OP_CAT Makes Bitcoin Quantum Secure [was CheckSigFromStack for Arithmetic Values]
6 messages, 7 to 9 Jul 2021. Started by Jeremy Rubin.
Jeremy Rubin (2), ZmnSCPxj (2), Ethan Heilman (2)
Dear Bitcoin Devs,
As mentioned previously, OP_CAT (or similar operation) can be used to make Bitcoin "quantum safe" by signing an EC signature. This should work in both Segwit V0 and Tapscript, although you have to use HASH160 for it to fit in Segwit V0.
See [my blog](https://rubin.io/blog/2021/07/06/quantum-bitcoin/) for the specific construction, reproduced below.
Yet another entry to the "OP_CAT can do that too" list.
Best,
Jeremy -----
I recently published [a blog post](https://rubin.io/blog/2021/07/02/signing-5-bytes/) about signing up to a 5 byte value using Bitcoin script arithmetic and Lamport signatures.
By itself, this is neat, but a little limited. What if we could sign longer messages? If we can sign up to 20 bytes, we could sign a HASH160 digest which is most likely quantum safe...
What would it mean if we signed the HASH160 digest of a signature? What the what? Why would we do that?
Well, as it turns out, even if a quantum computer were able to crack ECDSA, it would yield revealing the private key but not the ability to malleate the content of what was actually signed. I asked my good friend and cryptographer [Madars Virza](https://madars.org/) if my intuition was correct, and he confirmed that it should be sufficient, but it's definitely worth closer analysis before relying on this. While the ECDSA signature can be malleated to a different, negative form, if the signature is otherwise made immalleable there should only be one value the commitment can be opened to.
If we required the ECDSA signature be signed with a quantum proof signature algorithm, then we'd have a quantum proof Bitcoin! And the 5 byte signing scheme we discussed previously is a Lamport signature, which is quantum secure. Unfortunately, we need at least 20 contiguous bytes... so we need some sort of OP\_CAT like operation.
OP\_CAT can't be directly soft forked to Segwit v0 because it modifies the stack, so instead we'll (for simplicity) also show how to use a new opcode that uses verify semantics, OP\_SUBSTRINGEQUALVERIFY that checks a splice of a string for equality.
```
... FOR j in 0..=5
<0>
... FOR i in 0..=31
SWAP hash160 DUP <H(K_j_i_1)> EQUAL IF DROP <2**i> ADD ELSE
<H(K_j_i_0)> EQUALVERIFY ENDIF
... END FOR
TOALTSTACK
... END FOR
DUP HASH160
... IF CAT AVAILABLE
FROMALTSTACK
... FOR j in 0..=5
FROMALTSTACK
CAT
... END FOR
EQUALVERIFY
... ELSE SUBSTRINGEQUALVERIFY AVAILABLE
... FOR j in 0..=5
FROMALTSTACK <0+j*4> <4+j*4> SUBSTRINGEQUALVERIFY DROP DROP DROP
... END FOR
DROP
... END IF
<pk> CHECKSIG ```
That's a long script... but will it fit? We need to verify 20 bytes of message each bit takes around 10 bytes script, an average of 3.375 bytes per number (counting pushes), and two 21 bytes keys = 55.375 bytes of program space and 21 bytes of witness element per bit.
It fits! `20*8*55.375 = 8860`, which leaves 1140 bytes less than the limit for the rest of the logic, which is plenty (around 15-40 bytes required for the rest of the logic, leaving 1100 free for custom signature checking). The stack size is 160 elements for the hash gadget, 3360 bytes.
This can probably be made a bit more efficient by expanding to a ternary representation.
```
SWAP hash160 DUP <H(K_j_i_0)> EQUAL IF DROP ELSE <3**i> SWAP DUP
<H(K_j_i_T)> EQUAL IF DROP SUB ELSE <H(K_j_i_1)> EQUALVERIFY ADD ENDIF
ENDIF
```
This should bring it up to roughly 85 bytes per trit, and there should be 101 trits (`log(2**160)/log(3) == 100.94`), so about 8560 bytes... a bit cheaper! But the witness stack is "only" `2121` bytes...
As a homework exercise, maybe someone can prove the optimal choice of radix for this protocol... My guess is that base 4 is optimal!
## Taproot?
What about Taproot? As far as I'm aware the commitment scheme (`Q = pG + hash(pG || m)G`) can be securely opened to m even with a quantum computer (finding `q` such that `qG = Q` might be trivial, but suppose key path was disabled, then finding m and p such that the taproot equation holds should be difficult because of the hash, but I'd need to certify that claim better). Therefore this script can nest inside of a Tapscript path -- Tapscript also does not impose a length limit, 32 byte hashes could be used as well.
