Independent Submission F. Hao, Ed.
Request for Comments: 8235 Newcastle University (UK)
Category: Informational August 2017
ISSN: 2070-1721
Schnorr NIZK Proof: Non-interactive Zero-Knowledge Proof for a Discrete Logarithm
Abstract
This document describes the Schnorr NIZK non-interactive zero-knowledge
(NIZK) proof, a non-interactive variant of the three-pass Schnorr
identification scheme. The Schnorr NIZK proof allows one to prove
the knowledge of a discrete logarithm without leaking any information
about its value. It can serve as a useful building block for many
cryptographic protocols to ensure that participants follow the
protocol specification honestly. This document specifies the Schnorr
NIZK proof in both the finite field and the elliptic curve settings.
Status of This Memo
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1. Requirements Language . . . . . . . . . . . . . . . . . . 3
1.2. Notation . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Schnorr NIZK Proof over Finite Field . . . . . . . . . . . . 4
2.1. Group Parameters . . . . . . . . . . . . . . . . . . . . 4
2.2. Schnorr Identification Scheme . . . . . . . . . . . . . . 4
2.3. Non-interactive Zero-Knowledge Proof . . . . . . . . . . 5
2.4. Computation Cost . . . . . . . . . . . . . . . . . . . . 6
3. Schnorr NIZK Proof over Elliptic Curve . . . . . . . . . . . 6
3.1. Group Parameters . . . . . . . . . . . . . . . . . . . . 6
3.2. Schnorr Identification Scheme . . . . . . . . . . . . . . 7
3.3. Non-interactive Zero-Knowledge Proof . . . . . . . . . . 8
3.4. Computation Cost . . . . . . . . . . . . . . . . . . . . 8
4. Variants of Schnorr NIZK proof . . . . . . . . . . . . . . . 8
5. Applications of Schnorr NIZK proof . . . . . . . . . . . . . 9
6. Security Considerations . . . . . . . . . . . . . . . . . . . 9 10
7. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 11
8. References . . . . . . . . . . . . . . . . . . . . . . . . . 11
8.1. Normative References . . . . . . . . . . . . . . . . . . 11
8.2. Informative References . . . . . . . . . . . . . . . . . 12
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . 12 13
Author's Address . . . . . . . . . . . . . . . . . . . . . . . . 12 13
1. Introduction
A well-known principle for designing robust public key protocols is
as follows: "Do not assume that a message you receive has a
particular form (such as g^r for known r) unless you can check this"
[AN95]. This is the sixth of the eight principles defined by Ross
Anderson and Roger Needham at Crypto '95. Hence, it is also known as
the "sixth principle". In the past thirty years, many public key
protocols failed to prevent attacks, which can be explained by the
violation of this principle [Hao10].
While there may be several ways to satisfy the sixth principle, this
document describes one technique that allows one to prove the
knowledge of a discrete logarithm (e.g., r for g^r) without revealing
its value. This technique is called the Schnorr NIZK proof, which is
a non-interactive variant of the three-pass Schnorr identification
scheme [Stinson06]. The original Schnorr identification scheme is
made non-interactive through a Fiat-Shamir transformation [FS86],
assuming that there exists a secure cryptographic hash function
(i.e., the so-called random oracle model).
The Schnorr NIZK proof can be implemented over a finite field or an
elliptic curve (EC). The technical specification is basically the
same, except that the underlying cyclic group is different. For
completeness, this document describes the Schnorr NIZK proof in both
the finite field and the EC settings.
1.1. Requirements Language
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.
