{45}$ point	$T_{j} = t_{j} g + τ_{j} h$	97016421…f1bc262c
w95 · 3040–3071	$T_{46}$ point	$T_{j} = t_{j} g + τ_{j} h$	2f5a05ec…8c5bddf9
w96 · 3072–3103	$T_{47}$ point	$T_{j} = t_{j} g + τ_{j} h$	b3f218c7…b7c7831d
w97 · 3104–3135	$T_{48}$ point	$T_{j} = t_{j} g + τ_{j} h$	f4e71971…7242d242
w98 · 3136–3167	$T_{49}$ point	$T_{j} = t_{j} g + τ_{j} h$	565c27e9…31da22ce
w99 · 3168–3199	$T_{50}$ point	$T_{j} = t_{j} g + τ_{j} h$	18f32841…08d68493
w100 · 3200–3231	$T_{51}$ point	$T_{j} = t_{j} g + τ_{j} h$	774a32cf…425733e2
w101 · 3232–3263	$T_{52}$ point	$T_{j} = t_{j} g + τ_{j} h$	3d37190f…75645b0a
w102 · 3264–3295	$τ_{x}$ scalar	$τ_{x} = \sum_{j} τ_{j} x^{j}$	41eef7a2…b57f2029
w103 · 3296–3327	$u$ scalar	$u = α x^{c + 1} + β x^{n^{'}} + ρ x^{n^{'} + 1} + \sum_{i} μ_{i} x^{n^{'} - 1 - i}$	29c36735…ca783467
w104 · 3328–3359	$\hat{t}$ scalar	$\hat{t} = ⟨ l, r ⟩$	4f210549…483e3323
w105 · 3360–3391	$L_{0}$ point	$L = ⟨ a_{1}, G_{2} ⟩ + ⟨ b_{2}, H_{1}^{'} ⟩ + ⟨ a_{1}, b_{2} ⟩ U$	40d1a668…effe83b1
w106 · 3392–3423	$R_{0}$ point	$R = ⟨ a_{2}, G_{1} ⟩ + ⟨ b_{1}, H_{2}^{'} ⟩ + ⟨ a_{2}, b_{1} ⟩ U$	6e4b4068…958386f9
w107 · 3424–3455	$L_{1}$ point	$L = ⟨ a_{1}, G_{2} ⟩ + ⟨ b_{2}, H_{1}^{'} ⟩ + ⟨ a_{1}, b_{2} ⟩ U$	b582e8ef…19e3052f
w108 · 3456–3487	$R_{1}$ point	$R = ⟨ a_{2}, G_{1} ⟩ + ⟨ b_{1}, H_{2}^{'} ⟩ + ⟨ a_{2}, b_{1} ⟩ U$	64cbd322…7358bc34
w109 · 3488–3519	$L_{2}$ point	$L = ⟨ a_{1}, G_{2} ⟩ + ⟨ b_{2}, H_{1}^{'} ⟩ + ⟨ a_{1}, b_{2} ⟩ U$	924d4e04…d28e4dda
w110 · 3520–3551	$R_{2}$ point	$R = ⟨ a_{2}, G_{1} ⟩ + ⟨ b_{1}, H_{2}^{'} ⟩ + ⟨ a_{2}, b_{1} ⟩ U$	519ef1db…3274a68d
w111 · 3552–3583	$L_{3}$ point	$L = ⟨ a_{1}, G_{2} ⟩ + ⟨ b_{2}, H_{1}^{'} ⟩ + ⟨ a_{1}, b_{2} ⟩ U$	cd99c020…3e22574f
w112 · 3584–3615	$R_{3}$ point	$R = ⟨ a_{2}, G_{1} ⟩ + ⟨ b_{1}, H_{2}^{'} ⟩ + ⟨ a_{2}, b_{1} ⟩ U$	26f8b9ff…3b03d5f2
w113 · 3616–3647	$L_{4}$ point	$L = ⟨ a_{1}, G_{2} ⟩ + ⟨ b_{2}, H_{1}^{'} ⟩ + ⟨ a_{1}, b_{2} ⟩ U$	5b252bf6…9931e796
w114 · 3648–3679	$R_{4}$ point	$R = ⟨ a_{2}, G_{1} ⟩ + ⟨ b_{1}, H_{2}^{'} ⟩ + ⟨ a_{2}, b_{1} ⟩ U$	d7229d53…f6b3be80
w115 · 3680–3711	$L_{5}$ point	$L = ⟨ a_{1}, G_{2} ⟩ + ⟨ b_{2}, H_{1}^{'} ⟩ + ⟨ a_{1}, b_{2} ⟩ U$	51e6f7e6…72831bdf
w116 · 3712–3743	$R_{5}$ point	$R = ⟨ a_{2}, G_{1} ⟩ + ⟨ b_{1}, H_{2}^{'} ⟩ + ⟨ a_{2}, b_{1} ⟩ U$	68573891…623d7ed0
w117 · 3744–3775	$L_{6}$ point	$L = ⟨ a_{1}, G_{2} ⟩ + ⟨ b_{2}, H_{1}^{'} ⟩ + ⟨ a_{1}, b_{2} ⟩ U$	61847c76…078b0686