Further, to make keys reusable, there could be many Lamport keys comitted inside a taproot tree so that an address could be used for thousands of times before expiring. This could be used as a measure to protect accidental use rather than to support it.
Lastly, Schnorr actually has a stronger non-malleability property than ECDSA, the signatures will be binding to the approved transaction and once Lamport signed, even a quantum computer could not steal the funds.
Good morning Jeremy,
Yes, quite neat indeed, too bad Lamport signatures are so huge (a couple kilobytes)... blocksize increase *cough*
Since a quantum computer can derive the EC privkey from the EC pubkey and this scheme is resistant to that, I think you can use a single well-known EC privkey, you just need a unique Lamport keypair for each UTXO (uniqueness being mandatory due to Lamport requiring preimage revelation).
Regards, ZmnSCPxj
Quoted text (117 lines)
> Dear Bitcoin Devs, > > As mentioned previously, OP_CAT (or similar operation) can be used to make Bitcoin "quantum safe" by signing an EC signature. This should work in both Segwit V0 and Tapscript, although you have to use HASH160 for it to fit in Segwit V0. > > See [my blog](https://rubin.io/blog/2021/07/06/quantum-bitcoin/) for the specific construction, reproduced below. > > Yet another entry to the "OP_CAT can do that too" list. > > Best, > > Jeremy > ----- > > I recently published [a blog > post](https://rubin.io/blog/2021/07/02/signing-5-bytes/) about signing up to a > 5 byte value using Bitcoin script arithmetic and Lamport signatures. > > By itself, this is neat, but a little limited. What if we could sign longer > messages? If we can sign up to 20 bytes, we could sign a HASH160 digest which > is most likely quantum safe... > > What would it mean if we signed the HASH160 digest of a signature? What the > what? Why would we do that? > > Well, as it turns out, even if a quantum computer were able to crack ECDSA, it > would yield revealing the private key but not the ability to malleate the > content of what was actually signed. I asked my good friend and cryptographer > [Madars Virza](https://madars.org/) if my intuition was correct, and he > confirmed that it should be sufficient, but it's definitely worth closer > analysis before relying on this. While the ECDSA signature can be malleated to a > different, negative form, if the signature is otherwise made immalleable there > should only be one value the commitment can be opened to. > > If we required the ECDSA signature be signed with a quantum proof signature > algorithm, then we'd have a quantum proof Bitcoin! And the 5 byte signing scheme > we discussed previously is a Lamport signature, which is quantum secure. > Unfortunately, we need at least 20 contiguous bytes... so we need some sort of > OP\_CAT like operation. > > OP\_CAT can't be directly soft forked to Segwit v0 because it modifies the > stack, so instead we'll (for simplicity) also show how to use a new opcode that > uses verify semantics, OP\_SUBSTRINGEQUALVERIFY that checks a splice of a string > for equality. > > ``` > ... FOR j in 0..=5 > <0> > ... FOR i in 0..=31 > SWAP hash160 DUP <H(K_j_i_1)> EQUAL IF DROP <2**i> ADD ELSE <H(K_j_i_0)> EQUALVERIFY ENDIF > ... END FOR > TOALTSTACK > ... END FOR > > DUP HASH160 > > ... IF CAT AVAILABLE > FROMALTSTACK > ... FOR j in 0..=5 > FROMALTSTACK > CAT > ... END FOR > EQUALVERIFY > ... ELSE SUBSTRINGEQUALVERIFY AVAILABLE > ... FOR j in 0..=5 > FROMALTSTACK <0+j*4> <4+j*4> SUBSTRINGEQUALVERIFY DROP DROP DROP > ... END FOR > DROP > ... END IF > > <pk> CHECKSIG > ``` > > That's a long script... but will it fit? We need to verify 20 bytes of message > each bit takes around 10 bytes script, an average of 3.375 bytes per number > (counting pushes), and two 21 bytes keys = 55.375 bytes of program space and 21 > bytes of witness element per bit. > > It fits! `20*8*55.375 = 8860`, which leaves 1140 bytes less than the limit for > the rest of the logic, which is plenty (around 15-40 bytes required for the rest > of the logic, leaving 1100 free for custom signature checking). The stack size > is 160 elements for the hash gadget, 3360 bytes. > > This can probably be made a bit more efficient by expanding to a ternary > representation. > > ``` > SWAP hash160 DUP <H(K_j_i_0)> EQUAL IF DROP ELSE <3**i> SWAP DUP <H(K_j_i_T)> EQUAL IF DROP SUB ELSE <H(K_j_i_1)> EQUALVERIFY ADD ENDIF ENDIF > ``` > > This