Internet Engineering Task Force F. Hao, Ed.
Internet-Draft Newcastle University (UK)
Intended status: Informational July 6, 2016
Expires: January 7, 2017
Schnorr NIZK Proof: Non-interactive Zero Knowledge Proof for Discrete
Logarithm
draft-hao-schnorr-04
Abstract
This document describes Schnorr 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 the
participants follow the protocol specification honestly. This
document specifies the Schnorr NIZK proof in both the finite field
and the elliptic curve settings.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1. Requirements Language . . . . . . . . . . . . . . . . . . 3
1.2. Notations . . . . . . . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . 5
3. Schnorr NIZK Proof over Elliptic Curve . . . . . . . . . . . 6
3.1. Group Parameters . . . . . . . . . . . . . . . . . . . . 6
3.2. Schnorr Identification Scheme . . . . . . . . . . . . . . 6
3.3. Non-Interactive Zero-Knowledge Proof . . . . . . . . . . 7
3.4. Computation Cost . . . . . . . . . . . . . . . . . . . . 7
4. Applications of Schnorr NIZK proof . . . . . . . . . . . . . 8
5. Security Considerations . . . . . . . . . . . . . . . . . . . 8
6. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 9
7. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 9
8. References . . . . . . . . . . . . . . . . . . . . . . . . . 9
8.1. Normative References . . . . . . . . . . . . . . . . . . 9
8.2. Informative References . . . . . . . . . . . . . . . . . 10
8.3. URIs . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Author's Address . . . . . . . . . . . . . . . . . . . . . . . . 10
1. Introduction
A well-known principle for designing robust public key protocols
states 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],
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assuming that there exists a secure cryptographic hash function
(i.e., 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", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in RFC 2119 [RFC2119].
1.2. Notations
The following notations are 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: concatenation of 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 raised to the power of x
o a mod b: a modulo b
o Fq: a finite field of q elements where q is a prime
o E(Fq): an elliptic curve defined over Fq
o G: a generator of the subgroup over E(Fq) with prime order n
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o n: the order of G
o h: the cofactor of the subgroup generated by G, as defined by h
= |E(Fq)|/n
o P x [b]: multiplication of a point P with a scalar b over E(Fq)
o P.x: the x coordinate of a point P over E(Fq)
2. Schnorr NIZK Proof over Finite Field
2.1. Group Parameters
When implemented over a finite field, the Schnorr NIZK Proof uses the
same group setting as DSA. 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 NIST [1] for values of (p, q,
g) that satisfy different security levels.
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 = g^x mod p where x is the private key
chosen uniformly at random from [0, q-1]. The value X must be an
element in the subgroup Gq, which anyone can verify. This is to
ensure that the discrete logarithm of X with respect to the base g
actually exists.
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 = 80). Bob
sends c to Alice.
3. Alice computes b = v - x * c mod q and sends it to Bob.
At the end of the protocol, Bob checks if the following equality
holds: V = g^b * X^c mod p. The verification succeeds only if the
equality holds. The process is summarized in the following diagram.
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Information Flows in Schnorr Identification Scheme
Alice Bob
------- -----
choose random v from [0, q-1]
compute V = g^v mod p -- V ->
compute b = v-x*c mod q <- c -- choose random c from [0, 2^t-1]
-- b -> check if V = g^b * X^c mod p?
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 || g^x || UserID ||
OtherInfo), where UserID is a unique identifier for the prover and
OtherInfo is optional data. The OtherInfo is included here for
generality, as 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 the
concatenated items. Usually, the boundary is implicitly defined once
the length of each item is publicly known. However, in the general
case, it is safer to define the boundary explicitly. It is
recommended that one should always prepend each item with a 4-byte
integer that represents the byte length of the item. The OtherInfo
may contain multiple sub-items. In that case, the same rule shall
apply to ensure a clear boundary between adjacent sub-items.
2.4. Computation Cost
In summary, to prove the knowledge of the exponent for X = g^x, Alice
generates a Schnorr NIZK proof that contains: {UserID, OtherInfo, V =
g^v mod p, r = v - x*c mod q}, where c = H(g || g^v || g^x ||
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 pre-computed in the off-line manner to optimize
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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 following computations shall be
performed.
1. To verify X is within [1, p-1] and X^q = 1 mod p
2. To verify V = g^r * X^c mod p
Hence, the cost of verifying a Schnorr NIZK proof is approximately
two exponentiations: one for computing X^q mod p and the other for
computing g^r * X^c mod p. (It takes roughly one exponentiation to
compute the latter using a simultaneous exponentiation technique as
described in [MOV96].)
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 is within [2, p-1] instead of [1, p-1]. Also note that in
the DSA-like group setting, it requires a full modular exponentiation
to validate a public key, but in the ECDSA-like setting, the public
key validation incurs almost negligible cost due to the cofactor
being very small (see [MOV96]).
