| Internet-Draft | RSA Blind Signatures | June 2023 |
| Denis, et al. | Expires 17 December 2023 | [Page] |
This document specifies an RSA-based blind signature protocol. RSA blind signatures were first introduced by Chaum for untraceable payments. A signature that is output from this protocol can be verified as an RSA-PSS signature.¶
This document is a product of the Crypto Forum Research Group (CFRG) in the IRTF.¶
This note is to be removed before publishing as an RFC.¶
Source for this draft and an issue tracker can be found at https://github.com/chris-wood/draft-wood-cfrg-blind-signatures.¶
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Originally introduced in the context of digital cash systems by Chaum for untraceable payments [Chaum83], RSA blind signatures turned out to have a wide range of applications ranging from privacy-preserving digital payments to authentication mechanisms [GoogleVPN] [ApplePrivateRelay] [PrettyGoodPhonePrivacy].¶
Recently, interest in blind signatures has grown to address operational shortcomings from applications that use Verifiable Oblivious Pseudorandom Functions (VOPRFs) [VOPRF], such as Privacy Pass [PRIVACY-PASS]. Specifically, VOPRFs are not necessarily publicly verifiable, meaning that a verifier needs access to the VOPRF private key to verify that the output of a VOPRF protocol is valid for a given input. This limitation complicates deployments where it is not desirable to distribute private keys to entities performing verification. Additionally, if the private key is kept in a Hardware Security Module, the number of operations on the key is doubled compared to a scheme where only the public key is required for verification.¶
In contrast, digital signatures provide a primitive that is publicly verifiable and does not require access to the private key for verification. Moreover, [JKK14] shows that one can realize a VOPRF in the Random Oracle Model by hashing a signature-message pair, where the signature is computed using a deterministic blind signature protocol.¶
This document specifies a protocol for computing RSA blind signatures using RSA-PSS encoding, and a family of variants for this protocol, denoted RSABSSA (RSA Blind Signature with Appendix). In order to facilitate deployment, it is defined in such a way that the resulting (unblinded) signature can be verified with a standard RSA-PSS library.¶
This document represents the consensus of the Crypto Forum Research Group (CFRG). It is not an IETF product and is not a standard.¶
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.¶
The following terms are used throughout this document to describe the protocol operations in this document:¶
x starting from offset i and ending at
offset j, inclusive. For example, slice(0x010203040506, 1, 5) = 0x0203040506.¶
The RSA Blind Signature Protocol is a two-party protocol between a client and server
where they interact to compute sig = Sign(sk, input_msg), where input_msg = Prepare(msg)
is a prepared version of the private message msg provided by the client, and sk is the
private signing key provided by the server. See Section 6.2 for details on how sk is generated
and used in this protocol. Upon completion of this protocol, the server learns nothing,
whereas the client learns sig. In particular, this means the server learns nothing of msg
or input_msg and the client learns nothing of sk.¶
The protocol consists of four functions -- Prepare, Blind, BlindSign, and Finalize -- and requires
one round of interaction between client and server. Let msg be the client's private input
message, and (sk, pk) be the server's private and public key pair.¶
The protocol begins by the client preparing the message to be signed by computing:¶
input_msg = Prepare(msg)¶
The client then initiates the blind signature protocol by computing:¶
blinded_msg, inv = Blind(pk, input_msg)¶
The client then sends blinded_msg to the server, which then processes the message
by computing:¶
blind_sig = BlindSign(sk, blinded_msg)¶
The server then sends blind_sig to the client, which then finalizes the protocol by computing:¶
sig = Finalize(pk, input_msg, blind_sig, inv)¶
The output of the protocol is input_msg and sig. Upon completion, correctness requires that
clients can verify signature sig over the prepared message input_msg using the server
public key pk by invoking the RSASSA-PSS-VERIFY routine defined in
Section 8.1.2 of [RFC8017]. The Finalize function performs this check before returning the signature.
See Section 4.5 for more details about verifying signatures produced through this protocol.¶
In pictures, the protocol runs as follows:¶
Client(pk, msg) Server(sk, pk)
-----------------------------------------------------
input_msg = Prepare(msg)
blinded_msg, inv = Blind(pk, input_msg)
blinded_msg
---------->
blind_sig = BlindSign(sk, blinded_msg)
blind_sig
<----------
sig = Finalize(pk, input_msg, blind_sig, inv)
¶
In the remainder of this section, we specify the Prepare, Blind, BlindSign, and Finalize functions that are used in this protocol.¶
Message preparation, denoted by the Prepare function, is the process by which the message
to be signed and verified is prepared for input to the blind signing protocol.
There are two types of preparation functions: an identity preparation function,
and a randomized preparation function. The identity preparation function returns
the input message without transformation, i.e., msg = PrepareIdentity(msg).¶
The randomized preparation function augments the input message with fresh randomness.
We denote this process by the function PrepareRandomize(msg), which takes as input a message
msg and produces a randomized message input_msg. Its implementation is shown below.¶
PrepareRandomize(msg) Inputs: - msg, message to be signed, a byte string Outputs: - input_msg, a byte string that is 32 bytes longer than msg Steps: 1. msg_prefix = random(32) 2. input_msg = concat(msg_prefix, msg) 3. output input_msg¶
The Blind function encodes an input message and blinds it with the server's public key. It outputs the blinded message to be sent to the server, encoded as a byte string, and the corresponding inverse, an integer. RSAVP1 and EMSA-PSS-ENCODE are as defined in Sections 5.2.2 and 9.1.1 of [RFC8017], respectively.¶
If this function fails with an "blinding error" error, implementations SHOULD retry the function again. The probability of one or more such errors in sequence is negligible. This function can also fail with an "invalid input" error, which indicates that one of the inputs (likely the public key) was invalid. Implementations SHOULD update the public key before calling this function again. See Section 6.1 for more information about dealing with such errors.¶
Note that this function invokes RSAVP1, which is defined to throw an optional error for invalid inputs. However, this error cannot occur based on how RSAVP1 is invoked, so this error is not included in the list of errors for Blind.¶
Blind(pk, msg) Parameters: - modulus_len, the length in bytes of the RSA modulus n - Hash, the hash function used to hash the message - MGF, the mask generation function - salt_len, the length in bytes of the salt (denoted sLen in RFC8017) Inputs: - pk, server public key (n, e) - msg, message to be signed, a byte string Outputs: - blinded_msg, a byte string of length modulus_len - inv, an integer used to unblind the signature in Finalize Errors: - "message too long": Raised when the input message is too long (raised by EMSA-PSS-ENCODE). - "encoding error": Raised when the input message fails encoding (raised by EMSA-PSS-ENCODE). - "blinding error": Raised when the inverse of r cannot be found. - "invalid input": Raised when the message is not co-prime with n. Steps: 1. encoded_msg = EMSA-PSS-ENCODE(msg, bit_len(n)) with Hash, MGF, and salt_len as defined in the parameters 2. If EMSA-PSS-ENCODE raises an error, raise the error and stop 3. m = bytes_to_int(encoded_msg) 4. c = is_coprime(m, n) 5. If c is false, raise an "invalid input" error and stop 6. r = random_integer_uniform(1, n) 7. inv = inverse_mod(r, n) 8. If inverse_mod fails, raise an "blinding error" error and stop 9. x = RSAVP1(pk, r) 10. z = (m * x) mod n 11. blinded_msg = int_to_bytes(z, modulus_len) 12. output blinded_msg, inv¶
The blinding factor r MUST be randomly chosen from a uniform distribution. This is typically done via rejection sampling.¶
BlindSign performs the RSA private key operation on the client's blinded message input and returns the output encoded as a byte string. RSASP1 is as defined in Section 5.2.1 of [RFC8017].¶
BlindSign(sk, blinded_msg) Parameters: - modulus_len, the length in bytes of the RSA modulus n Inputs: - sk, server private key - blinded_msg, encoded and blinded message to be signed, a byte string Outputs: - blind_sig, a byte string of length modulus_len Errors: - "signing failure": Raised when the signing operation fails - "message representative out of range": Raised when the message representative to sign is not an integer between 0 and n - 1 (raised by RSASP1) Steps: 1. m = bytes_to_int(blinded_msg) 2. s = RSASP1(sk, m) 3. m' = RSAVP1(pk, s) 4. If m != m', raise "signing failure" and stop 5. blind_sig = int_to_bytes(s, modulus_len) 6. output blind_sig¶
Finalize validates the server's response, unblinds the message to produce a signature, verifies it for correctness, and outputs the signature upon success. Note that this function will internally hash the input message as is done in Blind.¶
Finalize(pk, msg, blind_sig, inv) Parameters: - modulus_len, the length in bytes of the RSA modulus n - Hash, the hash function used to hash the message - MGF, the mask generation function - salt_len, the length in bytes of the salt (denoted sLen in RFC8017) Inputs: - pk, server public key (n, e) - msg, message to be signed, a byte string - blind_sig, signed and blinded element, a byte string of length modulus_len - inv, inverse of the blind, an integer Outputs: - sig, a byte string of length modulus_len Errors: - "invalid signature": Raised when the signature is invalid - "unexpected input size": Raised when a byte string input doesn't have the expected length. Steps: 1. If len(blind_sig) != modulus_len, raise "unexpected input size" and stop 2. z = bytes_to_int(blind_sig) 3. s = (z * inv) mod n 4. sig = int_to_bytes(s, modulus_len) 5. result = RSASSA-PSS-VERIFY(pk, msg, sig) with Hash, MGF, and salt_len as defined in the parameters 6. If result = "valid signature", output sig, else raise "invalid signature" and stop¶
As described in Section 4, the output of the protocol is the prepared
message input_msg and the signature sig. The message that applications
consume is msg, from which input_msg is derived. Clients verify the
msg signature using the server's public key pk by invoking the
RSASSA-PSS-VERIFY routine defined in Section 8.1.2 of [RFC8017]
with (n, e) as pk, M as input_msg, and S as sig.¶
Verification and the message that applications consume therefore depends on
which preparation function is used. In particular, if the PrepareIdentity
function is used, then the application message is input_msg.
