## Winternitz One-Time-Signature

Python implementation of Winternitz one-time-signature schemes

### Description

Winternitz one-time-signature is an extension of lamport one-time-signature. This python package can be used to execute WOTS operations, including key generation, signature generation and signature verification. Currently WOTS and WOTS+ are implemented.

### Introduction

Lamport invented an algorithm in 1979 which allowed one to create one-time-signatures using a cryptographically secure one-way function. It is the basis for the Winternitz one-time-signature algorithm. Winternitz added the possibility to adjust the tradeoff between time- and space-complexity.

#### Lamport one-time-signature scheme

Lamport suggested to create two secret keys for each bit of a message which will be signed. One for each value the bit can take. To derive the verification key, each secret key is hashed once. Now you have a secret key and a verification key, which consists of m 2-tuples of values, where m is the number of bits of the message. The verification key is published. The signature consists of m values. For each bit of the message you release a secret key from the corresponding secret keys, depending on which value the bit has. All those secret keys form the signature for the message. The verifier hashes each of your secret keys once and compares it to one verification key for this position, depending on the value of the bit. The signature is valid, if and only if all derived verification keys match with your published verification key at the correct position of the 2-tuple, which is determined by the value of the bit. This algorithm is quite fast (comparing it to existing PQC-algorithms), but the signature sizes are huge.

#### Winternitz extension

Winternitz extended lamports algorithm by offering the possiblity to decide how many bits will be signed together. The amount of numbers those bits can represent is called the Winternitz parameter (w = 2^{bits}). This method offers the huge advantage that the user of this algorithm can choose the time and space tradeoff (whether speed or storage capacity is more relevant). A fingerprint of the message which will be signed is split into groups of ceil(log_2(w)) bits. Each of these groups gets one secret key. Each verification key is derived by hashing the secret key for each group 2^{w-1} times. All verification keys will be published and represent one unified verification key. When signing a message, the fingerprint of the message is split into groups of ceil(log2(w)) bits. To create the signature, the private key for each bit group is hashed bitgroup_value times, where bitgroup_value is the value of the bitgroup. Additionally a (inverse sum) checksum is appended, which denies man-in-the-middle attacks. The checksum is calculated from the signature, split into bit groups of ceil(log2(w)) bits, and signed. To verify the signature, the fingerprint of the message is first split into bit groups of ceil(log2(w) bits each. The basic idea is to take the signature of each bit group, calculate the verification key from it and finally compare it to the published verification key. Since the signature was hashed bitgroup_value times, all you have to do to calculate the verification key from the signature is to hash the signature 2^{w-1} - bitgroup_value - 1 times. Besides verifing the message, the verifier must also calculate the checksum and verify it.

### Setup

Requires: Python >= 3.4

`pip install winternitz`

`pip install winternitz[TEST]`

`pip install winternitz[LINT]`

`pip install winternitz[DOCS]`

`pip install winternitz[ALL]`

#### Test

`python setup.py test`

`python -m tox`

#### Generate documentation

`python setup.py docs`

### Usage

The package *winternitz* contains a module called *signatures*.
Within this package you can find the classes WOTS and WOTSPLUS.
Those classes can be used out of the box to sign or verify
messages

#### WOTS

```
import winternitz.signatures
# Create signature and verify it with the same object
wots = winternitz.signatures.WOTS()
message = "My message in bytes format".encode("utf-8")
sig = wots.sign(message)
success = wots.verify(message=message, signature=sig["signature"])
print("Verification success: " + str(success))
# Output: Verification success: True
```

If you don't specify any values in the constructor of WOTS, it will use
the winternitz parameter 16 and the hash function *sha512* as default parameters.
The private key will be generated from entropy. After you received the public key,
either through `wots.pubkey`

or inside the dict that is returned by the
`wots.sign(message)`

function call, you publish it. Verify that it was not modified.
In the best case a man-in-the-middle attack to modify your public key is impossible
by the design of the application. The last step is to publish your message and every
information in the dict that is returned by `wots.sign(message)`

, except the public
key (since it was already published). Publishing the fingerprint is optional, since it
is not essential for the signature verification. The signature dict contains the following
values:

```
{
"w": winternitz parameter (Type: int),
"fingerprint": message hash (Type: bytes),
"hashalgo": hash algorithm (Type: str),
"digestsize": hash byte count (Type: int),
"pubkey": public key (Type: List[bytes]),
"signature": signature (Type: List[bytes])
}
```