3. Schnorr NIZK Proof over Elliptic Curve
3.1. Group Parameters
When implemented over an elliptic curve, the Schnorr NIZK proof uses
essentially the same EC setting as ECDSA, e.g., NIST P-256, P-384,
and P-521 [NISTCurve]. Let E(Fq) be an elliptic curve defined over a
finite field Fq where q is a large prime. Let G be a base point on
the curve that serves as a generator for the subgroup over E(Fq) 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].
3.2. Schnorr Identification Scheme
In the setup of the scheme, Alice publishes her public key Q = G x
[x] where x is the private key chosen uniformly at random from [1,
n-1]. The value Q must be an element in the subgroup over the
elliptic curve, which anyone can verify.
The protocol works in three passes:
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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 = v - x * c mod n and sends it to Bob.
At the end of the protocol, Bob checks if the following equality
holds: V = G x [b] + Q x [c]. The verification succeeds only if the
equality holds. The process is summarized in the following diagram.
Information Flows in Schnorr Identification Scheme
Alice Bob
------- -----
choose random v from [1, n-1]
compute V = G x [v] -- V ->
compute b = v - x * c mod n <- c -- choose random c from [0, 2^t-1]
-- b -> check if V = G x [b] + Q x [c]?
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. Note that G, V and Q
are points on the curve. In practice, it is sufficient to include
only the x coordinate of the point into the hash function. Hence,
let G.x, V.x and Q.x be the x coordinates of these points
respectively. The challenge c is defined as c = H(G.x || V.x ||
Q.x || 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 =
G x [x] 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 mod n}, where c = H(G.x || V.x || Q.x ||
UserID || OtherInfo).
To generate a Schnorr NIZK proof, the cost is one scalar
multiplication: that is to compute G x [v].
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To verify the Schnorr NIZK proof in the EC setting, the following
computations shall be performed.
1. To verify Q is a valid public key in the subgroup over E(Fq)
2. To verify V = G x [r] + Q x [c]
In the EC setting where the cofactor is small (say 1, 2 or 4),
validating the public key Q is essentially free (see [MOV96]). The
cost of verifying a Schnorr NIZK proof in the EC setting is
approximately one multiplication over the elliptic curve: i.e.,
computing G x [r] + Q x [c] (using the same simultaneous computation
technique as before).
4. 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 in the protocol
follow the specification honestly. Hence, 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 sufficiently 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 Certificate Authority (CA) during the public
key registration phrase. Accordingly, the OtherInfo should include
extra information such as the CA name, the expiry date, the
applicant's email contact and so on. In this case, the Schnorr NIZK
proof is essentially no different from a self-signed Certificate
Signing Request generated by using DSA (or ECDSA).
5. 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 information to the honest verifier: whether the prover knows
the discrete logarithm.
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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. The assumption of an honest verifier naturally holds
because the verifier can be anyone.
A non-interactive Zero Knowledge Proof is often called a signature
scheme. However, it should be noted that the Schnorr NIZK proof
described in this document is different from the original Schnorr
signature scheme (see [Stinson06]) in that it is specifically
designed as a proof of knowledge of the discrete logarithm rather
than a general-purpose digital signing algorithm.
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 into the hash. The verifier shall
check the prover's UserID is a valid identity and is different from
its own. Depending the context of specific protocols, other forms of
replay attacks should be considered, and appropriate contextual
information included into OtherInfo whenever necessary.
6. IANA Considerations
This document has no actions for IANA.
7. Acknowledgements
The editor of this document would like to thank Dylan Clarke, Robert
Ransom, Siamak Shahandashti and Robert Cragie for useful comments.
This work is supported by the EPSRC First Grant (EP/J011541/1) and
the ERC Starting Grant (No. 306994).
8. References
8.1. Normative References
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
.
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[AN95] Anderson, R. and R. Needham, "Robustness principles for
public key protocols", Proceedings of the 15th Annual
International Cryptology Conference on Advances in
Cryptology, 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, 1986.
[MOV96] Menezes, A., Oorschot, P., and S. Vanstone, "Handbook of
Applied Cryptography", 1996.
[Stinson06]
Stinson, D., "Cryptography: Theory and Practice (3rd
Edition)", CRC, 2006.
8.2. Informative References
[NISTCurve]
"Recommended Elliptic Curves for Federal Government use",
July 1999,
.
[HR08] Hao, F. and P. Ryan, "Password Authenticated Key Exchange
by Juggling", the 16th Workshop on Security Protocols,
May 2008.
[Hao10] Hao, F., "On Robust Key Agreement Based on Public Key
Authentication", the 14th International Conference on
Financial Cryptography and Data Security, February 2010.
8.3. URIs
[1] http://csrc.nist.gov/groups/ST/toolkit/documents/Examples/
DSA2_All.pdf
Author's Address
Feng Hao (editor)
Newcastle University (UK)
Claremont Tower, School of Computing Science, Newcastle University
Newcastle Upon Tyne
United Kingdom
Phone: +44 (0)191-208-6384
EMail: feng.hao@ncl.ac.uk
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