Abstract
This chapter deals with multivariate signature schemes following the concept of Oil and Vinegar. After introducing the basic (balanced) Oil and Vinegar (OV) signature scheme, we describe its cryptanalysis by the incvariant subspace attack of Kipnis and Shamir and how this attack is prevented in the unbalanced Oil and Vinegar scheme (UOV). We then introduce the signature scheme Rainbow, which can be seen as multi-layer version of UOV and discuss its security and efficiency. Finally, we present techniques to reduce the public key size of UOV and Rainbow.
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Notes
- 1.
An encryption scheme of the SingleField type is presented in Chap. 7.
- 2.
The probability that the linear system of o equations in o variables does not have a solution is about 1∕q, where q is the size of the underlying field. Therefore, for reasonably large q (e.g. q ∈{31, 256}), we usually find a solution at the first try.
- 3.
If T (1) is not invertible, we can switch rows and columns of T by renumbering the variables until we get an invertible matrix.
- 4.
It may happen, that one of the linear systems does not have a solution. If so, one has to choose other values of \(x_1, \dots x_{v_1}\) and try again. However, the probability of this is very small. Therefore, in most cases, one gets a pre-image x at the first try.
- 5.
If S (4) is not invertible, we can switch the rows and columns of S by renumbering the equations of \(\mathcal F\) until we get an invertible matrix.
- 6.
Again it is possible to switch rows and columns of T by renumbering the variables.
- 7.
Experiments have shown that this is the case with high probability. Therefore, in most cases, we need only one run of the loop in line 1 to 4 of Algorithm 5.7.
- 8.
Experiments have shown that, for randomly chosen matrices S and T, the conditions are fulfilled with high probability. Therefore, in most cases, we need only one run of the loop in line 1 to 5 of Algorithm 5.8.
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Ding, J., Petzoldt, A., Schmidt, D.S. (2020). Oil and Vinegar. In: Multivariate Public Key Cryptosystems. Advances in Information Security, vol 80. Springer, New York, NY. https://doi.org/10.1007/978-1-0716-0987-3_5
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