In contrast, if the PrepareRandomize function is used, then the application
message is slice(input_msg, 32, len(input_msg)), i.e., the prepared message
with the random prefix removed.¶
In this section we define different named variants of RSABSSA. Each variant specifies RSASSA-PSS parameters as defined in Section 9.1.1 of [RFC8017] and the type of message preparation function applied (as described in Section 4.1). Each variant uses the MGF1 Mask Generation Function defined in Appendix B.2.1. of [RFC8017]. Future specifications can introduce other variants as desired. The named variants are as follows:¶
msg.¶
The RECOMMENDED variants are RSABSSA-SHA384-PSS-Randomized or RSABSSA-SHA384-PSSZERO-Randomized.¶
Not all named variants can be used interchangeably. In particular, applications that provide high-entropy input messages can safely use named variants without randomized message preparation, as the additional message randomization does not offer security advantages. See [Lys22] and Section 7.3 for more information. For all other applications, the variants that use the randomized preparation function protect clients from malicious signers. A verifier that accepts randomized messages needs to remove the random component from the signed part of messages before processing.¶
Applications that require deterministic signatures can use the RSABSSA-SHA384-PSSZERO-Deterministic variant, but only if their input messages have high entropy. Applications that use RSABSSA-SHA384-PSSZERO-Deterministic SHOULD carefully analyze the security implications, taking into account the possibility of adversarially generated signer keys as described in Section 7.3. When it is not clear whether an application requires deterministic or randomized signatures, applications SHOULD use one of the variants with randomized message preparation.¶
This section documents considerations for interfaces to implementations of the protocol in this document. This includes error handling and API considerations.¶
The high-level functions specified in Section 4 are all fallible. The explicit errors generated throughout this specification, along with the conditions that lead to each error, are listed in the definitions for Blind, BlindSign, and Finalize. These errors are meant as a guide for implementors. They are not an exhaustive list of all the errors an implementation might emit. For example, implementations might run out of memory.¶
Moreover, implementations can handle errors as needed or desired. Where applicable, this document provides guidance for how to deal with explicit errors that are generated in the protocol. For example, "blinding error" is generated in Blind when the client produces a prime factor of the server's public key. Section 4.2 indicates that implementations SHOULD retry the Blind function when this error occurs, but an implementation could also handle this exceptional event differently, e.g., by informing the server that the key has been factored.¶
The RECOMMENDED method for generating the server signing key pair is as specified in FIPS 186-4 [DSS].¶
A server signing key MUST NOT be reused for any other protocol beyond RSABSSA. Moreover, a server signing key MUST NOT be reused for different RSABSSA encoding options. That is, if a server supports two different encoding options, then it MUST have a distinct key pair for each option.¶
If the server public key is carried in an X.509 certificate, it MUST use the RSASSA-PSS OID [RFC5756]. It MUST NOT use the rsaEncryption OID [RFC5280].¶
Lysyanskaya proved one-more-forgery polynomial security of RSABSSA variants in the random oracle model under the one-more-RSA assumption in [Lys22]. This means the adversary cannot output n+1 valid message and signature tuples, where all messages are distinct, after interacting with the server (signer) as a client only n times, for some n which is polynomial in the protocol's security parameter. Lysyanskaya also proved that the RSABSSA variants which use the PrepareRandomize function achieve blindness in [Lys22]. Blindness means that the malicious signer learns nothing about the client input and output after the protocol execution. However, additional assumptions on the message inputs are required for blindness to hold for RSABSSA variants that use the PrepareIdentity function; see Section 7.3 for more discussion on those results.¶
BlindSign is functionally a remote procedure call for applying the RSA private key operation. As such, side channel resistance is paramount to protect the private key from exposure [RemoteTimingAttacks]. Implementations SHOULD implement some form of side channel attack mitigation, such as RSA blinding as described in Section 10 of [TimingAttacks]. Failure to apply such mitigations can lead to side channel attacks that leak the private signing key.¶
Moreover, we assume that the server does not initiate the protocol and therefore has no knowledge of when the Prepare and Blind operations take place. If this were not the case, additional side-channel mitigations might be required to prevent timing side channels through Prepare and Blind.¶
Beyond timing side channels, [FAULTS] describes the importance of implementation safeguards that protect against fault attacks that can also leak the private signing key. These safeguards require that implementations check that the result of the private key operation when signing is correct, i.e., given s = RSASP1(sk, m), verify that m = RSAVP1(pk, s), as is required by BlindSign. Applying this (or equivalent) safeguard is necessary to mitigate fault attacks, even for implementations that are not based on the Chinese remainder theorem.¶
An essential property of blind signature protocols is that the signer learns nothing of the message being signed. In some circumstances, this may raise concerns of arbitrary signing oracles. Applications using blind signature protocols should take precautions to ensure that such oracles do not cause cross-protocol attacks. Ensuring that the signing key used for RSABSSA is distinct from other protocols prevents such cross-protocol attacks.¶
An alternative solution to this problem of message blindness is to give signers proof that the message being signed is well-structured. Depending on the application, zero knowledge proofs could be useful for this purpose. Defining such a proof is out of scope for this document.¶
Verifiers should check that, in addition to signature validity, the signed message is well-structured for the relevant application. For example, if an application of this protocol requires messages to be structures of a particular form, then verifiers should check that messages adhere to this form.¶
As discussed in [Lys22], a malicious signer can construct an invalid public key and use it to learn information about low-entropy input messages. Note that some invalid public keys may not yield valid signatures when run with the protocol, e.g., because the signature fails to verify. However, if an attacker can coerce the client to use these invalid public keys with low-entropy inputs, they can learn information about the client inputs before the protocol completes.¶
A client that uses this protocol might be vulnerable to attack from a malicious signer unless it is able to ensure that either:¶
The named variants that use the PrepareRandomize function -- RSABSSA-SHA384-PSS-Randomized and RSABSSA-SHA384-PSSZERO-Randomized -- explicitly inject fresh entropy alongside each message to satisfy condition (2). As such, these variants are safe for all application use cases.¶
Note that these variants effectively mean that the resulting signature is always randomized. As such, this interface is not suitable for applications that require deterministic signatures.¶
All random values in the protocol, including the salt, message randomizer prefix, and random blind
value in Blind, MUST be generated from a cryptographically secure random number generator [RFC4086].