should bring it up to roughly 85 bytes per trit, and there should be 101 > trits (`log(2**160)/log(3) == 100.94`), so about 8560 bytes... a bit cheaper! > But the witness stack is "only" `2121` bytes... > > As a homework exercise, maybe someone can prove the optimal choice of radix for > this protocol... My guess is that base 4 is optimal! > > ## Taproot? > > What about Taproot? As far as I'm aware the commitment scheme (`Q = pG + hash(pG > || m)G`) can be securely opened to m even with a quantum computer (finding `q` > such that `qG = Q` might be trivial, but suppose key path was disabled, then > finding m and p such that the taproot equation holds should be difficult because > of the hash, but I'd need to certify that claim better). Therefore this > script can nest inside of a Tapscript path -- Tapscript also does not impose a > length limit, 32 byte hashes could be used as well. > > Further, to make keys reusable, there could be many Lamport keys comitted inside > a taproot tree so that an address could be used for thousands of times before > expiring. This could be used as a measure to protect accidental use rather than > to support it. > > Lastly, Schnorr actually has a stronger non-malleability property than ECDSA, > the signatures will be binding to the approved transaction and once Lamport > signed, even a quantum computer could not steal the funds. > > -- > @JeremyRubin
Quoted text (1 line)
>Yes, quite neat indeed, too bad Lamport signatures are so huge (a couple kilobytes)... blocksize increase *cough*
Couldn't you significantly compress the signatures by using either Winternitz OTS or by using OP_CAT to build a merkle tree so that the full signature can be derived during script execution from a much shorter set of seed values?
On Thu, Jul 8, 2021 at 4:12 AM ZmnSCPxj via bitcoin-dev <bitcoin-dev@lists•linuxfoundation•org> wrote:
Quoted text (135 lines)
> > > Good morning Jeremy, > > Yes, quite neat indeed, too bad Lamport signatures are so huge (a couple kilobytes)... blocksize increase *cough* > > Since a quantum computer can derive the EC privkey from the EC pubkey and this scheme is resistant to that, I think you can use a single well-known EC privkey, you just need a unique Lamport keypair for each UTXO (uniqueness being mandatory due to Lamport requiring preimage revelation). > > Regards, > ZmnSCPxj > > > > Dear Bitcoin Devs, > > > > As mentioned previously, OP_CAT (or similar operation) can be used to make Bitcoin "quantum safe" by signing an EC signature. This should work in both Segwit V0 and Tapscript, although you have to use HASH160 for it to fit in Segwit V0. > > > > See [my blog](https://rubin.io/blog/2021/07/06/quantum-bitcoin/) for the specific construction, reproduced below. > > > > Yet another entry to the "OP_CAT can do that too" list. > > > > Best, > > > > Jeremy > > ----- > > > > I recently published [a blog > > post](https://rubin.io/blog/2021/07/02/signing-5-bytes/) about signing up to a > > 5 byte value using Bitcoin script arithmetic and Lamport signatures. > > > > By itself, this is neat, but a little limited. What if we could sign longer > > messages? If we can sign up to 20 bytes, we could sign a HASH160 digest which > > is most likely quantum safe... > > > > What would it mean if we signed the HASH160 digest of a signature? What the > > what? Why would we do that? > > > > Well, as it turns out, even if a quantum computer were able to crack ECDSA, it > > would yield revealing the private key but not the ability to malleate the > > content of what was actually signed. I asked my good friend and cryptographer > > [Madars Virza](https://madars.org/) if my intuition was correct, and he > > confirmed that it should be sufficient, but it's definitely worth closer > > analysis before relying on this. While the ECDSA signature can be malleated to a > > different, negative form, if the signature is otherwise made immalleable there > > should only be one value the commitment can be opened to. > > > > If we required the ECDSA signature be signed with a quantum proof signature > > algorithm, then we'd have a quantum proof Bitcoin! And the 5 byte signing scheme > > we discussed previously is a Lamport signature, which is quantum secure. > > Unfortunately, we need at least 20 contiguous bytes... so we need some sort of > > OP\_CAT like operation. > > > > OP\_CAT can't be directly soft forked to Segwit v0 because it modifies the > > stack, so instead we'll (for simplicity) also show how to use a new opcode that > > uses verify semantics, OP\_SUBSTRINGEQUALVERIFY that checks a splice of a string > > for equality. > > > > ``` > > ... FOR j in 0..