w118 · 3776–3807	$R_{6}$ point	$R = ⟨ a_{2}, G_{1} ⟩ + ⟨ b_{1}, H_{2}^{'} ⟩ + ⟨ a_{2}, b_{1} ⟩ U$	efa01185…dbae4b98
w119 · 3808–3839	$L_{7}$ point	$L = ⟨ a_{1}, G_{2} ⟩ + ⟨ b_{2}, H_{1}^{'} ⟩ + ⟨ a_{1}, b_{2} ⟩ U$	9fd3d25d…dbe7f132
w120 · 3840–3871	$R_{7}$ point	$R = ⟨ a_{2}, G_{1} ⟩ + ⟨ b_{1}, H_{2}^{'} ⟩ + ⟨ a_{2}, b_{1} ⟩ U$	dac3ff86…2d77ec80
w121 · 3872–3903	$a$ scalar	$a = l$	66aaf0ef…35741e3d
w122 · 3904–3935	$b$ scalar	$b = r$	1845c797…7fc45402
Helios (C2) Bulletproof
w123 · 3936–3967	$A_{I}$ point	$A_{I} = ⟨ a_{L}, G ⟩ + ⟨ a_{R}, H ⟩ + α h$	76447122…3d498a0b
w124 · 3968–3999	$A_{O}$ point	$A_{O} = ⟨ a_{O}, G ⟩ + β h$	4fc25b25…ab9ab3b8
w125 · 4000–4031	$S$ point	$S = ⟨ s_{L}, G ⟩ + ⟨ s_{R}, H ⟩ + ρ h$	04d0ef2a…ec5eb187
w126 · 4032–4063	$T_{0}$ point	$T_{j} = t_{j} g + τ_{j} h$	e52f758b…1fc72f9e
w127 · 4064–4095	$T_{1}$ point	$T_{j} = t_{j} g + τ_{j} h$	b0218f00…f9e4698f
w128 · 4096–4127	$T_{2}$ point	$T_{j} = t_{j} g + τ_{j} h$	1e311a4a…e2f8a4f1
w129 · 4128–4159	$T_{3}$ point	$T_{j} = t_{j} g + τ_{j} h$	41e8e1ed…e769eda1
w130 · 4160–4191	$T_{4}$ point	$T_{j} = t_{j} g + τ_{j} h$	cdca3f71…543752b0
w131 · 4192–4223	$T_{5}$ point	$T_{j} = t_{j} g + τ_{j} h$	3cd80389…7cf0361d
w132 · 4224–4255	$T_{6}$ point	$T_{j} = t_{j} g + τ_{j} h$	5c814e08…c6c59170
w133 · 4256–4287	$T_{7}$ point	$T_{j} = t_{j} g + τ_{j} h$	3ab9cb06…fb45b99b
w134 · 4288–4319	$T_{8}$ point	$T_{j} = t_{j} g + τ_{j} h$	55d503a7…2f3b043e
w135 · 4320–4351	$T_{9}$ point	$T_{j} = t_{j} g + τ_{j} h$	dcbf6aab…640a33f2
w136 · 4352–4383	$T_{10}$ point	$T_{j} = t_{j} g + τ_{j} h$	cb74ebde…b88bcfd1
w137 · 4384–4415	$T_{11}$ point	$T_{j} = t_{j} g + τ_{j} h$	d812ca67…4b51c6df
w138 · 4416–4447	$T_{12}$ point	$T_{j} = t_{j} g + τ_{j} h$	df990711…928d8261
w139 · 4448–4479	$T_{13}$ point	$T_{j} = t_{j} g + τ_{j} h$	5cae22ff…c4bf1032
w140 · 4480–4511	$T_{14}$ point	$T_{j} = t_{j} g + τ_{j} h$	48085f24…47b44ee2
w141 · 4512–4543	$T_{15}$ point	$T_{j} = t_{j} g + τ_{j} h$	2ab40a56…a346bc9a
w142 · 4544–4575	$T_{16}$ point	$T_{j} = t_{j} g + τ_{j} h$	39b327b5…e3f265cb
w143 · 4576–4607	$T_{17}$ point	$T_{j} = t_{j} g + τ_{j} h$	62e1fccb…f02a0cbe
w144 · 4608–4639	$T_{18}$ point	$T_{j} = t_{j} g + τ_{j} h$	11095751…67d6ca65
w145 · 4640–4671	$T_{19}$ point	$T_{j} = t_{j} g + τ_{j} h$	58634080…07bb0af3