If these values are not generated randomly, or are otherwise constructed maliciously, it might be
possible for them to encode information that is not present in the signed message. For example,
the PSS salt might be maliciously constructed to encode the local IP address of the client. As a result,
implementations SHOULD NOT allow clients to provide these values directly.¶
Note that malicious implementations could also encode client information in the message being signed, but since clients can verify the resulting message signature using the public key this can be detected.¶
RSA is well known to permit key substitution attacks, wherein an attacker generates a key pair (skA, pkA) that verifies some known (message, signature) pair produced under a different (sk, pk) key pair [WM99]. This means it may be possible for an attacker to use a (message, signature) pair from one context in another. Entities that verify signatures must take care to ensure a (message, signature) pair verifies with a valid public key from the expected issuer.¶
This document document uses PSS encoding as specified in [RFC8017] for a number of reasons. First, it is recommended in recent standards, including TLS 1.3 [RFC8446], X.509v3 [RFC4055], and even PKCS#1 itself. According to [RFC8017], "Although no attacks are known against RSASSA-PKCS#1 v1.5, in the interest of increased robustness, RSA-PSS [RFC8017] is recommended for eventual adoption in new applications." While RSA-PSS is more complex than RSASSA-PKCS#1 v1.5 encoding, ubiquity of RSA-PSS support influenced the design decision in this draft, despite PKCS#1 v1.5 having equivalent security properties for digital signatures [JKM18].¶
Full Domain Hash (FDH) [RSA-FDH] encoding is also possible, and this variant has equivalent security to PSS [KK18]. However, FDH is less standard and not used widely in related technologies. Moreover, FDH is deterministic, whereas PSS supports deterministic and probabilistic encodings.¶
The blind signature protocol specified in this document is not post-quantum ready since it is based on RSA. Shor's polynomial-time factorization algorithm readily applies.¶
This document makes no IANA requests.¶
This section includes test vectors for the blind signature protocol defined in Section 4. The following parameters are specified for each test vector:¶
p = e1f4d7a34802e27c7392a3cea32a262a34dc3691bd87f3f310dc756734889305 59c120fd0410194fb8a0da55bd0b81227e843fdca6692ae80e5a5d414116d4803fca 7d8c30eaaae57e44a1816ebb5c5b0606c536246c7f11985d731684150b63c9a3ad9e 41b04c0b5b27cb188a692c84696b742a80d3cd00ab891f2457443dadfeba6d6daf10 8602be26d7071803c67105a5426838e6889d77e8474b29244cefaf418e381b312048 b457d73419213063c60ee7b0d81820165864fef93523c9635c22210956e53a8d9632 2493ffc58d845368e2416e078e5bcb5d2fd68ae6acfa54f9627c42e84a9d3f277401 7e32ebca06308a12ecc290c7cd1156dcccfb2311 q = c601a9caea66dc3835827b539db9df6f6f5ae77244692780cd334a006ab353c8 06426b60718c05245650821d39445d3ab591ed10a7339f15d83fe13f6a3dfb20b945 2c6a9b42eaa62a68c970df3cadb2139f804ad8223d56108dfde30ba7d367e9b0a7a8 0c4fdba2fd9dde6661fc73fc2947569d2029f2870fc02d8325acf28c9afa19ecf962 daa7916e21afad09eb62fe9f1cf91b77dc879b7974b490d3ebd2e95426057f35d0a3 c9f45f79ac727ab81a519a8b9285932d9b2e5ccd347e59f3f32ad9ca359115e7da00 8ab7406707bd0e8e185a5ed8758b5ba266e8828f8d863ae133846304a2936ad7bc7c 9803879d2fc4a28e69291d73dbd799f8bc238385 n = aec4d69addc70b990ea66a5e70603b6fee27aafebd08f2d94cbe1250c556e047 a928d635c3f45ee9b66d1bc628a03bac9b7c3f416fe20dabea8f3d7b4bbf7f963be3 35d2328d67e6c13ee4a8f955e05a3283720d3e1f139c38e43e0338ad058a9495c533 77fc35be64d208f89b4aa721bf7f7d3fef837be2a80e0f8adf0bcd1eec5bb040443a 2b2792fdca522a7472aed74f31a1ebe1eebc1f408660a0543dfe2a850f106a617ec6 685573702eaaa21a5640a5dcaf9b74e397fa3af18a2f1b7c03ba91a6336158de420d 63188ee143866ee415735d155b7c2d854d795b7bc236cffd71542df34234221a0413 e142d8c61355cc44d45bda94204974557ac2704cd8b593f035a5724b1adf442e78c5 42cd4414fce6f1298182fb6d8e53cef1adfd2e90e1e4deec52999bdc6c29144e8d52 a125232c8c6d75c706ea3cc06841c7bda33568c63a6c03817f722b50fcf898237d78 8a4400869e44d90a3020923dc646388abcc914315215fcd1bae11b1c751fd52443aa c8f601087d8d42737c18a3fa11ecd4131ecae017ae0a14acfc4ef85b83c19fed33cf d1cd629da2c4c09e222b398e18d822f77bb378dea3cb360b605e5aa58b20edc29d00 0a66bd177c682a17e7eb12a63ef7c2e4183e0d898f3d6bf567ba8ae84f84f1d23bf8 b8e261c3729e2fa6d07b832e07cddd1d14f55325c6f924267957121902dc19b3b329 48bdead5 e = 10001 d = d43242aefe1fb2c13fbc66e20b678c4336d20b1808c558b6e62ad16a28707718 0b177e1f01b12f9c6cd6c52630257ccef26a45135a990928773f3bd2fc01a313f1da c97a51cec71cb1fd7efc7adffdeb05f1fb04812c924ed7f4a8269925dad88bd7dcfb c4ef01020ebfc60cb3e04c54f981fdbd273e69a8a58b8ceb7c2d83fbcbd6f784d052 201b88a9848186f2a45c0d2826870733e6fd9aa46983e0a6e82e35ca20a439c5ee7b 502a9062e1066493bdadf8b49eb30d9558ed85abc7afb29b3c9bc644199654a46766 81af4babcea4e6f71fe4565c9c1b85d9985b84ec1abf1a820a9bbebee0df1398aae2 c85ab580a9f13e7743afd3108eb32100b870648fa6bc17e8abac4d3c99246b1f0ea9 f7f93a5dd5458c56d9f3f81ff2216b3c3680a13591673c43194d8e6fc93fc1e37ce2 986bd628ac48088bc723d8fbe293861ca7a9f4a73e9fa63b1b6d0074f5dea2a624c5 249ff3ad811b6255b299d6bc5451ba7477f19c5a0db690c3e6476398b1483d10314a fd38bbaf6e2fbdbcd62c3ca9797a420ca6034ec0a83360a3ee2adf4b9d4ba29731d1 31b099a38d6a23cc463db754603211260e99d19affc902c915d7854554aabf608e3a c52c19b8aa26ae042249b17b2d29669b5c859103ee53ef9bdc73ba3c6b537d5c34b6 d8f034671d7f3a8a6966cc4543df223565343154140fd7391c7e7be03e241f4ecfeb 877a051 msg = 8f3dc6fb8c4a02f4d6352edf0907822c1210a9b32f9bdda4c45a698c80023a a6b59f8cfec5fdbb36331372ebefedae7d msg_prefix = 190db6270a235a568a5f397012c635cb5603f9ec1498e8ab2cf4908 c00a0cc20 random_msg = 190db6270a235a568a5f397012c635cb5603f9ec1498e8ab2cf4908 c00a0cc208f3dc6fb8c4a02f4d6352edf0907822c1210a9b32f9bdda4c45a698c800 23aa6b59f8cfec5fdbb36331372ebefedae7d salt = 051722b35f458781397c3a671a7d3bd3096503940e4c4f1aaa269d60300ce 449555cd7340100df9d46944c5356825abf encoded_msg = 6e0c464d9c2f9fbc147b43570fc4f238e0d0b38870b3addcf7a421 7df912ccef17a7f629aa850f63a063925f312d61d6437be954b45025e8282f9c0b11 31bc8ff19a8a928d859b37113db1064f92a27f64761c181c1e1f9b251ae5a2f8a404 7573b67a270584e089beadcb13e7c82337797119712e9b849ff56e04385d144d3ca9 