Schnorr NIZK Proof: Non-interactive Zero Knowledge Proof for Discrete Logarithm

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" . 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 . 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 . The original Schnorr identification scheme is made non-interactive through a Fiat-Shamir transformation , 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.

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 .

The following notations are used in this document: Alice: the assumed identity of the prover in the protocol Bob: the assumed identity of the verifier in the protocol a || b: concatenation of a and b t: the bit length of the challenge chosen by Bob H: a secure cryptographic hash function p: a large prime q: a large prime divisor of p-1, i.e., q | p-1 Zp*: a multiplicative group of integers modulo p Gq: a subgroup of Zp* with prime order q g: a generator of Gq g^x: g raised to the power of x a mod b: a modulo b Fq: a finite field of q elements where q is a prime E(Fq): an elliptic curve defined over Fq G: a generator of the subgroup over E(Fq) with prime order n n: the order of G h: the cofactor of the subgroup generated by G, as defined by h = |E(Fq)|/n P x [b]: multiplication of a point P with a scalar b over E(Fq) P.x: the x coordinate of a point P over E(Fq)

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 for values of (p, q, g) that satisfy different security levels.

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: Alice chooses a number v uniformly at random from [0, q-1] and computes V = g^v mod p. She sends V to Bob. 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. 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.

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? ]]>

The Schnorr NIZK proof is obtained from the interactive Schnorr identification scheme through a Fiat-Shamir transformation . 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.

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 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. To verify X is within [1, p-1] and X^q = 1 mod p 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 .) 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 ).

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 . 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 .

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: Alice chooses a number v uniformly at random from [1, n-1] and computes V = G x [v]. She sends V to Bob. 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. 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.

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]? ]]>

Same as before, the non-interactive variant is obtained through a Fiat-Shamir transformation , 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.

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]. To verify the Schnorr NIZK proof in the EC setting, the following computations shall be performed. To verify Q is a valid public key in the subgroup over E(Fq) 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 ). 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).

Some key exchange protocols, such as J-PAKE and YAK , 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).

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 ). Completeness -- a prover who knows the discrete logarithm is always able to pass the verification challenge. Soundness -- an adversary who does not know the discrete logarithm has only a negligible probability (i.e., 2^(-t)) to pass the verification challenge. Honest verifier zero-knowledge -- a prover leaks no more than one bit 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. 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 ) 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.

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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).