w146 · 4672–4703	$T_{20}$ point	$T_{j} = t_{j} g + τ_{j} h$	7595882a…cffb45eb
w147 · 4704–4735	$T_{21}$ point	$T_{j} = t_{j} g + τ_{j} h$	d8d277b6…cf91e24d
w148 · 4736–4767	$T_{22}$ point	$T_{j} = t_{j} g + τ_{j} h$	edfe64cb…a9084224
w149 · 4768–4799	$T_{23}$ point	$T_{j} = t_{j} g + τ_{j} h$	498606a5…02e6bdf4
w150 · 4800–4831	$T_{24}$ point	$T_{j} = t_{j} g + τ_{j} h$	b7d115ed…297f9c03
w151 · 4832–4863	$T_{25}$ point	$T_{j} = t_{j} g + τ_{j} h$	e0380e3c…e17c0f43
w152 · 4864–4895	$T_{26}$ point	$T_{j} = t_{j} g + τ_{j} h$	e03fe1ad…dcd07be0
w153 · 4896–4927	$T_{27}$ point	$T_{j} = t_{j} g + τ_{j} h$	d435a2b5…231ce9fb
w154 · 4928–4959	$T_{28}$ point	$T_{j} = t_{j} g + τ_{j} h$	c1f528cd…2e6901b6
w155 · 4960–4991	$T_{29}$ point	$T_{j} = t_{j} g + τ_{j} h$	6b036ad9…82f301b6
w156 · 4992–5023	$T_{30}$ point	$T_{j} = t_{j} g + τ_{j} h$	c364c08e…4324880f
w157 · 5024–5055	$T_{31}$ point	$T_{j} = t_{j} g + τ_{j} h$	bddff62d…5613e980
w158 · 5056–5087	$T_{32}$ point	$T_{j} = t_{j} g + τ_{j} h$	6bea324e…937b9ca8
w159 · 5088–5119	$T_{33}$ point	$T_{j} = t_{j} g + τ_{j} h$	8780fbe4…922423c9
w160 · 5120–5151	$T_{34}$ point	$T_{j} = t_{j} g + τ_{j} h$	4086e4b3…a0e2d0ed
w161 · 5152–5183	$T_{35}$ point	$T_{j} = t_{j} g + τ_{j} h$	da6a66f6…7233fe09
w162 · 5184–5215	$T_{36}$ point	$T_{j} = t_{j} g + τ_{j} h$	80e68547…98c23040
w163 · 5216–5247	$T_{37}$ point	$T_{j} = t_{j} g + τ_{j} h$	6ed8527d…263ebd6f
w164 · 5248–5279	$T_{38}$ point	$T_{j} = t_{j} g + τ_{j} h$	d3d88507…fe1523d9
w165 · 5280–5311	$T_{39}$ point	$T_{j} = t_{j} g + τ_{j} h$	533ca28d…d5c7687d
w166 · 5312–5343	$T_{40}$ point	$T_{j} = t_{j} g + τ_{j} h$	1a0a2e47…9e327926
w167 · 5344–5375	$T_{41}$ point	$T_{j} = t_{j} g + τ_{j} h$	383c3605…e83d5746
w168 · 5376–5407	$T_{42}$ point	$T_{j} = t_{j} g + τ_{j} h$	aed5658c…5e8da057
w169 · 5408–5439	$T_{43}$ point	$T_{j} = t_{j} g + τ_{j} h$	bc167b70…04a6f069
w170 · 5440–5471	$T_{44}$ point	$T_{j} = t_{j} g + τ_{j} h$	e19a9958…59de76bf
w171 · 5472–5503	$T_{45}$ point	$T_{j} = t_{j} g + τ_{j} h$	0fcead3e…bda24d5d
w172 · 5504–5535	$T_{46}$ point	$T_{j} = t_{j} g + τ_{j} h$	da3b9aff…8823796a
w173 · 5536–5567	$T_{47}$ point	$T_{j} = t_{j} g + τ_{j} h$	0b5db31c…595ce4c9
w174 · 5568–5599	$T_{48}$ point	$T_{j} = t_{j} g + τ_{j} h$	00cd032c…73e701ba
w175 · 5600–5631	$T_{49}$ point	$T_{j} = t_{j} g + τ_{j} h$	667b316c…2d479924