d8d92bf78adb20b5bbeb3685f17038ec6afade3ef354429c51c687b45a7018ee3a69 66b3af15c9ba8f40e6461ba0a17ef5a799672ad882bab02b518f9da7c1a962945c2e 9b0f02f29b31b9cdf3e633f9d9d2a22e96e1de28e25241ca7dd04147112f57897340 3e0f4fd80865965475d22294f065e17a1c4a201de93bd14223e6b1b999fd548f2f75 9f52db71964528b6f15b9c2d7811f2a0a35d534b8216301c47f4f04f412cae142b48 c4cdff78bc54df690fd43142d750c671dd8e2e938e6a440b2f825b6dbb3e19f1d7a3 c0150428a47948037c322365b7fe6fe57ac88d8f80889e9ff38177bad8c8d8d98db4 2908b389cb59692a58ce275aa15acb032ca951b3e0a3404b7f33f655b7c7d83a2f8d 1b6bbff49d5fcedf2e030e80881aa436db27a5c0dea13f32e7d460dbf01240c2320c 2bb5b3225b17145c72d61d47c8f84d1e19417ebd8ce3638a82d395cc6f7050b6209d 9283dc7b93fecc04f3f9e7f566829ac41568ef799480c733c09759aa9734e2013d76 40dc6151018ea902bc inv = 80682c48982407b489d53d1261b19ec8627d02b8cda5336750b8cee332ae26 0de57b02d72609c1e0e9f28e2040fc65b6f02d56dbd6aa9af8fde656f70495dfb723 ba01173d4707a12fddac628ca29f3e32340bd8f7ddb557cf819f6b01e445ad96f874 ba235584ee71f6581f62d4f43bf03f910f6510deb85e8ef06c7f09d9794a008be7ff 2529f0ebb69decef646387dc767b74939265fec0223aa6d84d2a8a1cc912d5ca25b4 e144ab8f6ba054b54910176d5737a2cff011da431bd5f2a0d2d66b9e70b39f4b050e 45c0d9c16f02deda9ddf2d00f3e4b01037d7029cd49c2d46a8e1fc2c0c17520af1f4 b5e25ba396afc4cd60c494a4c426448b35b49635b337cfb08e7c22a39b256dd032c0 0adddafb51a627f99a0e1704170ac1f1912e49d9db10ec04c19c58f420212973e0cb 329524223a6aa56c7937c5dffdb5d966b6cd4cbc26f3201dd25c80960a1a111b3294 7bb78973d269fac7f5186530930ed19f68507540eed9e1bab8b00f00d8ca09b3f099 aae46180e04e3584bd7ca054df18a1504b89d1d1675d0966c4ae1407be325cdf623c f13ff13e4a28b594d59e3eadbadf6136eee7a59d6a444c9eb4e2198e8a974f27a39e b63af2c9af3870488b8adaad444674f512133ad80b9220e09158521614f1faadfe85 05ef57b7df6813048603f0dd04f4280177a11380fbfc861dbcbd7418d62155248dad 5fdec0991f blinded_msg = 10c166c6a711e81c46f45b18e5873cc4f494f003180dd7f115585d 871a28930259654fe28a54dab319cc5011204c8373b50a57b0fdc7a678bd74c52325 9dfe4fd5ea9f52f170e19dfa332930ad1609fc8a00902d725cfe50685c95e5b2968c 9a2828a21207fcf393d15f849769e2af34ac4259d91dfd98c3a707c509e1af55647e faa31290ddf48e0133b798562af5eabd327270ac2fb6c594734ce339a14ea4fe1b9a 2f81c0bc230ca523bda17ff42a377266bc2778a274c0ae5ec5a8cbbe364fcf0d2403 f7ee178d77ff28b67a20c7ceec009182dbcaa9bc99b51ebbf13b7d542be337172c64 74f2cd3561219fe0dfa3fb207cff89632091ab841cf38d8aa88af6891539f263adb8 eac6402c41b6ebd72984e43666e537f5f5fe27b2b5aa114957e9a580730308a5f5a9 c63a1eb599f093ab401d0c6003a451931b6d124180305705845060ebba6b0036154f cef3e5e9f9e4b87e8f084542fd1dd67e7782a5585150181c01eb6d90cb9588383738 4a5b91dbb606f266059ecc51b5acbaa280e45cfd2eec8cc1cdb1b7211c8e14805ba6 83f9b78824b2eb005bc8a7d7179a36c152cb87c8219e5569bba911bb32a1b923ca83 de0e03fb10fba75d85c55907dda5a2606bf918b056c3808ba496a4d95532212040a5 f44f37e1097f26dc27b98a51837daa78f23e532156296b64352669c94a8a855acf30 533d8e0594ace7c442 blind_sig = 364f6a40dbfbc3bbb257943337eeff791a0f290898a6791283bba581 d9eac90a6376a837241f5f73a78a5c6746e1306ba3adab6067c32ff69115734ce014 d354e2f259d4cbfb890244fd451a497fe6ecf9aa90d19a2d441162f7eaa7ce3fc4e8 9fd4e76b7ae585be2a2c0fd6fb246b8ac8d58bcb585634e30c9168a434786fe5e0b7 4bfe8187b47ac091aa571ffea0a864cb906d0e28c77a00e8cd8f6aba4317a8cc7bf3 2ce566bd1ef80c64de041728abe087bee6cadd0b7062bde5ceef308a23bd1ccc154f d0c3a26110df6193464fc0d24ee189aea8979d722170ba945fdcce9b1b4b63349980 f3a92dc2e5418c54d38a862916926b3f9ca270a8cf40dfb9772bfbdd9a3e0e089236 9c18249211ba857f35963d0e05d8da98f1aa0c6bba58f47487b8f663e395091275f8 2941830b050b260e4767ce2fa903e75ff8970c98bfb3a08d6db91ab1746c86420ee2 e909bf681cac173697135983c3594b2def673736220452fde4ddec867d40ff42dd3d a36c84e3e52508b891a00f50b4f62d112edb3b6b6cc3dbd546ba10f36b03f06c0d82 aeec3b25e127af545fac28e1613a0517a6095ad18a98ab79f68801e05c175e15bae2 1f821e80c80ab4fdec6fb34ca315e194502b8f3dcf7892b511aee45060e3994cd15e 003861bc7220a2babd7b40eda03382548a34a7110f9b1779bf3ef6011361611e6bc5 c0dc851e1509de1a sig = 6fef8bf9bc182cd8cf7ce45c7dcf0e6f3e518ae48f06f3c670c649ac737a8b 8119a34d51641785be151a697ed7825fdfece82865123445eab03eb4bb91cecf4d69 51738495f8481151b62de869658573df4e50a95c17c31b52e154ae26a04067d5ecdc 1592c287550bb982a5bb9c30fd53a768cee6baabb3d483e9f1e2da954c7f4cf492fe 3944d2fe456c1ecaf0840369e33fb4010e6b44bb1d721840513524d8e9a3519f40d1 b81ae34fb7a31ee6b7ed641cb16c2ac999004c2191de0201457523f5a4700dd64926 7d9286f5c1d193f1454c9f868a57816bf5ff76c838a2eeb616a3fc9976f65d4371de ecfbab29362caebdff69c635fe5a2113da4d4d8c24f0b16a0584fa05e80e607c5d9a 2f765f1f069f8d4da21f27c2a3b5c984b4ab24899bef46c6d9323df4862fe51ce300 fca40fb539c3bb7fe2dcc9409e425f2d3b95e70e9c49c5feb6ecc9d43442c33d5000 3ee936845892fb8be475647da9a080f5bc7f8a716590b3745c2209fe05b17992830c e15f32c7b22cde755c8a2fe50bd814a0434130b807dc1b7218d4e85342d70695a5d7 f29306f25623ad1e8aa08ef71b54b8ee447b5f64e73d09bdd6c3b7ca224058d7c67c c7551e9241688ada12d859cb7646fbd3ed8b34312f3b49d69802f0eaa11bc4211c2f 7a29cd5c01ed01a39001c5856fab36228f5ee2f2e1110811872fe7c865c42ed59029 c706195d52¶
p = e1f4d7a34802e27c7392a3cea32a262a34dc3691bd87f3f310dc756734889305 59c120fd0410194fb8a0da55bd0b81227e843fdca6692ae80e5a5d414116d4803fca 7d8c30eaaae57e44a1816ebb5c5b0606c536246c7f11985d731684150b63c9a3ad9e 41b04c0b5b27cb188a692c84696b742a80d3cd00ab891f2457443dadfeba6d6daf10 8602be26d7071803c67105a5426838e6889d77e8474b29244cefaf418e381b312048 b457d73419213063c60ee7b0d81820165864fef93523c9635c22210956e53a8d9632 2493ffc58d845368e2416e078e5bcb5d2fd68ae6acfa54f9627c42e84a9d3f277401 7e32ebca06308a12ecc290c7cd1156dcccfb2311 q = c601a9caea66dc3835827b539db9df6f6f5ae77244692780cd334a006ab353c8 06426b60718c05245650821d39445d3ab591ed10a7339f15d83fe13f6a3dfb20b945 2c6a9b42eaa62a68c970df3cadb2139f804ad8223d56108dfde30ba7d367e9b0a7a8 0c4fdba2fd9dde6661fc73fc2947569d2029f2870fc02d8325acf28c9afa19ecf962 daa7916e21afad09eb62fe9f1cf91b77dc879b7974b490d3ebd2e95426057f35d0a3 c9f45f79ac727ab81a519a8b9285932d9b2e5ccd347e59f3f32ad9ca359115e7da00 8ab7406707bd0e8e185a5ed8758b5ba266e8828f8d863ae133846304a2936ad7bc7c 9803879d2fc4a28e69291d73dbd799f8bc238385 n = aec4d69addc70b990ea66a5e70603b6fee27aafebd08f2d94cbe1250c556e047 a928d635c3f45ee9b66d1bc628a03bac9b7c3f416fe20dabea8f3d7b4bbf7f963be3 35d2328d67e6c13ee4a8f955e05a3283720d3e1f139c38e43e0338ad058a9495c533 