=5 > > <0> > > ... FOR i in 0..=31 > > SWAP hash160 DUP <H(K_j_i_1)> EQUAL IF DROP <2**i> ADD ELSE <H(K_j_i_0)> EQUALVERIFY ENDIF > > ... END FOR > > TOALTSTACK > > ... END FOR > > > > DUP HASH160 > > > > ... IF CAT AVAILABLE > > FROMALTSTACK > > ... FOR j in 0..=5 > > FROMALTSTACK > > CAT > > ... END FOR > > EQUALVERIFY > > ... ELSE SUBSTRINGEQUALVERIFY AVAILABLE > > ... FOR j in 0..=5 > > FROMALTSTACK <0+j*4> <4+j*4> SUBSTRINGEQUALVERIFY DROP DROP DROP > > ... END FOR > > DROP > > ... END IF > > > > <pk> CHECKSIG > > ``` > > > > That's a long script... but will it fit? We need to verify 20 bytes of message > > each bit takes around 10 bytes script, an average of 3.375 bytes per number > > (counting pushes), and two 21 bytes keys = 55.375 bytes of program space and 21 > > bytes of witness element per bit. > > > > It fits! `20*8*55.375 = 8860`, which leaves 1140 bytes less than the limit for > > the rest of the logic, which is plenty (around 15-40 bytes required for the rest > > of the logic, leaving 1100 free for custom signature checking). The stack size > > is 160 elements for the hash gadget, 3360 bytes. > > > > This can probably be made a bit more efficient by expanding to a ternary > > representation. > > > > ``` > > SWAP hash160 DUP <H(K_j_i_0)> EQUAL IF DROP ELSE <3**i> SWAP DUP <H(K_j_i_T)> EQUAL IF DROP SUB ELSE <H(K_j_i_1)> EQUALVERIFY ADD ENDIF ENDIF > > ``` > > > > This should bring it up to roughly 85 bytes per trit, and there should be 101 > > trits (`log(2**160)/log(3) == 100.94`), so about 8560 bytes... a bit cheaper! > > But the witness stack is "only" `2121` bytes... > > > > As a homework exercise, maybe someone can prove the optimal choice of radix for > > this protocol... My guess is that base 4 is optimal! > > > > ## Taproot? > > > > What about Taproot? As far as I'm aware the commitment scheme (`Q = pG + hash(pG > > || m)G`) can be securely opened to m even with a quantum computer (finding `q` > > such that `qG = Q` might be trivial, but suppose key path was disabled, then > > finding m and p such that the taproot equation holds should be difficult because > > of the hash, but I'd need to certify that claim better). Therefore this > > script can nest inside of a Tapscript path -- Tapscript also does not impose a > > length limit, 32 byte hashes could be used as well. > > > > Further, to make keys reusable, there could be many Lamport keys comitted inside > > a taproot tree so that an address could be used for thousands of times before > > expiring. This could be used as a measure to protect accidental use rather than > > to support it. > > > > Lastly, Schnorr actually has a stronger non-malleability property than ECDSA, > > the signatures will be binding to the approved transaction and once Lamport > > signed, even a quantum computer could not steal the funds. > > > > -- > > @JeremyRubin > > > _______________________________________________ > bitcoin-dev mailing list > bitcoin-dev@lists•linuxfoundation•org > https://lists.linuxfoundation.org/mailman/listinfo/bitcoin-dev
Good morning Ethan,
Quoted text (6 lines)
> > Yes, quite neat indeed, too bad Lamport signatures are so huge (a couple kilobytes)... blocksize increase cough > > Couldn't you significantly compress the signatures by using either > Winternitz OTS or by using OP_CAT to build a merkle tree so that the > full signature can be derived during script execution from a much > shorter set of seed values?
To implement Winternitz we need some kind of limited-repeat construct, which is not available in SCRIPT, but may be emulatable with enough `OP_IF` and sheer brute force. But what you gain in smaller signatures, you lose in a more complex and longer SCRIPT, and there are limits to SCRIPT size (in order to limit the processing done in each node).
Merkle signatures trade off shorter pubkeys for longer signatures (signatures need to provide the hash of the *other* preimage you are not revealing), but in the modern post-SegWit Bitcoin context both pubkeys and signatures are stored in the witness area, which have the same weight, thus it is actually a loss compared to Lamport.
So yes, maybe Winternitz (could be a replacement for the "trinary" Jeremy refers to), Merkle not so much.
