As special types of factorization of finite groups, logarithmic signature and cover have been used as the main components of cryptographic keys for secret key cryptosystems such as
With the interdisciplinary development of information science, physical science, and biological science, a lot of new technology appeared in the field of cryptography and has made new progress. The new branches of cryptography mainly consist of quantum cryptography, chaotic cryptography, DNA cryptography, and so forth. The security of quantum cryptography is based on the Heisenberg uncertainty principle. Quantum cryptography is the only one that can realize unconditional security at present [
Meanwhile, cryptographers look forward to applying new intractable mathematical problems in classical cryptography. Currently, most public cryptographic primitives are based on the perceived intractability of certain mathematical problems in very large finite abelian groups [
Our main contribution is to devise a digital signature scheme based on random covers and logarithmic signatures. In this process, we also construct a secure and more efficient encryption scheme based on
The rest of contents are organized as follows. Necessary preliminaries are given in Section
Let
Suppose that
Let a a
The sequences
More generally, if
Let
Then the surjective (bijection)
In [
Furthermore, motivated by attacks in [
Until now, the only instantiation of
Furthermore, the inverse of an element in group
Through comparison and analysis, we find that it is rather difficult to devise signature schemes based on the two
(1) Choose a tame logarithmic signature (2) Select a random cover (3) Choose (4) Construct a homomorphism (5) Compute (6) Output public key
(1) Choose a random (2) Compute
(3) Output
(1) Compute (2) Compute (3) Output
For then using Consequently, using
(a) In general, the adversary tries to obtain
The adversary mainly attempts to compute enough values
(b) In this attack, an adversary mainly wants to utilize equivalent secret key
In this section, we utilize the encryption scheme above to construct a digital signature scheme based on random covers and logarithmic signatures.
(1) Choose a tame logarithmic signature (2) Select a random cover (3) Choose (4) Construct a homomorphisms (5) Compute (6) Define a hash function (7) Output public key
(1) Randomly select (2) Compute (3) Output
(1) Compute (2) If
Meanwhile,
Hence,
(a) Compared with the encryption scheme in Section
(b) In our signature scheme, we construct a ciphertext pair
Suppose that Eve attempts to forge a message-signature pair
Eve chooses a random number
Eve randomly selects two elements
Eve randomly chooses one pair
In this subsection, we compare
Table
Number of basic operations for one encryption/decryption.
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Factor | ||
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Encryption1 |
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— | 1 | — |
Our scheme |
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— | 1 | — | |
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Decryption2 |
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1 | — | 1 |
Our scheme |
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— | — | 1 |
Table
Number of basic operations for public key generation.
Secret key |
Secret key |
Public key |
Public key |
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— |
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— |
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— |
5
For example, when
Table
Number of basic operations for digital signature scheme.
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Factor | |
---|---|---|---|---|---|
Signature |
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— | 1 | 1 |
Verification |
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— | — | — |
Parameter size.
Public key | Secret key | Encryption | Signature | |
---|---|---|---|---|
Number of elements |
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|
2 | 2 |
In the community of cryptography based on chaos theory, a lot of efforts were focused on secret key cryptography in early years [
In this subsection, we present a toy example of signing a random element
Input: a Suzuki 2-group Output: public key
In general, let a pair
(i) For simplicity, we use a one-way function
(ii) A factorizable logarithmic signature
(1) We first construct canonical logarithmic signature
(2) Fuse blocks
(i) A random cover
(ii) Select
(iii) Construct a homomorphism
(iv) Compute
(i) Choose a message
(ii) Sample
(iii) Signature
(i) Compute
(ii) Compute
Since
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is partially supported by the National Natural Science Foundation of China (NSFC) (nos. 61103198, 61121061, 61370194) and the NSFC A3 Foresight Program (no. 61161140320).