77fc35be64d208f89b4aa721bf7f7d3fef837be2a80e0f8adf0bcd1eec5bb040443a 2b2792fdca522a7472aed74f31a1ebe1eebc1f408660a0543dfe2a850f106a617ec6 685573702eaaa21a5640a5dcaf9b74e397fa3af18a2f1b7c03ba91a6336158de420d 63188ee143866ee415735d155b7c2d854d795b7bc236cffd71542df34234221a0413 e142d8c61355cc44d45bda94204974557ac2704cd8b593f035a5724b1adf442e78c5 42cd4414fce6f1298182fb6d8e53cef1adfd2e90e1e4deec52999bdc6c29144e8d52 a125232c8c6d75c706ea3cc06841c7bda33568c63a6c03817f722b50fcf898237d78 8a4400869e44d90a3020923dc646388abcc914315215fcd1bae11b1c751fd52443aa c8f601087d8d42737c18a3fa11ecd4131ecae017ae0a14acfc4ef85b83c19fed33cf d1cd629da2c4c09e222b398e18d822f77bb378dea3cb360b605e5aa58b20edc29d00 0a66bd177c682a17e7eb12a63ef7c2e4183e0d898f3d6bf567ba8ae84f84f1d23bf8 b8e261c3729e2fa6d07b832e07cddd1d14f55325c6f924267957121902dc19b3b329 48bdead5 e = 10001 d = d43242aefe1fb2c13fbc66e20b678c4336d20b1808c558b6e62ad16a28707718 0b177e1f01b12f9c6cd6c52630257ccef26a45135a990928773f3bd2fc01a313f1da c97a51cec71cb1fd7efc7adffdeb05f1fb04812c924ed7f4a8269925dad88bd7dcfb c4ef01020ebfc60cb3e04c54f981fdbd273e69a8a58b8ceb7c2d83fbcbd6f784d052 201b88a9848186f2a45c0d2826870733e6fd9aa46983e0a6e82e35ca20a439c5ee7b 502a9062e1066493bdadf8b49eb30d9558ed85abc7afb29b3c9bc644199654a46766 81af4babcea4e6f71fe4565c9c1b85d9985b84ec1abf1a820a9bbebee0df1398aae2 c85ab580a9f13e7743afd3108eb32100b870648fa6bc17e8abac4d3c99246b1f0ea9 f7f93a5dd5458c56d9f3f81ff2216b3c3680a13591673c43194d8e6fc93fc1e37ce2 986bd628ac48088bc723d8fbe293861ca7a9f4a73e9fa63b1b6d0074f5dea2a624c5 249ff3ad811b6255b299d6bc5451ba7477f19c5a0db690c3e6476398b1483d10314a fd38bbaf6e2fbdbcd62c3ca9797a420ca6034ec0a83360a3ee2adf4b9d4ba29731d1 31b099a38d6a23cc463db754603211260e99d19affc902c915d7854554aabf608e3a c52c19b8aa26ae042249b17b2d29669b5c859103ee53ef9bdc73ba3c6b537d5c34b6 d8f034671d7f3a8a6966cc4543df223565343154140fd7391c7e7be03e241f4ecfeb 877a051 msg = 8f3dc6fb8c4a02f4d6352edf0907822c1210a9b32f9bdda4c45a698c80023a a6b59f8cfec5fdbb36331372ebefedae7d msg_prefix = dfe133d11c7ebf54943e86f56323bd16d89a43a8d2aaaa257cbe5a9 4f59d436c random_msg = dfe133d11c7ebf54943e86f56323bd16d89a43a8d2aaaa257cbe5a9 4f59d436c8f3dc6fb8c4a02f4d6352edf0907822c1210a9b32f9bdda4c45a698c800 23aa6b59f8cfec5fdbb36331372ebefedae7d salt = 051722b35f458781397c3a671a7d3bd3096503940e4c4f1aaa269d60300ce 449555cd7340100df9d46944c5356825abf encoded_msg = 6e0c464d9c2f9fbc147b43570fc4f238e0d0b38870b3addcf7a421 7df912ccef17a7f629aa850f63a063925f312d61d6437be954b45025e8282f9c0b11 31bc8ff19a8a928d859b37113db1064f92a27f64761c181c1e1f9b251ae5a2f8a404 7573b67a270584e089beadcb13e7c82337797119712e9b849ff56e04385d144d3ca9 d8d92bf78adb20b5bbeb3685f17038ec6afade3ef354429c51c687b45a7018ee3a69 66b3af15c9ba8f40e6461ba0a17ef5a799672ad882bab02b518f9da7c1a962945c2e 9b0f02f29b31b9cdf3e633f9d9d2a22e96e1de28e25241ca7dd04147112f57897340 3e0f4fd80865965475d22294f065e17a1c4a201de93bd14223e6b1b999fd548f2f75 9f52db71964528b6f15b9c2d7811f2a0a35d534b8216301c47f4f04f412cae142b48 c4cdff78bc54df690fd43142d750c671dd8e2e938e6a440b2f825b6dbb3e19f1d7a3 c0150428a47948037c322365b7fe6fe57ac88d8f80889e9ff38177bad8c8d8d98db4 2908b389cb59692a58ce275aa15acb032ca951b3e0a3404b7f33f655b7c7d83a2f8d 1b6bbff49d5fcedf2e030e80881aa436db27a5c0dea13f32e7d460dbf01240c2320c 2bb5b3225b17145c72d61d47c8f84d1e19417ebd8ce3638a82d395cc6f7050b6209d 9283dc7b93fecc04f3f9e7f566829ac41568ef799480c733c09759aa9734e2013d76 40dc6151018ea902bc inv = 80682c48982407b489d53d1261b19ec8627d02b8cda5336750b8cee332ae26 0de57b02d72609c1e0e9f28e2040fc65b6f02d56dbd6aa9af8fde656f70495dfb723 ba01173d4707a12fddac628ca29f3e32340bd8f7ddb557cf819f6b01e445ad96f874 ba235584ee71f6581f62d4f43bf03f910f6510deb85e8ef06c7f09d9794a008be7ff 2529f0ebb69decef646387dc767b74939265fec0223aa6d84d2a8a1cc912d5ca25b4 e144ab8f6ba054b54910176d5737a2cff011da431bd5f2a0d2d66b9e70b39f4b050e 45c0d9c16f02deda9ddf2d00f3e4b01037d7029cd49c2d46a8e1fc2c0c17520af1f4 b5e25ba396afc4cd60c494a4c426448b35b49635b337cfb08e7c22a39b256dd032c0 0adddafb51a627f99a0e1704170ac1f1912e49d9db10ec04c19c58f420212973e0cb 329524223a6aa56c7937c5dffdb5d966b6cd4cbc26f3201dd25c80960a1a111b3294 7bb78973d269fac7f5186530930ed19f68507540eed9e1bab8b00f00d8ca09b3f099 aae46180e04e3584bd7ca054df18a1504b89d1d1675d0966c4ae1407be325cdf623c f13ff13e4a28b594d59e3eadbadf6136eee7a59d6a444c9eb4e2198e8a974f27a39e b63af2c9af3870488b8adaad444674f512133ad80b9220e09158521614f1faadfe85 05ef57b7df6813048603f0dd04f4280177a11380fbfc861dbcbd7418d62155248dad 5fdec0991f blinded_msg = 10c166c6a711e81c46f45b18e5873cc4f494f003180dd7f115585d 871a28930259654fe28a54dab319cc5011204c8373b50a57b0fdc7a678bd74c52325 9dfe4fd5ea9f52f170e19dfa332930ad1609fc8a00902d725cfe50685c95e5b2968c 9a2828a21207fcf393d15f849769e2af34ac4259d91dfd98c3a707c509e1af55647e faa31290ddf48e0133b798562af5eabd327270ac2fb6c594734ce339a14ea4fe1b9a 2f81c0bc230ca523bda17ff42a377266bc2778a274c0ae5ec5a8cbbe364fcf0d2403 f7ee178d77ff28b67a20c7ceec009182dbcaa9bc99b51ebbf13b7d542be337172c64 74f2cd3561219fe0dfa3fb207cff89632091ab841cf38d8aa88af6891539f263adb8 eac6402c41b6ebd72984e43666e537f5f5fe27b2b5aa114957e9a580730308a5f5a9 c63a1eb599f093ab401d0c6003a451931b6d124180305705845060ebba6b0036154f cef3e5e9f9e4b87e8f084542fd1dd67e7782a5585150181c01eb6d90cb9588383738 4a5b91dbb606f266059ecc51b5acbaa280e45cfd2eec8cc1cdb1b7211c8e14805ba6 83f9b78824b2eb005bc8a7d7179a36c152cb87c8219e5569bba911bb32a1b923ca83 de0e03fb10fba75d85c55907dda5a2606bf918b056c3808ba496a4d95532212040a5 f44f37e1097f26dc27b98a51837daa78f23e532156296b64352669c94a8a855acf30 533d8e0594ace7c442 blind_sig = 364f6a40dbfbc3bbb257943337eeff791a0f290898a6791283bba581 d9eac90a6376a837241f5f73a78a5c6746e1306ba3adab6067c32ff69115734ce014 d354e2f259d4cbfb890244fd451a497fe6ecf9aa90d19a2d441162f7eaa7ce3fc4e8 9fd4e76b7ae585be2a2c0fd6fb246b8ac8d58bcb585634e30c9168a434786fe5e0b7 4bfe8187b47ac091aa571ffea0a864cb906d0e28c77a00e8cd8f6aba4317a8cc7bf3 2ce566bd1ef80c64de041728abe087bee6cadd0b7062bde5ceef308a23bd1ccc154f d0c3a26110df6193464fc0d24ee189aea8979d722170ba945fdcce9b1b4b63349980 f3a92dc2e5418c54d38a862916926b3f9ca270a8cf40dfb9772bfbdd9a3e0e089236 9c18249211ba857f35963d0e05d8da98f1aa0c6bba58f47487b8f663e395091275f8 2941830b050b260e4767ce2fa903e75ff8970c98bfb3a08d6db91ab1746c86420ee2 e909bf681cac173697135983c3594b2def673736220452fde4ddec867d40ff42dd3d