1.2. Notation
The following notation is used in this document:
o Alice: the assumed identity of the prover in the protocol
o Bob: the assumed identity of the verifier in the protocol
o a | b: a divides b
o a || b: concatenation of a and b
o [a, b]: the interval of integers between and including a and b
o t: the bit length of the challenge chosen by Bob
o H: a secure cryptographic hash function
o p: a large prime
o q: a large prime divisor of p-1, i.e., q | p-1
o Zp*: a multiplicative group of integers modulo p
o Gq: a subgroup of Zp* with prime order q
o g: a generator of Gq
o g^x: g^d: g raised to the power of x d
o a mod b: a modulo b
o Fp: a finite field of p elements, where p is a prime
o E(Fp): an elliptic curve defined over Fp
o G: a generator of the subgroup over E(Fp) with prime order n
o n: the order of G
o h: the cofactor of the subgroup generated by G, which is equal to
the order of the elliptic curve divided by n
o P x [b]: multiplication of a point P with a scalar b over E(Fp)
2. Schnorr NIZK Proof over Finite Field
2.1. Group Parameters
When implemented over a finite field, the Schnorr NIZK proof may use
the same group setting as DSA [FIPS186-4]. Let p and q be two large
primes with q | p-1. Let Gq denote the subgroup of Zp* of prime
order q, and g be a generator for the subgroup. Refer to the DSA
examples in the NIST Cryptographic Toolkit [NIST_DSA] for values of
(p, q, g) that provide different security levels. A level of 128-bit
security or above is recommended. Here, DSA groups are used only as
an example. Other multiplicative groups where the discrete logarithm
problem (DLP) is intractable are also suitable for the implementation
of the Schnorr NIZK proof.
2.2. Schnorr Identification Scheme
The Schnorr identification scheme runs interactively between Alice
(prover) and Bob (verifier). In the setup of the scheme, Alice
publishes her public key X A = g^x g^a mod p, where x a is the private key
chosen uniformly at random from [0, q-1].
The protocol works in three passes:
1. Alice chooses a number v uniformly at random from [0, q-1] and
computes V = g^v mod p. She sends V to Bob.
2. Bob chooses a challenge c uniformly at random from [0, 2^t-1],
where t is the bit length of the challenge (say, t = 160). Bob
sends c to Alice.
3. Alice computes b r = v - x a * c mod q and sends it to Bob.
At the end of the protocol, Bob performs the following checks. If
any check fails, the identification is unsuccessful.
1. To verify X A is within [1, p-1] and X^q A^q = 1 mod p;
2. To verify V = g^r * X^c A^c mod p.
The first check ensures that X A is a valid public key, hence the
discrete logarithm of X A with respect to the base g actually exists.
It is worth noting that some applications may specifically exclude
the identity element as a valid public key. In that case, one shall
check X A is within [2, p-1] instead of [1, p-1].
The process is summarized in the following diagram.
Alice Bob
------- -----
choose random v from [0, q-1]
compute V = g^v mod p -- V ->
compute b r = v-x*c v-a*c mod q <- c -- choose random c from [0, 2^t-1]
-- b -> check 1) X A is a valid public key
2) V = g^b g^r * X^c A^c mod p
Information Flows in Schnorr Identification Scheme over Finite Field
2.3. Non-interactive Zero-Knowledge Proof
The Schnorr NIZK proof is obtained from the interactive Schnorr
identification scheme through a Fiat-Shamir transformation [FS86].
This transformation involves using a secure cryptographic hash
function to issue the challenge instead. More specifically, the
challenge is redefined as c = H(g || g^v V || g^x A || UserID || OtherInfo),
where UserID is a unique identifier for the prover and OtherInfo is
optional data. Here, the hash function H shall be a secure
cryptographic hash function, e.g., SHA-256, SHA-384, SHA-512,
SHA3-256, SHA3-384, or SHA3-512. The bit length of the hash output
should be at least equal to that of the order q of the considered
subgroup.
The
OtherInfo is defined to allow flexible inclusion of contextual
information (also known as "labels" in [ABM15]) in the Schnorr NIZK
proof so that the technique defined in this document can be generally
useful. For example, some security protocols built on top of the
Schnorr NIZK proof may wish to include more contextual information
such as the protocol name, timestamp, and so on. The exact items (if
any) in OtherInfo shall be left to specific protocols to define.
However, the format of OtherInfo in any specific protocol must be
fixed and explicitly defined in the protocol specification.
Within the hash function, there must be a clear boundary between any
two concatenated items. It is recommended that one should always
prepend each item with a 4-byte integer that represents the byte
length of that item. The OtherInfo may contain multiple subitems. In
that case, the same rule shall apply to ensure a clear boundary
between adjacent subitems.
2.4. Computation Cost
In summary, to prove the knowledge of the exponent for X A = g^x, g^a, Alice
generates a Schnorr NIZK proof that contains: {UserID, OtherInfo, V =
g^v mod p, r = v - x*c a*c mod q}, where c = H(g || g^v V || g^x A || UserID ||
OtherInfo).