Regards, ZmnSCPxj
Quoted text (113 lines)
> On Thu, Jul 8, 2021 at 4:12 AM ZmnSCPxj via bitcoin-dev > bitcoin-dev@lists•linuxfoundation•org wrote: > > > Good morning Jeremy, > > Yes, quite neat indeed, too bad Lamport signatures are so huge (a couple kilobytes)... blocksize increase cough > > Since a quantum computer can derive the EC privkey from the EC pubkey and this scheme is resistant to that, I think you can use a single well-known EC privkey, you just need a unique Lamport keypair for each UTXO (uniqueness being mandatory due to Lamport requiring preimage revelation). > > Regards, > > ZmnSCPxj > > > > > Dear Bitcoin Devs, > > > As mentioned previously, OP_CAT (or similar operation) can be used to make Bitcoin "quantum safe" by signing an EC signature. This should work in both Segwit V0 and Tapscript, although you have to use HASH160 for it to fit in Segwit V0. > > > See my blog for the specific construction, reproduced below. > > > Yet another entry to the "OP_CAT can do that too" list. > > > Best, > > > > > > Jeremy > > > > > > ------- > > > > > > I recently published a blogpost about signing up to a5 byte value using Bitcoin script arithmetic and Lamport signatures. > > > By itself, this is neat, but a little limited. What if we could sign longer > > > messages? If we can sign up to 20 bytes, we could sign a HASH160 digest which > > > is most likely quantum safe... > > > What would it mean if we signed the HASH160 digest of a signature? What the > > > what? Why would we do that? > > > Well, as it turns out, even if a quantum computer were able to crack ECDSA, it > > > would yield revealing the private key but not the ability to malleate the > > > content of what was actually signed. I asked my good friend and cryptographer > > > Madars Virza if my intuition was correct, and he > > > confirmed that it should be sufficient, but it's definitely worth closer > > > analysis before relying on this. While the ECDSA signature can be malleated to a > > > different, negative form, if the signature is otherwise made immalleable there > > > should only be one value the commitment can be opened to. > > > If we required the ECDSA signature be signed with a quantum proof signature > > > algorithm, then we'd have a quantum proof Bitcoin! And the 5 byte signing scheme > > > we discussed previously is a Lamport signature, which is quantum secure. > > > Unfortunately, we need at least 20 contiguous bytes... so we need some sort of > > > OP\_CAT like operation. > > > OP\_CAT can't be directly soft forked to Segwit v0 because it modifies the > > > stack, so instead we'll (for simplicity) also show how to use a new opcode that > > > uses verify semantics, OP\_SUBSTRINGEQUALVERIFY that checks a splice of a string > > > for equality. > > > > > > ... FOR j in 0..=5 > > > <0> > > > ... FOR i in 0..=31 > > > SWAP hash160 DUP <H(K_j_i_1)> EQUAL IF DROP <2**i> ADD ELSE <H(K_j_i_0)> EQUALVERIFY ENDIF > > > ... END FOR > > > TOALTSTACK > > > ... END FOR > > > > > > DUP HASH160 > > > > > > ... IF CAT AVAILABLE > > > FROMALTSTACK > > > ... FOR j in 0..=5 > > > FROMALTSTACK > > > CAT > > > ... END FOR > > > EQUALVERIFY > > > ... ELSE SUBSTRINGEQUALVERIFY AVAILABLE > > > ... FOR j in 0..=5 > > > FROMALTSTACK <0+j*4> <4+j*4> SUBSTRINGEQUALVERIFY DROP DROP DROP > > > ... END FOR > > > DROP > > > ... END IF > > > > > > <pk> CHECKSIG > > > > > > > > > That's a long script... but will it fit? We need to verify 20 bytes of message > > > each bit takes around 10 bytes script, an average of 3.375 bytes per number > > > (counting pushes), and two 21 bytes keys = 55.375 bytes of program space and 21 > > > bytes of witness element per bit. > > > It fits! `20*8*55.375 = 8860`, which leaves 1140 bytes less than the limit for > > > the rest of the logic, which is plenty (around 15-40 bytes required for the rest > > > of the logic, leaving 1100 free for custom signature checking). The stack size > > > is 160 elements for the hash gadget, 3360 bytes. > > > This can probably be made a bit more efficient by expanding to a ternary > > > representation. > > > > > > SWAP hash160 DUP <H(K_j_i_0)> EQUAL IF DROP ELSE <3**i> SWAP DUP <H(K_j_i_T)> EQUAL IF DROP SUB ELSE <H(K_j_i_1)> EQUALVERIFY ADD ENDIF ENDIF > > > > > > > > > This should bring it up to roughly 85 bytes per trit, and there