a36c84e3e52508b891a00f50b4f62d112edb3b6b6cc3dbd546ba10f36b03f06c0d82 aeec3b25e127af545fac28e1613a0517a6095ad18a98ab79f68801e05c175e15bae2 1f821e80c80ab4fdec6fb34ca315e194502b8f3dcf7892b511aee45060e3994cd15e 003861bc7220a2babd7b40eda03382548a34a7110f9b1779bf3ef6011361611e6bc5 c0dc851e1509de1a sig = 6fef8bf9bc182cd8cf7ce45c7dcf0e6f3e518ae48f06f3c670c649ac737a8b 8119a34d51641785be151a697ed7825fdfece82865123445eab03eb4bb91cecf4d69 51738495f8481151b62de869658573df4e50a95c17c31b52e154ae26a04067d5ecdc 1592c287550bb982a5bb9c30fd53a768cee6baabb3d483e9f1e2da954c7f4cf492fe 3944d2fe456c1ecaf0840369e33fb4010e6b44bb1d721840513524d8e9a3519f40d1 b81ae34fb7a31ee6b7ed641cb16c2ac999004c2191de0201457523f5a4700dd64926 7d9286f5c1d193f1454c9f868a57816bf5ff76c838a2eeb616a3fc9976f65d4371de ecfbab29362caebdff69c635fe5a2113da4d4d8c24f0b16a0584fa05e80e607c5d9a 2f765f1f069f8d4da21f27c2a3b5c984b4ab24899bef46c6d9323df4862fe51ce300 fca40fb539c3bb7fe2dcc9409e425f2d3b95e70e9c49c5feb6ecc9d43442c33d5000 3ee936845892fb8be475647da9a080f5bc7f8a716590b3745c2209fe05b17992830c e15f32c7b22cde755c8a2fe50bd814a0434130b807dc1b7218d4e85342d70695a5d7 f29306f25623ad1e8aa08ef71b54b8ee447b5f64e73d09bdd6c3b7ca224058d7c67c c7551e9241688ada12d859cb7646fbd3ed8b34312f3b49d69802f0eaa11bc4211c2f 7a29cd5c01ed01a39001c5856fab36228f5ee2f2e1110811872fe7c865c42ed59029 c706195d52¶
p = e1f4d7a34802e27c7392a3cea32a262a34dc3691bd87f3f310dc756734889305 59c120fd0410194fb8a0da55bd0b81227e843fdca6692ae80e5a5d414116d4803fca 7d8c30eaaae57e44a1816ebb5c5b0606c536246c7f11985d731684150b63c9a3ad9e 41b04c0b5b27cb188a692c84696b742a80d3cd00ab891f2457443dadfeba6d6daf10 8602be26d7071803c67105a5426838e6889d77e8474b29244cefaf418e381b312048 b457d73419213063c60ee7b0d81820165864fef93523c9635c22210956e53a8d9632 2493ffc58d845368e2416e078e5bcb5d2fd68ae6acfa54f9627c42e84a9d3f277401 7e32ebca06308a12ecc290c7cd1156dcccfb2311 q = c601a9caea66dc3835827b539db9df6f6f5ae77244692780cd334a006ab353c8 06426b60718c05245650821d39445d3ab591ed10a7339f15d83fe13f6a3dfb20b945 2c6a9b42eaa62a68c970df3cadb2139f804ad8223d56108dfde30ba7d367e9b0a7a8 0c4fdba2fd9dde6661fc73fc2947569d2029f2870fc02d8325acf28c9afa19ecf962 daa7916e21afad09eb62fe9f1cf91b77dc879b7974b490d3ebd2e95426057f35d0a3 c9f45f79ac727ab81a519a8b9285932d9b2e5ccd347e59f3f32ad9ca359115e7da00 8ab7406707bd0e8e185a5ed8758b5ba266e8828f8d863ae133846304a2936ad7bc7c 9803879d2fc4a28e69291d73dbd799f8bc238385 n = aec4d69addc70b990ea66a5e70603b6fee27aafebd08f2d94cbe1250c556e047 a928d635c3f45ee9b66d1bc628a03bac9b7c3f416fe20dabea8f3d7b4bbf7f963be3 35d2328d67e6c13ee4a8f955e05a3283720d3e1f139c38e43e0338ad058a9495c533 77fc35be64d208f89b4aa721bf7f7d3fef837be2a80e0f8adf0bcd1eec5bb040443a 2b2792fdca522a7472aed74f31a1ebe1eebc1f408660a0543dfe2a850f106a617ec6 685573702eaaa21a5640a5dcaf9b74e397fa3af18a2f1b7c03ba91a6336158de420d 63188ee143866ee415735d155b7c2d854d795b7bc236cffd71542df34234221a0413 e142d8c61355cc44d45bda94204974557ac2704cd8b593f035a5724b1adf442e78c5 42cd4414fce6f1298182fb6d8e53cef1adfd2e90e1e4deec52999bdc6c29144e8d52 a125232c8c6d75c706ea3cc06841c7bda33568c63a6c03817f722b50fcf898237d78 8a4400869e44d90a3020923dc646388abcc914315215fcd1bae11b1c751fd52443aa c8f601087d8d42737c18a3fa11ecd4131ecae017ae0a14acfc4ef85b83c19fed33cf d1cd629da2c4c09e222b398e18d822f77bb378dea3cb360b605e5aa58b20edc29d00 0a66bd177c682a17e7eb12a63ef7c2e4183e0d898f3d6bf567ba8ae84f84f1d23bf8 b8e261c3729e2fa6d07b832e07cddd1d14f55325c6f924267957121902dc19b3b329 48bdead5 e = 10001 d = d43242aefe1fb2c13fbc66e20b678c4336d20b1808c558b6e62ad16a28707718 0b177e1f01b12f9c6cd6c52630257ccef26a45135a990928773f3bd2fc01a313f1da c97a51cec71cb1fd7efc7adffdeb05f1fb04812c924ed7f4a8269925dad88bd7dcfb c4ef01020ebfc60cb3e04c54f981fdbd273e69a8a58b8ceb7c2d83fbcbd6f784d052 201b88a9848186f2a45c0d2826870733e6fd9aa46983e0a6e82e35ca20a439c5ee7b 502a9062e1066493bdadf8b49eb30d9558ed85abc7afb29b3c9bc644199654a46766 81af4babcea4e6f71fe4565c9c1b85d9985b84ec1abf1a820a9bbebee0df1398aae2 c85ab580a9f13e7743afd3108eb32100b870648fa6bc17e8abac4d3c99246b1f0ea9 f7f93a5dd5458c56d9f3f81ff2216b3c3680a13591673c43194d8e6fc93fc1e37ce2 986bd628ac48088bc723d8fbe293861ca7a9f4a73e9fa63b1b6d0074f5dea2a624c5 249ff3ad811b6255b299d6bc5451ba7477f19c5a0db690c3e6476398b1483d10314a fd38bbaf6e2fbdbcd62c3ca9797a420ca6034ec0a83360a3ee2adf4b9d4ba29731d1 31b099a38d6a23cc463db754603211260e99d19affc902c915d7854554aabf608e3a c52c19b8aa26ae042249b17b2d29669b5c859103ee53ef9bdc73ba3c6b537d5c34b6 d8f034671d7f3a8a6966cc4543df223565343154140fd7391c7e7be03e241f4ecfeb 877a051 msg = 8f3dc6fb8c4a02f4d6352edf0907822c1210a9b32f9bdda4c45a698c80023a a6b59f8cfec5fdbb36331372ebefedae7d random_msg = 8f3dc6fb8c4a02f4d6352edf0907822c1210a9b32f9bdda4c45a698 c80023aa6b59f8cfec5fdbb36331372ebefedae7d salt = encoded_msg = 159499b90471b496c2639ec482e99feaba525c0420c565d17dc60c 1bb1f47703f04436cceaa8f69811e1bf8546fa971226c9e71421b32b571ed5ea0e03 2269d4219b4404316eb17a58f277634aeed394b7f3888153b5bb163e40807e605daf dd1789dd473b0846bdcb6524417bc3a35366fab4261708c0e4b4beba07a1a64bbccb 4b1ac215d1350a50a501e8e96612028b535ad731abf1f117ee07d07a4de9cef3d70f 5845ba84c29d5d92c6e66a1f9489a5f527b846825360fd6e90f40ed041c682e489f3 acde984a3ea580181418c1d15017af2657bc4b70485cdc0f1ebc3693e0d70a5d01f3 7ff640993fa071274fb9ee44e0c24dcb58ffa21a9a6540d87f24379beaafcc3b4bd4 2c45ec6820e03738ce98bea11c71685f31db63429fab8658bdb816f1ecccb1888f24 02de0bd2f0f9646decdcad4c11b41428eec1ed25f2a86d43bb04f95726bfbd98ea34 ca091b7adbabd0e28f17fa0345b89542d23c3530554987508a23641bd4f9e52962b0 bee3ac9ffe005322d26a39941c5847774300411c69635f96903e8d593530908bd92a 4fa6a2d52f88073a647a4b3894b7e4ebb80699e60227397bfa93f41b1c97e107b632 f68e70409372ead2f072c11cf99be4486fcbf763dde28ee156db26cd358a69fcb796 44f1f2fcc166f41a4c80f5851ee08be051f14b601418d6e56e61733b9b210c6bef17 edac121a754d19b9bc inv = 80682c48982407b489d53d1261b19ec8627d02b8cda5336750b8cee332ae26 0de57b02d72609c1e0e9f28e2040fc65b6f02d56dbd6aa9af8fde656f70495dfb723 ba01173d4707a12fddac628ca29f3e32340bd8f7ddb557cf819f6b01e445ad96f874 ba235584ee71f6581f62d4f43bf03f910f6510deb85e8ef06c7f09d9794a008be7ff 2529f0ebb69decef646387dc767b74939265fec0223aa6d84d2a8a1cc912d5ca25b4 e144ab8f6ba054b54910176d5737a2cff011da431bd5f2a0d2d66b9e70b39f4b050e 45c0d9c16f02deda9ddf2d00f3e4b01037d7029cd49c2d46a8e1fc2c0c17520af1f4 