To generate a Schnorr NIZK proof, the cost is roughly one modular
exponentiation: that is to compute g^v mod p. In practice, this
exponentiation may be precomputed in the offline manner to optimize
efficiency. The cost of the remaining operations (random number
generation, modular multiplication, and hashing) is negligible as
compared with the modular exponentiation.
To verify the Schnorr NIZK proof, the cost is approximately two
exponentiations: one for computing X^q A^q mod p and the other for
computing g^r * X^c A^c mod p. (It takes roughly one exponentiation to
compute the latter using a simultaneous exponentiation technique as
described in [MOV96].)
3. Schnorr NIZK Proof over Elliptic Curve
3.1. Group Parameters
When implemented over an elliptic curve, the Schnorr NIZK proof may
use the same EC setting as ECDSA [FIPS186-4]. For the illustration
purpose, only curves over the prime fields (e.g., NIST P-256) are
described here. Other curves over the binary fields (see
[FIPS186-4]) that are suitable for ECDSA can also be used for
implementing the Schnorr NIZK proof. Let E(Fp) be an elliptic curve
defined over a finite field Fp, where p is a large prime. Let G be a
base point on the curve that serves as a generator for the subgroup
over E(Fp) of prime order n. The cofactor of the subgroup is denoted
h, which is usually a small value (not more than 4). Details on EC
operations, such as addition, negation and scalar multiplications,
can be found in [MOV96]. Date Data types and conversions including
elliptic-curve-point-to-octet-string and vice versa can be found in
Section 2.3 of [SEC1]. Here, the NIST curves are used only as an
example. Other secure curves such as Curve25519 are also suitable
for the implementation as long as the elliptic curve discrete
logarithm problem (ECDLP) remains intractable.
3.2. Schnorr Identification Scheme
In the setup of the scheme, Alice publishes her public key
Q
A = G x [x], [a], where x a is the private key chosen uniformly at random
from [1, n-1].
The protocol works in three passes:
1. Alice chooses a number v uniformly at random from [1, n-1] and
computes V = G x [v]. She sends V to Bob.
2. Bob chooses a challenge c uniformly at random from [0, 2^t-1],
where t is the bit length of the challenge (say, t = 80). Bob
sends c to Alice.
3. Alice computes b r = v - x a * c mod n and sends it to Bob.
At the end of the protocol, Bob performs the following checks. If
any check fails, the verification is unsuccessful.
1. To verify X A is a valid point on the curve and X A x [h] is not the
point at infinity;
2. To verify V = G x [b] [r] + Q A x [c].
The first check ensures that X A is a valid public key, hence the
discrete logarithm of X A with respect to the base G actually exists.
Unlike in the DSA-like group setting where a full modular
exponentiation is required to validate a public key, in the ECDSA-
like setting, the public key validation incurs almost negligible cost
due to the cofactor being small (e.g., 1, 2, or 4).
The process is summarized in the following diagram.
Alice Bob
------- -----
choose random v from [1, n-1]
compute V = G x [v] -- V ->
compute b r = v - x a * c mod n <- c -- choose random c from [0, 2^t-1]
-- b -> check 1) X A is a valid public key
2) V = G x [b] [r] + Q A x [c]
Information Flows in Schnorr Identification Scheme
over Elliptic Curve
3.3. Non-interactive Zero-Knowledge Proof
Same as before, the non-interactive variant is obtained through a
Fiat-Shamir transformation [FS86], by using a secure cryptographic
hash function to issue the challenge instead. The challenge c is
defined as c = H(G || V || Q A || UserID || OtherInfo), where UserID is
a unique identifier for the prover and OtherInfo is optional data as
explained earlier.
3.4. Computation Cost
In summary, to prove the knowledge of the discrete logarithm for Q A =
G x [x] [a] with respect to base G over the elliptic curve, Alice
generates a Schnorr NIZK proof that contains: {UserID, OtherInfo, V =
G x [v], r = v - x*c a*c mod n}, where c = H(G || V || Q A || UserID ||
OtherInfo).
To generate a Schnorr NIZK proof, the cost is one scalar
multiplication: that is to compute G x [v].
To verify the Schnorr NIZK proof in the EC setting, the cost is
approximately one multiplication over the elliptic curve: i.e.,
computing G x [r] + Q A x [c] (using the same simultaneous computation
technique as before). The cost of public key validation in the EC
setting is essentially free.
