GPS is an efficient identification (ID) scheme based on Schnorr ID scheme designed for applications where low cost devices with limited resources are used and a very-short authentication time is required. Let
P and
V be a prover and a verifier in GPS and <
g> be a multiplicative group.
P holds a secret key
s ∈ [0,
S) and publishes
I =
g-s. In each elementary round: (1)
P sends to
V x =
gr where
r is chosen randomly from [0, A), (2)
V sends to
P a random
c ∈ [0,
B), and (3)
P sends
y =
r +
cs (no modulus computation). Since there is no modular reduction on
y, a key issue is whether GPS leaks information about
s. It has been proved that GPS is
statistical zero-knowledge, if in
asymptotic sense, $\\ell$
BS/
A is
negligible, where $\\ell$ is the number of elementary rounds in one complete identification trial. In this paper, first we will show the followings. (1) We can construct a concrete attack procedure which reveals one bit of secret key
s from the specified value range of
y unless
BS/
A is negligible. We reconfirm that we must set
A extremely large compared to
BS. (2) This drawback can be avoided by modifying GPS into a new scheme, GPS
+., in which
P does not send the value of
y in the specified range where
y reveals some information about
s. GPS
+ ensures
perfect ZK only by requiring both
A >
BS and
A being a multiple of the order of
g, while it allows an honest
P to be rejected with probability at most
BS/(2
A) in one elementary round. Under the standard recommended parameters for 80-bit security where $\\ell$ = 1, |
S| = 160, and |
B| = 35, |
A| = 275 is recommended for GPS in GPS' paper. On the other hand, GPS
+ can guarantee 80-bit security and less than one false rejection on average in 100 identifications with only |
A| = 210 with the same parameters as above. In practice, this implies 275-210 = 65 bits (≈ 24%) reductions on storage requirement. We have confirmed that the reduce of
A also reduces approximately 4% of running time for online response using a certain implementation technique for GPS
+ by machine experiment.
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