b5e25ba396afc4cd60c494a4c426448b35b49635b337cfb08e7c22a39b256dd032c0 0adddafb51a627f99a0e1704170ac1f1912e49d9db10ec04c19c58f420212973e0cb 329524223a6aa56c7937c5dffdb5d966b6cd4cbc26f3201dd25c80960a1a111b3294 7bb78973d269fac7f5186530930ed19f68507540eed9e1bab8b00f00d8ca09b3f099 aae46180e04e3584bd7ca054df18a1504b89d1d1675d0966c4ae1407be325cdf623c f13ff13e4a28b594d59e3eadbadf6136eee7a59d6a444c9eb4e2198e8a974f27a39e b63af2c9af3870488b8adaad444674f512133ad80b9220e09158521614f1faadfe85 05ef57b7df6813048603f0dd04f4280177a11380fbfc861dbcbd7418d62155248dad 5fdec0991f blinded_msg = 982790826556aabe6004467671a864397eea3b95740e9a11c8b80b 99ee0cf4dbc50af860bda81b601a2eceaa6943ef104f13325ad0be2e37f42030b312 0e87cfee8cfe59cde1acfb25485a43275ebe777292e2518181ae531e596f988ff16f 458daa5a42408939cbe60e7271391a21657276427d195bee6a20054101d4ceb892ec dea402ea1a866acf0e451a3336f07e7589330d96c3883fd5bc1a829a715b618b74a8 6b2a898764246ad081d4c9f1edb8ab5077e315fde2417ec2dd33cad93e120340b49b e89c18a63e62c6bb289037283d3bf18608be11ee4c823c710b0c6b89235fed3f03a7 b96ddd25a8f54f20dac37ce8905093ad8e066810f354fb1773236e3d3788ba755de2 c9bce8d340078bb1831ddc7314a5018673427ced65cb356281aae08b5e6636f3eb24 17e09d6ae476a9abcc410bc8c90813d0740e39ae75efae4c02eed49dbb7aa51258bb 71197445d17a6029bf566ba6b36282173af2c42e9b9631366f22eb6a19ef1d92bd3c e0631d3a7fb3288195b0ba380a3828d5411cefd5eba83e52198c001ac9946a333a33 d89d4d235fc833239d59837f04eaf065e9563659b00c7624a6263b727d8f2c07959b a2bb592e7ff251b8f09c85995fd2e4474e743586576b518230986b6076b762ae7708 8a37e4bffd2ef41ae68d6d4e79205290b4f76c42ef039638c41cdc6fe8af9b429c0d ee45b2942e3861da2a blind_sig = 362ef369f9b8c1487e285514702a7cd6fe03e4a2fb854881f3d3f986 b7742a0c9bfab6562a6cd5ed71c574af67d7e77e71b33420c08ebb0ff37886b85829 7f9562fc366066c6d8e77bad1918b04756ba03f5c385d44f06759daf1b7a38b2a642 48dee95d0e3886c8afa1f74afd8ac3c56520d0f3fd206df8e0d257312756803b09a7 9d0cc38112592c3aec32de5a9bc3284c5a0a2d0808b102deafa5cc60f04e3d71c028 4cba04f17f88aa8e07d5544fe0265807d515877f79d30ed26d522b9d9c56597647b0 dbca5a69d6418f8d1b51481723f272c2a3d48f6f4fd6beeac3576c3edb00e8779964 548aeab8e004c7c4f8ef9cb6e680e2d2d49792004bb3e6974fa48f241a361ca449c0 2bd4c0ad4e66252c55e656f16049908efe59acbafa1171895dfac64d909808e54204 69d622c7253ec1de7522b41634d383bf8786bf881cbf1561627f1e62b2d93300ec30 ec0f5f0ab32036fce068bc76b0b0c6452079537f8d7f8dcee4b42bbf2d9ad7499d38 35cd93cfc7e8ebea3554ab5241e181e5d73241b7bebf0a281b63594a35f4993e2b41 6d60db966b58b648cfcba2c4bee4c2830aae4a70ff55012480298f549c13b1b26842 77bca12f592471b8a99285174f1c0ebb38fc80e74a10b3f02ec3e6682ba873f7ff0e 1e79718b470927c74ed754d4f7c3d9a55e22246e829cdb5a1c6fb2a0a6c896df3030 63c918bcf5eb0017 sig = 4454b6983ff01cb28545329f394936efa42ed231e15efbc025fdaca00277ac f0c8e00e3d8b0ecebd35b057b8ebfc14e1a7097368a4abd20b555894ccef3d1b9528 c6bcbda6b95376bef230d0f1feff0c1064c62c60a7ae7431d1fdfa43a81eed9235e3 63e1ffa0b2797aba6aad6082fcd285e14fc8b71de6b9c87cb4059c7dc1e96ae1e637 95a1e9af86b9073d1d848aef3eca8a03421bcd116572456b53bcfd4dabb0a9691f1f abda3ed0ce357aee2cfee5b1a0eb226f69716d4e011d96eede5e38a9acb531a64336 a0d5b0bae3ab085b658692579a376740ff6ce69e89b06f360520b864e33d82d029c8 08248a19e18e31f0ecd16fac5cd4870f8d3ebc1c32c718124152dc905672ab0b7af4 8bf7d1ac1ff7b9c742549c91275ab105458ae37621757add83482bbcf779e777bbd6 1126e93686635d4766aedf5103cf7978f3856ccac9e28d21a850dbb03c811128616d 315d717be1c2b6254f8509acae862042c034530329ce15ca2e2f6b1f5fd59272746e 3918c748c0eb810bf76884fa10fcf749326bbfaa5ba285a0186a22e4f628dbf178d3 bb5dc7e165ca73f6a55ecc14c4f5a26c4693ce5da032264cbec319b12ddb9787d0ef a4fcf1e5ccee35ad85ecd453182df9ed735893f830b570faae8be0f6fe2e571a4e0d 927cba4debd368d3b4fca33ec6251897a137cf75474a32ac8256df5e5ffa518b88b4 3fb6f63a24¶
p = e1f4d7a34802e27c7392a3cea32a262a34dc3691bd87f3f310dc756734889305 59c120fd0410194fb8a0da55bd0b81227e843fdca6692ae80e5a5d414116d4803fca 7d8c30eaaae57e44a1816ebb5c5b0606c536246c7f11985d731684150b63c9a3ad9e 41b04c0b5b27cb188a692c84696b742a80d3cd00ab891f2457443dadfeba6d6daf10 8602be26d7071803c67105a5426838e6889d77e8474b29244cefaf418e381b312048 b457d73419213063c60ee7b0d81820165864fef93523c9635c22210956e53a8d9632 2493ffc58d845368e2416e078e5bcb5d2fd68ae6acfa54f9627c42e84a9d3f277401 7e32ebca06308a12ecc290c7cd1156dcccfb2311 q = c601a9caea66dc3835827b539db9df6f6f5ae77244692780cd334a006ab353c8 06426b60718c05245650821d39445d3ab591ed10a7339f15d83fe13f6a3dfb20b945 2c6a9b42eaa62a68c970df3cadb2139f804ad8223d56108dfde30ba7d367e9b0a7a8 0c4fdba2fd9dde6661fc73fc2947569d2029f2870fc02d8325acf28c9afa19ecf962 daa7916e21afad09eb62fe9f1cf91b77dc879b7974b490d3ebd2e95426057f35d0a3 c9f45f79ac727ab81a519a8b9285932d9b2e5ccd347e59f3f32ad9ca359115e7da00 8ab7406707bd0e8e185a5ed8758b5ba266e8828f8d863ae133846304a2936ad7bc7c 9803879d2fc4a28e69291d73dbd799f8bc238385 n = aec4d69addc70b990ea66a5e70603b6fee27aafebd08f2d94cbe1250c556e047 a928d635c3f45ee9b66d1bc628a03bac9b7c3f416fe20dabea8f3d7b4bbf7f963be3 35d2328d67e6c13ee4a8f955e05a3283720d3e1f139c38e43e0338ad058a9495c533 77fc35be64d208f89b4aa721bf7f7d3fef837be2a80e0f8adf0bcd1eec5bb040443a 2b2792fdca522a7472aed74f31a1ebe1eebc1f408660a0543dfe2a850f106a617ec6 685573702eaaa21a5640a5dcaf9b74e397fa3af18a2f1b7c03ba91a6336158de420d 63188ee143866ee415735d155b7c2d854d795b7bc236cffd71542df34234221a0413 e142d8c61355cc44d45bda94204974557ac2704cd8b593f035a5724b1adf442e78c5 42cd4414fce6f1298182fb6d8e53cef1adfd2e90e1e4deec52999bdc6c29144e8d52 a125232c8c6d75c706ea3cc06841c7bda33568c63a6c03817f722b50fcf898237d78 8a4400869e44d90a3020923dc646388abcc914315215fcd1bae11b1c751fd52443aa c8f601087d8d42737c18a3fa11ecd4131ecae017ae0a14acfc4ef85b83c19fed33cf d1cd629da2c4c09e222b398e18d822f77bb378dea3cb360b605e5aa58b20edc29d00 0a66bd177c682a17e7eb12a63ef7c2e4183e0d898f3d6bf567ba8ae84f84f1d23bf8 b8e261c3729e2fa6d07b832e07cddd1d14f55325c6f924267957121902dc19b3b329 48bdead5 e = 10001 d = d43242aefe1fb2c13fbc66e20b678c4336d20b1808c558b6e62ad16a28707718 0b177e1f01b12f9c6cd6c52630257ccef26a45135a990928773f3bd2fc01a313f1da c97a51cec71cb1fd7efc7adffdeb05f1fb04812c924ed7f4a8269925dad88bd7dcfb c4ef01020ebfc60cb3e04c54f981fdbd273e69a8a58b8ceb7c2d83fbcbd6f784d052 201b88a9848186f2a45c0d2826870733e6fd9aa46983e0a6e82e35ca20a439c5ee7b 502a9062e1066493bdadf8b49eb30d9558ed85abc7afb29b3c9bc644199654a46766 