4. Variants of Schnorr NIZK proof
In the finite field setting, the prover sends (V, b) r) (along with
UserID and OtherInfo), and the verifier first computes c, and then
checks for V = g^b g^r * X^c A^c mod p. This requires the transmission of an
element V of Zp, whose size is typically between 2048 and 3072 bits,
and an element b r of Zq whose size is typically between 224 and 256
bits. It is possible to reduce the amount of transmitted data to two
elements of Zq as below.
In the modified variant, the prover works exactly the same as before,
except that it sends (c, b) r) instead of (V, b). r). The verifier computes
V = g^b g^r * X^c A^c mod p and then checks whether H(g || V || g^x A || UserID
|| OtherInfo) = c. The security of this modified variant follows
from the fact that one can compute V from (c, b) r) and c from (V, b). r).
Therefore, sending (c, b) r) is equivalent to sending (V, c,
b), r), which
in turn is equivalent to sending (V, b). r). Thus, the size of the
Schnorr NIZK proof is significantly reduced. However, the
computation costs for both the prover and the verifier stay the same.
The same optimization technique also applies to the elliptic curve
setting by replacing (V, b) r) with (c, b), r), but the benefit is extremely
limited. When V is encoded in the compressed form, this optimization
only saves 1 bit. The computation costs for generating and verifying
the NIZK proof remain the same as before.
5. Applications of Schnorr NIZK proof
Some key exchange protocols, such as J-PAKE [HR08] and YAK [Hao10],
rely on the Schnorr NIZK proof to ensure participants have the
knowledge of discrete logarithms, hence following the protocol
specification honestly. The technique described in this document can
be directly applied to those protocols.
The inclusion of OtherInfo also makes the Schnorr NIZK proof
generally useful and flexible to cater for a wide range of
applications. For example, the described technique may be used to
allow a user to demonstrate the proof of possession (PoP) of a long-
term private key to a Certification Authority (CA) during the public
key registration phrase. It must be ensured that the hash contains
data that links the proof to one particular key registration
procedure, e.g.,
procedure (e.g., by including the CA name, the expiry date, the
applicant's email contact contact, and so on to the OtherInfo. on, in OtherInfo). In this case,
the Schnorr NIZK proof is functionally equivalent to a self-signed
Certificate Signing Request generated by using DSA or ECDSA.
6. Security Considerations
The Schnorr identification protocol has been proven to satisfy the
following properties, assuming that the verifier is honest and the
discrete logarithm problem is intractable (see [Stinson06]).
1. Completeness -- a prover who knows the discrete logarithm is
always able to pass the verification challenge.
2. Soundness -- an adversary who does not know the discrete
logarithm has only a negligible probability (i.e., 2^(-t)) to
pass the verification challenge.
3. Honest verifier zero-knowledge -- a prover leaks no more than one
bit of information to the honest verifier: whether the prover
knows the discrete logarithm.
The Fiat-Shamir transformation is a standard technique to transform a
three-pass interactive Zero-Knowledge Proof protocol (in which the
verifier chooses a random challenge) to a non-interactive one,
assuming that there exists a secure cryptographic hash function.
Since the hash function is publicly defined, the prover is able to
compute the challenge by itself, hence making the protocol non-
interactive. In this case, the hash function (more precisely, the
random oracle in the security proof) implements an honest verifier,
because it assigns a uniformly random challenge c to each commitment
(g^v or G x [v]) sent by the prover. This is exactly what an honest
verifier would do.
It is important to note that in Schnorr's identification scheme and
its non-interactive variant, a secure random number generator is
required. In particular, bad randomness in v may reveal the secret
discrete logarithm. For example, suppose the same random value V =
g^v mod p is used twice by the prover (e.g., because its random
number generator failed), but the verifier chooses different
challenges c and c' (or the hash function is used on two different
OtherInfo data, producing two different values c and c'). The
adversary now observes two proof transcripts (V, c, b) r) and (V, c',
b'),
r'), based on which he can compute the secret key x a by:
(b-b')/(c'-c)
(r-r')/(c'-c) = (v-x*c-v+x*c')/(c'-c) (v-a*c-v+a*c')/(c'-c) = x a mod q.
More generally, such an attack may even work for a slightly better
(but still bad) random number generator, where the value v is not
repeated, but the adversary knows a relation between two values v and
v' such as v' = v + w for some known value w. Suppose the adversary
observes two proof transcripts (V, c, b) r) and (V', c', b'). r'). He can
compute the secret key x a by:
(b-b'+w)/(c'-c)
(r-r'+w)/(c'-c) = (v-x*c-v-w+x*c'+w)/(c'-c) (v-a*c-v-w+a*c'+w)/(c'-c) = x a mod q.