With that data, another person can verify the authenticity of your message:

```
# Another person or machine wants to verify your signature:
# get required hash function by comparing the name
# published with local implementaitons
if sig["hashalgo"] == "openssl_sha512":
hashfunc = winternitz.signatures.openssl_sha512
elif sig["hashalgo"] == "openssl_sha256":
hashfunc = winternitz.signautres.openssl_sha256
else:
raise NotImplementedError("Hash function not implemented")
wots_other = winternitz.signatures.WOTS(w=sig["w"], hashfunction=hashfunc,
digestsize=sig["digestsize"], pubkey=sig["pubkey"])
success = wots_other.verify(message=message, signature=sig["signature"])
print("Verification success: " + str(success))
# Output: Verification success: True
```

In certain situations it might not be wanted to verify the derived public key with the
public key inside the executing WOTS object. For example, this can be this case when
the verification happens in a wrapping structure, like a XMSS tree. In such cases
the public key can be derived from a message and a signature with the function
`wots.getPubkeyFromSignature(message=message, signature=signature)`

#### WOTSPLUS

```
import winternitz.signatures
wotsplus = winternitz.signatures.WOTSPLUS()
message = "My message in bytes format".encode("utf-8")
sig = wotsplus.sign(message)
success = wotsplus.verify(message=message, signature=sig["signature"])
print("Verification success: " + str(success))
# Output: Verification success: True
```

If you don't specify any values in the constructor of WOTSPLUS, it will use the winternitz parameter
16 and the hash function defaults to *sha256*. It further requires a pseudo random function, which defaults
to *HMAC-sha256*, as well as a seed which is also generated from entropy. For further
informations about functions and their parameters, visit the module reference in
this the documentation.
Since WOTS+ uses a pseudo random function and a seed to derive signatures and public
keys, they have to be published as well. In addition to the signature of WOTS, the returned dict contains
the following values:

```
{
# ...
"prf": pseudo random function (Type: str),
"seed": Seed used in prf (Type: bytes)
}
```

Those arguments have to be specified in the constructor of WOTSPLUS in addition to those parameters specified in WOTS.

#### Misc

The WOTS classes come with some features that will be explained in the following sections.

##### Fully configurable

The WOTS classes are fully parameterizable. You can specify anything that is specified in the papers describing the algorithm, including the Winternitz parameter, the hash function, the pseudo random function (WOTSPLUS), the seed (WOTSPLUS), the private key and the public key. specifing both a private key and public key results in the public key beeing discarded.

##### On-demand generation of keys

If no private key or no public key is specified, they will be set to None. The same
goes for the seed in wots+. Only when they are required, they will be generated or
derived. This means that as long as you don't execute `repr(obj)`

, `str(obj)`

, `obj1 == obj2`

,
`obj1 != obj2`

, `obj.pubkey`

, `obj.privkey`

, `obj.sign(...)`

or `obj.verify(...)`

, where obj is a
WOTS object, the keys will stay None.

##### Code representation of WOTS objects

You can call `repr(obj)`

, where obj is a WOTS object, to get a line of code which contains
all information to initialize another object so that it is equal to obj. Executing `obj2 = eval(repr(obj))`

executes that code which is returned by `repr(obj)`

and ultimately stores a copy of it in `obj2`

.

##### Human readable string representation

You can call `str(obj)`

to get a string which contains a human readable representation of that object.

##### Comparison of objects

You can compare two objects from this class `obj1 == obj2`

and `obj1 != obj2`

##### Optimizations

The code was carefully written to reduce execution times. It surely is not perfect and can still be optimized,
further time-critical sections could be coded as C extensions, but nevertheless in the current state it should
offer quite an efficient implementation. It defines `__slots__`

to reduce execution times and storage requirements
within the class. Implementation of parallelization is planned, but it is only usefull when using huge winternitz
parameters, since python can only execute code in parallel if you spawn a new process and the overhead of forking
a new python interpreter is not negliable.

### Note

This project has been set up using PyScaffold 3.1. For details and usage information on PyScaffold see https://pyscaffold.org/.