should be 101 > > > trits (`log(2**160)/log(3) == 100.94`), so about 8560 bytes... a bit cheaper! > > > But the witness stack is "only" `2121` bytes... > > > As a homework exercise, maybe someone can prove the optimal choice of radix for > > > this protocol... My guess is that base 4 is optimal! > > > > > > Taproot? > > > > > > --------- > > > > > > What about Taproot? As far as I'm aware the commitment scheme (`Q = pG + hash(pG || m)G`) can be securely opened to m even with a quantum computer (finding `q` > > > such that `qG = Q` might be trivial, but suppose key path was disabled, then > > > finding m and p such that the taproot equation holds should be difficult because > > > of the hash, but I'd need to certify that claim better). Therefore this > > > script can nest inside of a Tapscript path -- Tapscript also does not impose a > > > length limit, 32 byte hashes could be used as well. > > > Further, to make keys reusable, there could be many Lamport keys comitted inside > > > a taproot tree so that an address could be used for thousands of times before > > > expiring. This could be used as a measure to protect accidental use rather than > > > to support it. > > > Lastly, Schnorr actually has a stronger non-malleability property than ECDSA, > > > the signatures will be binding to the approved transaction and once Lamport > > > signed, even a quantum computer could not steal the funds. > > > -- > > > @JeremyRubin > > > > bitcoin-dev mailing list > > bitcoin-dev@lists•linuxfoundation•org > > https://lists.linuxfoundation.org/mailman/listinfo/bitcoin-dev
I thought about this, but at the time of writing I couldn't come up with something I thought was substantially better. I spent a few more cycles thinking on it -- you can definitely do better. It's not clear how much better Winternitz might be, or if it would be secure in this context? Here's some exploration...
maybe you can do something like:
<x0> <H(x1)> <dir in {0,1}> || IF SWAP HASH SWAP ELSE HASH FROMALTSTACK
<2**n> TOALTSTACK ADD ENDIF CAT
you can process this (assume HASH160) into chunks of 26 bits, cat them all together, and then stash that hash. You would need 6 gadgets, and then 1 overflow + 4 bare hashes for the final key hash (e.g. your tree looks like) H(H(26x20) || H(26x20)...H(bit)|| H(bit) || H(bit) || H(bit)). It doesn't make sense to have a "nice" merkle tree, just fit in as much data as possible per call (520 bytes). If OP_SHASTREAM, this is even better since you can ignore structuring...
This would bring your cost down by about 20 bytes per bit, for 160 bits, so around a savings of 3200 bytes... not bad! 1/3 cheaper.
Script is about 15x160 = 2400 and change, witness is 43x160 = 6880
If you were to convert to 3-ary, you could cut this down to 101 gates with a script like:
witnesses: <H(xT)> <H(x1)> <0> <x0> <H(x0) || H(x1)> <1> <xT> <H(xT) || H(x0)> <2> <x1>
script:
HASH SWAP
IFDUP
NOTIF # 0 passed in (0)
SWAP CAT
ELSE
<3**n> TOALT
1SUB
IF # 2 passed in (+1)
FROMALT # do nothing
ELSE # 1 passed in (T)
SWAP # Swaps H(xT) to back
FROMALT NEGATE # negate
END
FROMALT ADD TOALT # add to accumulator
ENDIF
CAT
you would end up having to publish ~64x101 data in the witness, so only 6464 total (and about 24x101 = 2424 and change for the script)
Making the script smaller also means that choice of hash160/sha256 doesn't change script size much, just witness. And the witnesses are free to provide their own preimages, so it would be OK to use something > 20 bytes, < 32 for more variable security/length tradeoff.
At the cost of marginally bigger script (by about 6x101 bytes), you can save 20x101 off the witness stack by making each key H(H(xT) || H(x0)) || H(x1). 43x101 + 30x101 = 7373 + change for the final grouping.
witnesses: <H(xT)> <H(x1)> <0> <x0> <H(x0)> <H(x1)> <1> <xT> <H(H(xT) || H(x0))> <2> <x1>
script:
HASH SWAP
IFDUP
NOTIF # 0 passed in (0)
ROT SWAP CAT HASH
ELSE
<3**n> TOALT
1SUB
IF # 2 passed in (+1)
FROMALT # do nothing
ELSE # 1 passed in (T)
TOALTSTACK CAT HASH FROMALTSTACK SWAP # Swaps H(xT) to back
FROMALT NEGATE # negate
END
FROMALT ADD TOALT # add to accumulator
ENDIF
CATQuoted text (1 line)
>To implement Winternitz we need some kind of limited-repeat construct, which is not available in SCRIPT, but may be emulatable with enough `OP_IF` and sheer brute force.