81af4babcea4e6f71fe4565c9c1b85d9985b84ec1abf1a820a9bbebee0df1398aae2 c85ab580a9f13e7743afd3108eb32100b870648fa6bc17e8abac4d3c99246b1f0ea9 f7f93a5dd5458c56d9f3f81ff2216b3c3680a13591673c43194d8e6fc93fc1e37ce2 986bd628ac48088bc723d8fbe293861ca7a9f4a73e9fa63b1b6d0074f5dea2a624c5 249ff3ad811b6255b299d6bc5451ba7477f19c5a0db690c3e6476398b1483d10314a fd38bbaf6e2fbdbcd62c3ca9797a420ca6034ec0a83360a3ee2adf4b9d4ba29731d1 31b099a38d6a23cc463db754603211260e99d19affc902c915d7854554aabf608e3a c52c19b8aa26ae042249b17b2d29669b5c859103ee53ef9bdc73ba3c6b537d5c34b6 d8f034671d7f3a8a6966cc4543df223565343154140fd7391c7e7be03e241f4ecfeb 877a051 msg = 8f3dc6fb8c4a02f4d6352edf0907822c1210a9b32f9bdda4c45a698c80023a a6b59f8cfec5fdbb36331372ebefedae7d random_msg = 8f3dc6fb8c4a02f4d6352edf0907822c1210a9b32f9bdda4c45a698 c80023aa6b59f8cfec5fdbb36331372ebefedae7d salt = encoded_msg = 159499b90471b496c2639ec482e99feaba525c0420c565d17dc60c 1bb1f47703f04436cceaa8f69811e1bf8546fa971226c9e71421b32b571ed5ea0e03 2269d4219b4404316eb17a58f277634aeed394b7f3888153b5bb163e40807e605daf dd1789dd473b0846bdcb6524417bc3a35366fab4261708c0e4b4beba07a1a64bbccb 4b1ac215d1350a50a501e8e96612028b535ad731abf1f117ee07d07a4de9cef3d70f 5845ba84c29d5d92c6e66a1f9489a5f527b846825360fd6e90f40ed041c682e489f3 acde984a3ea580181418c1d15017af2657bc4b70485cdc0f1ebc3693e0d70a5d01f3 7ff640993fa071274fb9ee44e0c24dcb58ffa21a9a6540d87f24379beaafcc3b4bd4 2c45ec6820e03738ce98bea11c71685f31db63429fab8658bdb816f1ecccb1888f24 02de0bd2f0f9646decdcad4c11b41428eec1ed25f2a86d43bb04f95726bfbd98ea34 ca091b7adbabd0e28f17fa0345b89542d23c3530554987508a23641bd4f9e52962b0 bee3ac9ffe005322d26a39941c5847774300411c69635f96903e8d593530908bd92a 4fa6a2d52f88073a647a4b3894b7e4ebb80699e60227397bfa93f41b1c97e107b632 f68e70409372ead2f072c11cf99be4486fcbf763dde28ee156db26cd358a69fcb796 44f1f2fcc166f41a4c80f5851ee08be051f14b601418d6e56e61733b9b210c6bef17 edac121a754d19b9bc inv = 80682c48982407b489d53d1261b19ec8627d02b8cda5336750b8cee332ae26 0de57b02d72609c1e0e9f28e2040fc65b6f02d56dbd6aa9af8fde656f70495dfb723 ba01173d4707a12fddac628ca29f3e32340bd8f7ddb557cf819f6b01e445ad96f874 ba235584ee71f6581f62d4f43bf03f910f6510deb85e8ef06c7f09d9794a008be7ff 2529f0ebb69decef646387dc767b74939265fec0223aa6d84d2a8a1cc912d5ca25b4 e144ab8f6ba054b54910176d5737a2cff011da431bd5f2a0d2d66b9e70b39f4b050e 45c0d9c16f02deda9ddf2d00f3e4b01037d7029cd49c2d46a8e1fc2c0c17520af1f4 b5e25ba396afc4cd60c494a4c426448b35b49635b337cfb08e7c22a39b256dd032c0 0adddafb51a627f99a0e1704170ac1f1912e49d9db10ec04c19c58f420212973e0cb 329524223a6aa56c7937c5dffdb5d966b6cd4cbc26f3201dd25c80960a1a111b3294 7bb78973d269fac7f5186530930ed19f68507540eed9e1bab8b00f00d8ca09b3f099 aae46180e04e3584bd7ca054df18a1504b89d1d1675d0966c4ae1407be325cdf623c f13ff13e4a28b594d59e3eadbadf6136eee7a59d6a444c9eb4e2198e8a974f27a39e b63af2c9af3870488b8adaad444674f512133ad80b9220e09158521614f1faadfe85 05ef57b7df6813048603f0dd04f4280177a11380fbfc861dbcbd7418d62155248dad 5fdec0991f blinded_msg = 982790826556aabe6004467671a864397eea3b95740e9a11c8b80b 99ee0cf4dbc50af860bda81b601a2eceaa6943ef104f13325ad0be2e37f42030b312 0e87cfee8cfe59cde1acfb25485a43275ebe777292e2518181ae531e596f988ff16f 458daa5a42408939cbe60e7271391a21657276427d195bee6a20054101d4ceb892ec dea402ea1a866acf0e451a3336f07e7589330d96c3883fd5bc1a829a715b618b74a8 6b2a898764246ad081d4c9f1edb8ab5077e315fde2417ec2dd33cad93e120340b49b e89c18a63e62c6bb289037283d3bf18608be11ee4c823c710b0c6b89235fed3f03a7 b96ddd25a8f54f20dac37ce8905093ad8e066810f354fb1773236e3d3788ba755de2 c9bce8d340078bb1831ddc7314a5018673427ced65cb356281aae08b5e6636f3eb24 17e09d6ae476a9abcc410bc8c90813d0740e39ae75efae4c02eed49dbb7aa51258bb 71197445d17a6029bf566ba6b36282173af2c42e9b9631366f22eb6a19ef1d92bd3c e0631d3a7fb3288195b0ba380a3828d5411cefd5eba83e52198c001ac9946a333a33 d89d4d235fc833239d59837f04eaf065e9563659b00c7624a6263b727d8f2c07959b a2bb592e7ff251b8f09c85995fd2e4474e743586576b518230986b6076b762ae7708 8a37e4bffd2ef41ae68d6d4e79205290b4f76c42ef039638c41cdc6fe8af9b429c0d ee45b2942e3861da2a blind_sig = 362ef369f9b8c1487e285514702a7cd6fe03e4a2fb854881f3d3f986 b7742a0c9bfab6562a6cd5ed71c574af67d7e77e71b33420c08ebb0ff37886b85829 7f9562fc366066c6d8e77bad1918b04756ba03f5c385d44f06759daf1b7a38b2a642 48dee95d0e3886c8afa1f74afd8ac3c56520d0f3fd206df8e0d257312756803b09a7 9d0cc38112592c3aec32de5a9bc3284c5a0a2d0808b102deafa5cc60f04e3d71c028 4cba04f17f88aa8e07d5544fe0265807d515877f79d30ed26d522b9d9c56597647b0 dbca5a69d6418f8d1b51481723f272c2a3d48f6f4fd6beeac3576c3edb00e8779964 548aeab8e004c7c4f8ef9cb6e680e2d2d49792004bb3e6974fa48f241a361ca449c0 2bd4c0ad4e66252c55e656f16049908efe59acbafa1171895dfac64d909808e54204 69d622c7253ec1de7522b41634d383bf8786bf881cbf1561627f1e62b2d93300ec30 ec0f5f0ab32036fce068bc76b0b0c6452079537f8d7f8dcee4b42bbf2d9ad7499d38 35cd93cfc7e8ebea3554ab5241e181e5d73241b7bebf0a281b63594a35f4993e2b41 6d60db966b58b648cfcba2c4bee4c2830aae4a70ff55012480298f549c13b1b26842 77bca12f592471b8a99285174f1c0ebb38fc80e74a10b3f02ec3e6682ba873f7ff0e 1e79718b470927c74ed754d4f7c3d9a55e22246e829cdb5a1c6fb2a0a6c896df3030 63c918bcf5eb0017 sig = 4454b6983ff01cb28545329f394936efa42ed231e15efbc025fdaca00277ac f0c8e00e3d8b0ecebd35b057b8ebfc14e1a7097368a4abd20b555894ccef3d1b9528 c6bcbda6b95376bef230d0f1feff0c1064c62c60a7ae7431d1fdfa43a81eed9235e3 63e1ffa0b2797aba6aad6082fcd285e14fc8b71de6b9c87cb4059c7dc1e96ae1e637 95a1e9af86b9073d1d848aef3eca8a03421bcd116572456b53bcfd4dabb0a9691f1f abda3ed0ce357aee2cfee5b1a0eb226f69716d4e011d96eede5e38a9acb531a64336 a0d5b0bae3ab085b658692579a376740ff6ce69e89b06f360520b864e33d82d029c8 08248a19e18e31f0ecd16fac5cd4870f8d3ebc1c32c718124152dc905672ab0b7af4 8bf7d1ac1ff7b9c742549c91275ab105458ae37621757add83482bbcf779e777bbd6 1126e93686635d4766aedf5103cf7978f3856ccac9e28d21a850dbb03c811128616d 315d717be1c2b6254f8509acae862042c034530329ce15ca2e2f6b1f5fd59272746e 3918c748c0eb810bf76884fa10fcf749326bbfaa5ba285a0186a22e4f628dbf178d3 bb5dc7e165ca73f6a55ecc14c4f5a26c4693ce5da032264cbec319b12ddb9787d0ef a4fcf1e5ccee35ad85ecd453182df9ed735893f830b570faae8be0f6fe2e571a4e0d 927cba4debd368d3b4fca33ec6251897a137cf75474a32ac8256df5e5ffa518b88b4 3fb6f63a24¶
We would like to thank Bjoern Tackmann who provided an editorial and security review of this document.¶