w176 · 5632–5663	$τ_{x}$ scalar	$τ_{x} = \sum_{j} τ_{j} x^{j}$	85f3477f…1b345932
w177 · 5664–5695	$u$ scalar	$u = α x^{c + 1} + β x^{n^{'}} + ρ x^{n^{'} + 1} + \sum_{i} μ_{i} x^{n^{'} - 1 - i}$	62a0b528…89005f3a
w178 · 5696–5727	$\hat{t}$ scalar	$\hat{t} = ⟨ l, r ⟩$	7a0511c9…1dd51b05
w179 · 5728–5759	$L_{0}$ point	$L = ⟨ a_{1}, G_{2} ⟩ + ⟨ b_{2}, H_{1}^{'} ⟩ + ⟨ a_{1}, b_{2} ⟩ U$	01c75675…c1324dbd
w180 · 5760–5791	$R_{0}$ point	$R = ⟨ a_{2}, G_{1} ⟩ + ⟨ b_{1}, H_{2}^{'} ⟩ + ⟨ a_{2}, b_{1} ⟩ U$	57df3ce8…12319064
w181 · 5792–5823	$L_{1}$ point	$L = ⟨ a_{1}, G_{2} ⟩ + ⟨ b_{2}, H_{1}^{'} ⟩ + ⟨ a_{1}, b_{2} ⟩ U$	6211ddfd…f6dd61bf
w182 · 5824–5855	$R_{1}$ point	$R = ⟨ a_{2}, G_{1} ⟩ + ⟨ b_{1}, H_{2}^{'} ⟩ + ⟨ a_{2}, b_{1} ⟩ U$	4c68e191…1e47ec15
w183 · 5856–5887	$L_{2}$ point	$L = ⟨ a_{1}, G_{2} ⟩ + ⟨ b_{2}, H_{1}^{'} ⟩ + ⟨ a_{1}, b_{2} ⟩ U$	11988062…27a8b6e2
w184 · 5888–5919	$R_{2}$ point	$R = ⟨ a_{2}, G_{1} ⟩ + ⟨ b_{1}, H_{2}^{'} ⟩ + ⟨ a_{2}, b_{1} ⟩ U$	20a0574a…f3e430fc
w185 · 5920–5951	$L_{3}$ point	$L = ⟨ a_{1}, G_{2} ⟩ + ⟨ b_{2}, H_{1}^{'} ⟩ + ⟨ a_{1}, b_{2} ⟩ U$	04e16b30…c978179f
w186 · 5952–5983	$R_{3}$ point	$R = ⟨ a_{2}, G_{1} ⟩ + ⟨ b_{1}, H_{2}^{'} ⟩ + ⟨ a_{2}, b_{1} ⟩ U$	e96cf679…ff0e6c9f
w187 · 5984–6015	$L_{4}$ point	$L = ⟨ a_{1}, G_{2} ⟩ + ⟨ b_{2}, H_{1}^{'} ⟩ + ⟨ a_{1}, b_{2} ⟩ U$	65df6a6c…b995ffc2
w188 · 6016–6047	$R_{4}$ point	$R = ⟨ a_{2}, G_{1} ⟩ + ⟨ b_{1}, H_{2}^{'} ⟩ + ⟨ a_{2}, b_{1} ⟩ U$	1b5b4a3d…12a8d450
w189 · 6048–6079	$L_{5}$ point	$L = ⟨ a_{1}, G_{2} ⟩ + ⟨ b_{2}, H_{1}^{'} ⟩ + ⟨ a_{1}, b_{2} ⟩ U$	993e33dd…c5bc6918
w190 · 6080–6111	$R_{5}$ point	$R = ⟨ a_{2}, G_{1} ⟩ + ⟨ b_{1}, H_{2}^{'} ⟩ + ⟨ a_{2}, b_{1} ⟩ U$	13e49fe4…91ced441
w191 · 6112–6143	$L_{6}$ point	$L = ⟨ a_{1}, G_{2} ⟩ + ⟨ b_{2}, H_{1}^{'} ⟩ + ⟨ a_{1}, b_{2} ⟩ U$	f3ab91a4…fa194cf8
w192 · 6144–6175	$R_{6}$ point	$R = ⟨ a_{2}, G_{1} ⟩ + ⟨ b_{1}, H_{2}^{'} ⟩ + ⟨ a_{2}, b_{1} ⟩ U$	92091733…754e4de3
w193 · 6176–6207	$a$ scalar	$a = l$	ff88c4b7…bd00e111
w194 · 6208–6239	$b$ scalar	$b = r$	c73b3146…6e307c54
root anchor
w195 · 6240–6271	$R_{pok}$ point	$R_{pok} = r H$	0ddbf0f7…99fce08e
w196 · 6272–6303	$s$ scalar	$s = r + c b_{root}$	da23ae70…07c69b0d

← GBP

Anatomy of a real FCMP++ proof

The fcmp_pp field

What is not in fcmp_pp and where it lives

How the transaction scales

Every 32-byte word

The `fcmp_pp` field

What is not in `fcmp_pp` and where it lives