But what you gain in smaller signatures, you lose in a more complex and longer SCRIPT, and there are limits to SCRIPT size (in order to limit the processing done in each node).
Using depth 4 Winternitz would increase the number of instructions in SCRIPT by 4*(signature bits/2) instructions, but decrease the signature size by (signature bits/2) hash preimages. Given that instructions are significantly smaller in size than the hash preimages used, it seems like this would significantly reduce total size.
Quoted text (1 line)
>Merkle signatures trade off shorter pubkeys for longer signatures (signatures need to provide the hash of the *other* preimage you are not revealing), but in the modern post-SegWit Bitcoin context both pubkeys and signatures are stored in the witness area, which have the same weight, thus it is actually a loss compared to Lamport.
I wasn't proposing using plain merkle signatures, rather I was thinking about something where if particular chunks of the message fit a pattern you could release a seed higher in the commitment tree. For instance 1,1,1 could be signed as by releasing H(01||H(01||H(01||x))), H(11||H(11||H(11||x))), H(21||H(21||H(21||x))), or by releasing X. However, you would want to only release X in that one specific case (1,1,1) but no others. Again this would bloat the SCRIPT and decrease signature size but at a favorable ratio.
I am not convinced anyone should do these things, but they are fun to think about and I suspect with more thought such signature sizes and SCRIPT sizes could be even further reduced.
On Fri, Jul 9, 2021 at 6:38 PM ZmnSCPxj <ZmnSCPxj@protonmail•com> wrote:
Quoted text (136 lines)
> > Good morning Ethan, > > > > Yes, quite neat indeed, too bad Lamport signatures are so huge (a couple kilobytes)... blocksize increase cough > > > > Couldn't you significantly compress the signatures by using either > > Winternitz OTS or by using OP_CAT to build a merkle tree so that the > > full signature can be derived during script execution from a much > > shorter set of seed values? > > To implement Winternitz we need some kind of limited-repeat construct, which is not available in SCRIPT, but may be emulatable with enough `OP_IF` and sheer brute force. > But what you gain in smaller signatures, you lose in a more complex and longer SCRIPT, and there are limits to SCRIPT size (in order to limit the processing done in each node). > > Merkle signatures trade off shorter pubkeys for longer signatures (signatures need to provide the hash of the *other* preimage you are not revealing), but in the modern post-SegWit Bitcoin context both pubkeys and signatures are stored in the witness area, which have the same weight, thus it is actually a loss compared to Lamport. > > > So yes, maybe Winternitz (could be a replacement for the "trinary" Jeremy refers to), Merkle not so much. > > Regards, > ZmnSCPxj > > > On Thu, Jul 8, 2021 at 4:12 AM ZmnSCPxj via bitcoin-dev > > bitcoin-dev@lists•linuxfoundation•org wrote: > > > > > Good morning Jeremy, > > > Yes, quite neat indeed, too bad Lamport signatures are so huge (a couple kilobytes)... blocksize increase cough > > > Since a quantum computer can derive the EC privkey from the EC pubkey and this scheme is resistant to that, I think you can use a single well-known EC privkey, you just need a unique Lamport keypair for each UTXO (uniqueness being mandatory due to Lamport requiring preimage revelation). > > > Regards, > > > ZmnSCPxj > > > > > > > Dear Bitcoin Devs, > > > > As mentioned previously, OP_CAT (or similar operation) can be used to make Bitcoin "quantum safe" by signing an EC signature. This should work in both Segwit V0 and Tapscript, although you have to use HASH160 for it to fit in Segwit V0. > > > > See my blog for the specific construction, reproduced below. > > > > Yet another entry to the "OP_CAT can do that too" list. > > > > Best, > > > > > > > > Jeremy > > > > > > > > ------- > > > > > > > > I recently published a blogpost about signing up to a5 byte value using Bitcoin script arithmetic and Lamport signatures. > > > > By itself, this is neat, but a little limited. What if we could sign longer > > > > messages? If we can sign up to 20 bytes, we could sign a HASH160 digest which > > > > is most likely quantum safe... > > > > What would it mean if we signed the HASH160 digest of a signature? What the > > > > what? Why would we do that? > > > > Well, as it turns out, even if a quantum computer were able to crack