 All right, so the final presentation of this session will be side-channel attacks on post-quantum signature schemes based on multivariate quadratic equations. This is work by Azen Park, who will be presenting, Kangan Sheem, Naham Koo, and Dung Gak Han. And if anyone's coming in, again, there are still seats at the front, so please come up around. But with that, we'll pass it over to Azen. Thank you for your introduction. Hi, everyone. I'm Azen Park from Kung Min University. I'm so glad to talk in chess. Today, I will talk about the side-channel vulnerability on UOV variants. This is joint work with Nims. In 1999, UOV has been proposed in Eurocrypt, and then in 2005, General Rainbow, which is a layered MQ signature scheme based on UOV, has been proposed to improve efficiency and reduce the key sizes. Since the study on efficient software implementation was proposed in 2006, there have been many studies on efficient implementation in hardware and software. In chess 2012, CPEC-Aero demonstrated the feasibility of MQ signature scheme on an ABR microprocessor. They removed the constant part of linear maps and applied linear maps like this picture to reduce the key space and run time. I call this shape chess form in this presentation. In 2017, BLN's Adder proposed a repeated UOV for smaller UOV public key in UGS.t chess form. After poor culture proposed the methods of finding keys using time and power and so on, power analysis has been carried out for many crypto-algorithms. It is not surprising that implementation of post-content algorithms are vulnerable to power analysis. Therefore, many studies for power analysis vulnerability of post-content cryptography have been studied. However, the studies of power analysis against UOV variants lack. There is only one result, exact hours. In 2017, ERO recovered the central map F on ENTTS using a fault analysis. There is no attack using only power analysis. Today, I will introduce non-invisible attacks against rainbow and UOV. Here, I will briefly explain the signature generation of rainbow. There are three secret maps, S, F, and T. Signature generation consists of three steps. The first step is to perform an inversion linear map of the input message. At this step, a matrix vector product over a field is used. The second step is to invert the central map F with the transformed messages. At this step, random values are used and linear equations are solved. The last one is to apply an inversion of the other linear map. It also uses a matrix vector product over a field. Rainbow uses field multiplications and auditions as the basic operations. As I mentioned, the second step uses random values. Because of this, rainbow generates different signatures for the same message. This picture leads to difficulty in power analysis. Power analysis generally uses the position where the fixed secret value and the random public value are computed. So, in the first step, power analysis is easily applied. However, in the calculation of F, unknown random values are used. Therefore, the power analysis cannot be easy. Because the intermediate value of T can't be calculated. However, the methods of efficiency can be vulnerable to power analysis. Our goal is to recover the secret maps of the rainbow and UV only using CPA, correlation power analysis, and algebraic key recovery attacks. We propose two attacks. For easily explained, I describe three sub-attacks. The first sub-attack explores the general field multiplication vulnerabilities. The second one is used when the linear map T has the chess form. This attack can be used for UV as well as for rainbow. The last one is used to rainbow with random linear map T after S has been recovered. We implemented the MetaExpect product on a field GF2 to the power of 8. And experiment with chip is pro-right, which is developed for side-channel analysis by coin. The algorithm used in the experiment is implemented by multiplying each loaded Y by the Iced column to reduce the number of times while loaded. Power traces were collected using 500 random messages. First, this is sub-attack 1. CPA on the S inverse is very easy because the attacker can control the message. Intermediate results are chosen as the value multiplied by each element. For example, guess times Y1 can be used on intermediate results to recover S'11. Here, guess means a hypothetical key. In the same way, after recover S'11, S'11 times Y1 plus guess times Y2 is used as an intermediate result to recover S'12. This picture shows the result of CPA for S'11. This picture represents the maximum correlation coefficient according to an increased number of traces. Because the last step also uses a MetaExpect product, this part can also be an attack point. However, it is hard to compute X' which is calculated with T because we don't know even if S is recovered. It means that to compute the intermediate value is difficult. If the rainbow or UVO is used chess form, it is possible to compute the intermediate value, so CPA is possible. For example, suppose we use the T that looks like this. We can know the values from X'5 to X'8 because of the blue scale. That is X'8 equals X8, X'7 equals X6, X'5 equals X5. So now we can attack the green part. Unlike us, we cannot guess the exact value because we do not know X'3 and X'4. However, there are positions where T'IJ and X'J are multiplied. We target these positions as intermediate results. Now we have found the green part. We can compute X'3 and X'4. Therefore, T can be recovered by finding the blue part as the previous method. This picture shows the CPA result for T'45. As you can see, we could find it. Here, I will explain a brief description about the recovery of F and T using algebraic key recovery attack. We assume that S has been found and T has been used in the general form. Because we know S and public P, we can compute S inverse P. We know green P is S inverse P and tilde T is inverse T. Green P circles tilde T equals F and certain places with zero coefficient in FK are unknown. Because we have found S and rainbow has different central maps each layer. We obtain the following equality. We can find an equivalent key T prime with high probability by solving the equations by finding P1 times or O2 linear equations with O2 times P1 plus O1 variables. Therefore, we can make a positive signature. We could find an equivalent key using this parameter in less than 1.46 milliseconds on Intel Xon CPU. In summary, the first attack can be used when the linear map has a chess form. All of the secret key S, F, and T can be recovered by a combination of sub 1 and sub 2. There is, after recovering the two linear maps S and T using power analysis, we recovered the central map F by simple calculation with public P. The second one is hybrid attack, which uses CPA and algebraic key recovery attack. It can attack rainbow-like signature skin with random linear map. Our attack can apply to other MQ signature skin. First, LUOV, which is submit to NIST using chess form. So it can be applied to our proposed attack 1. Rainbow and high MQ3, which use ASA structure, and they are computed over GF2 to the power of N. Also, so they are vulnerable to our attack 2. To avoid our attacks, you will like single-layered skin should not use the chess form. In other words, we must use the T that is removed the relation between the signature value and the intermediate values. Rainbow-like multi-layered skins can be recovered or secret maps if S is recovered. Therefore, we must focus on implementing a secure matrix vector product against PA. We can use message randomization to prevent our in the first step. That is, multiply all elements of a message by a random number R and perform a general matrix vector product. And then, we multiply the output of the matrix vector product by the inversion R. We need two times and field multiplication and field inversion. We also need a random number generation step here and means size of the vector. This is a conclusion. We propose the CPA on rainbow UV with equivalent key in chess form. And we also propose the hybrid attack on rainbow implementation with random linear maps using CPA and algebraic key recovery attacks. Our attack can apply to other MQ signature scams and we propose a simple countermeasure against first order CPA. We will about more efficient countermeasures and about security analysis against high order and fault injection attacks. That's it. Thank you for your attention. Thank you very much for that presentation. So we have time for a few questions if there are any from the audience. I had a short question on the countermeasure. You, whoops, killing the chairs up here, sorry. On the countermeasure you had there, what sort of performance overhead? Overhead? Yeah, was it compared to the unprotected? Even roughly as a... We only calculated multiplication on two times and field multiplication and one inversion. Field inversion. Okay. Thank you. There's no other questions. I guess I'll thank the speaker again for this wonderful presentation. So we are now on break until 10.50. There's coffee break. There's the poster session in the back. So these are new posters, I believe. So please go check those out and be back at 10.50 for session nine post-quantum cryptography two.