This example reinforces the importance of using a secure random
number generator to generate the ephemeral secret v in Schnorr's
schemes.
Finally, when a security protocol relies on the Schnorr NIZK proof
for proving the knowledge of a discrete logarithm in a non-
interactive way, the threat of replay attacks shall be considered.
For example, the Schnorr NIZK proof might be replayed back to the
prover itself (to introduce some undesirable correlation between
items in a cryptographic protocol). This particular attack is
prevented by the inclusion of the unique UserID in the hash. The
verifier shall check the prover's UserID is a valid identity and is
different from its own. Depending on the context of specific
protocols, other forms of replay attacks should be considered, and
appropriate contextual information included in OtherInfo whenever
necessary.
7. IANA Considerations
This document does not require any IANA actions.
8. References
8.1. Normative References
[ABM15] Abdalla, M., Benhamouda, F., and P. MacKenzie, "Security
of the J-PAKE Password-Authenticated Key Exchange
Protocol", 2015 IEEE Symposium on Security and Privacy,
DOI 10.1109/sp.2015.41, May 2015.
[AN95] Anderson, R. and R. Needham, "Robustness principles for
public key protocols", Proceedings of the 15th Annual
International Cryptology Conference on Advances in
Cryptology, DOI 10.1007/3-540-44750-4_19, 1995.
[FS86] Fiat, A. and A. Shamir, "How to Prove Yourself: Practical
Solutions to Identification and Signature Problems",
Proceedings of the 6th Annual International Cryptology
Conference on Advances in Cryptology,
DOI 10.1007/3-540-47721-7_12, 1986.
[MOV96] Menezes, A., Oorschot, P., and S. Vanstone, "Handbook of
Applied Cryptography", 1996.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
<http://www.rfc-editor.org/info/rfc2119>.
<https://www.rfc-editor.org/info/rfc2119>.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, <http://www.rfc-editor.org/info/rfc8174>. <https://www.rfc-editor.org/info/rfc8174>.
[SEC1] "Standards for Efficient Cryptography. SEC 1: Elliptic
Curve Cryptography", SECG SEC1-v2, May 2009,
<http://www.secg.org/sec1-v2.pdf>.
[Stinson06]
Stinson, D., "Cryptography: Theory and Practice", 3rd
Edition, CRC, 2006.
8.2. Informative References
[FIPS186-4]
National Institute of Standards and Technology, "Digital
Signature Standard (DSS)", FIPS PUB 186-4,
DOI 10.6028/NIST.FIPS.186-4, July 2013,
<http://nvlpubs.nist.gov/nistpubs/FIPS/
NIST.FIPS.186-4.pdf>.
[Hao10] Hao, F., "On Robust Key Agreement Based on Public Key
Authentication", 14th International Conference on
Financial Cryptography and Data Security,
DOI 10.1007/978-3-642-14577-3_33, February 2010.
[HR08] Hao, F. and P. Ryan, "Password Authenticated Key Exchange
by Juggling", Lecture Notes in Computer Science Science, pp.
159-171, from 16th Security Protocols Workshop (SPW'08),
DOI 10.1007/978-3-642-22137-8_23, 2011.
[NIST_DSA]
NIST Cryptographic Toolkit, "DSA Examples",
<http://csrc.nist.gov/groups/ST/toolkit/documents/
Examples/DSA2_All.pdf>.
Acknowledgements
The editor of this document would like to thank Dylan Clarke, Robert
Ransom, Siamak Shahandashti, Robert Cragie, Stanislav Smyshlyaev, and
Tibor Jager for many useful comments. Tibor Jager contributed pointed out the
optimization technique and discussion the vulnerability issue when the ephemeral
secret v is not generated randomly. This work is supported by the
EPSRC First Grant (EP/J011541/1) and the ERC Starting Grant (No.
306994).
Author's Address
Feng Hao (editor)
Newcastle University (UK)
Claremont Tower,
Urban Sciences Building, School of Computing Science, Computing, Newcastle University
Newcastle Upon Tyne
United Kingdom
Phone: +44 (0)191-208-6384
Email: feng.hao@ncl.ac.uk