ECDSA, it > > > > would yield revealing the private key but not the ability to malleate the > > > > content of what was actually signed. I asked my good friend and cryptographer > > > > Madars Virza if my intuition was correct, and he > > > > confirmed that it should be sufficient, but it's definitely worth closer > > > > analysis before relying on this. While the ECDSA signature can be malleated to a > > > > different, negative form, if the signature is otherwise made immalleable there > > > > should only be one value the commitment can be opened to. > > > > If we required the ECDSA signature be signed with a quantum proof signature > > > > algorithm, then we'd have a quantum proof Bitcoin! And the 5 byte signing scheme > > > > we discussed previously is a Lamport signature, which is quantum secure. > > > > Unfortunately, we need at least 20 contiguous bytes... so we need some sort of > > > > OP\_CAT like operation. > > > > OP\_CAT can't be directly soft forked to Segwit v0 because it modifies the > > > > stack, so instead we'll (for simplicity) also show how to use a new opcode that > > > > uses verify semantics, OP\_SUBSTRINGEQUALVERIFY that checks a splice of a string > > > > for equality. > > > > > > > > ... FOR j in 0..=5 > > > > <0> > > > > ... FOR i in 0..=31 > > > > SWAP hash160 DUP <H(K_j_i_1)> EQUAL IF DROP <2**i> ADD ELSE <H(K_j_i_0)> EQUALVERIFY ENDIF > > > > ... END FOR > > > > TOALTSTACK > > > > ... END FOR > > > > > > > > DUP HASH160 > > > > > > > > ... IF CAT AVAILABLE > > > > FROMALTSTACK > > > > ... FOR j in 0..=5 > > > > FROMALTSTACK > > > > CAT > > > > ... END FOR > > > > EQUALVERIFY > > > > ... ELSE SUBSTRINGEQUALVERIFY AVAILABLE > > > > ... FOR j in 0..=5 > > > > FROMALTSTACK <0+j*4> <4+j*4> SUBSTRINGEQUALVERIFY DROP DROP DROP > > > > ... END FOR > > > > DROP > > > > ... END IF > > > > > > > > <pk> CHECKSIG > > > > > > > > > > > > That's a long script... but will it fit? We need to verify 20 bytes of message > > > > each bit takes around 10 bytes script, an average of 3.375 bytes per number > > > > (counting pushes), and two 21 bytes keys = 55.375 bytes of program space and 21 > > > > bytes of witness element per bit. > > > > It fits! `20*8*55.375 = 8860`, which leaves 1140 bytes less than the limit for > > > > the rest of the logic, which is plenty (around 15-40 bytes required for the rest > > > > of the logic, leaving 1100 free for custom signature checking). The stack size > > > > is 160 elements for the hash gadget, 3360 bytes. > > > > This can probably be made a bit more efficient by expanding to a ternary > > > > representation. > > > > > > > > SWAP hash160 DUP <H(K_j_i_0)> EQUAL IF DROP ELSE <3**i> SWAP DUP <H(K_j_i_T)> EQUAL IF DROP SUB ELSE <H(K_j_i_1)> EQUALVERIFY ADD ENDIF ENDIF > > > > > > > > > > > > This should bring it up to roughly 85 bytes per trit, and there should be 101 > > > > trits (`log(2**160)/log(3) == 100.94`), so about 8560 bytes... a bit cheaper! > > > > But the witness stack is "only" `2121` bytes... > > > > As a homework exercise, maybe someone can prove the optimal choice of radix for > > > > this protocol... My guess is that base 4 is optimal! > > > > > > > > Taproot? > > > > > > > > --------- > > > > > > > > What about Taproot? As far as I'm aware the commitment scheme (`Q = pG + hash(pG || m)G`) can be securely opened to m even with a quantum computer (finding `q` > > > > such that `qG = Q` might be trivial, but suppose key path was disabled, then > > > > finding m and p such that the taproot equation holds should be difficult because > > > > of the hash, but I'd need to certify that claim better). Therefore this > > > > script can nest inside of a Tapscript path -- Tapscript also does not impose a > > > > length limit, 32 byte hashes could be used as well. > > > > Further, to make keys reusable, there could be many Lamport keys comitted inside > > > > a taproot tree so that an address could be used for thousands of times before > > > > expiring. This could be used as a measure to protect accidental use rather than > > > > to support it. > > > > Lastly, Schnorr actually has a stronger non-malleability property than ECDSA, > > > > the signatures will be binding to the approved transaction and once Lamport > > > > signed, even a quantum computer could not steal the funds. > > > > -- > > > > @JeremyRubin > > > > > > bitcoin-dev mailing list > > > bitcoin-dev@lists•linuxfoundation•org > > > https://lists.linuxfoundation.org/mailman/listinfo/bitcoin-dev > >