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How much do you know about RSA Cryptography?

September 21, 2020

Whether you’re browsing TikTok, Instagram, or scrolling through random Wikipedia articles, do you ever wonder if your internet browsing is secure? Probably not – but that’s okay! The world can peacefully rely on the wonders of mathematics to keep their data and internet browsing safe from peeking eyes. In fact, the entire safety of the internet primarily relies on a single mathematical algorithm to secure it all, ranging from state secrets in secret government databases – which are probably all about UFOs – to your grandmother’s email account. This algorithm was first developed in 1977; it’s so powerful that it still remains the primary public cryptographic algorithm today! It’s called the Rivest-Shamir-Adleman (RSA) cryptosystem.

Background

Developed in 1977 from a paper released by Ron Rivest, Adi Shamir, and Leonard Adleman, while they were at the Massachusetts Institute of Technology, RSA quickly became adapted to securing digital communication because of its robust mathematical properties that makes it virtually impossible to break.

Interestingly, the actual mathematics behind the RSA cryptosystem is rather simple. It’s based on the idea of prime factorization – it’s really easy to find the prime factors for a small number but becomes much harder as the numbers grow. For example, if I ask you to give me the two prime factors that multiply together to give the number 15, you would probably answer really quickly with “3 and 5”. Easy, right? Now, without a calculator, what if I asked you to give me the prime factorization for the number 414,863? You would answer with “577 and 719” right away, right? Unless you’re a super math genius or a human calculator, probably not.

Implementation Idea

The actual RSA algorithm is used in public-key cryptography, which is the process that secures data over the internet. It’s a complicated process but the basic idea is this: if Alice wants to send Bob a secret gift, Bob can send her a padlock that only he has the key to. Then, Alice can put her gift in a box and lock it with Bob’s padlock. Then, when Alice sends her package, if anyone tries to steal the gift, they can’t because only Bob has the key. Once Bob receives his gift, all he has to do is unlock the padlock that he gave to Alice. The padlock that Bob gave to Alice is called Bob’s public key, while the key that only Bob has is his private key.

In the real world, if my friend wants to send me a message, I will give them my public key (which is basically just a really large number). They can then use my public key to encrypt, or hide, the message. Then, in transit, the message is completely safe because I’m the only person that holds the key that decrypts, or unlocks, the message. The only tricky part about the process is making keys that are not able to be broken.

The Magic Behind It All

The RSA algorithm is responsible for creating the public and private keys. To create the actual keys, RSA relies on some basic math: modular arithmetic, Euler’s Theorem, and prime numbers.

To create my public key that I can give to anyone who wants to send me secrets, I do two things:

  • First, I will choose two very large prime numbers, p and q. The multiplication of these two numbers, x = pq, is the first half of the key. The key take-away in this first portion is that the longer that x is, the more secure your key will be. As of 2020 practices, a length of 4096 bits (or characters) is recommended!
  • Then, I calculate another number y by multiplying from the following: y = (p-1)(q-1). From here, I choose a number that is relatively prime to y and call it e. If you don’t recall what it means for two numbers to be relatively prime, it just means that two numbers have a greatest common divisor of 1. In practice, e is typically chosen to be 65,537 – e then becomes the other half of my public key.

Now, to create my private key that no one else knows about, I use modular arithmetic. If you’re not familiar with modular arithmetic, think of it as the remainder produced after dividing a small number into a big number – for example, 10 modulo 3 = 1 because 3*3 is 9, which leaves us with 1 left over. To create my private key, I calculate the modular inverse of the second half of my public key with the first half of my public key, e modulo x. That’s it! I have my private key! Now I can share my public key with friends, and they can share juicy, secret memes with me without the fear of someone else seeing!

Convinced?

What makes RSA so robust and secure is that, even as computers gain more processing power and become faster, all we have to do is increase the length of the key! Think about how fast and powerful our current computers are: our computers are solving problems in astrophysics, mathematics, and are beginning to become ubiquitous in controlling cars. Even with how advanced technology currently is, if someone uses a 4096-bit RSA key, it would take tens of thousands of years for computer to break the code! And even when that’s broken, all we have to do is increase the number of bits in the RSA key and it’s secure for another few thousand years or so!

If you’re not convinced by how amazing (and even basic) cryptography is, then I don’t think you’re someone who can be easily impressed!

The Process in Pictures

If you prefer pictures, here is a simple illustration of the process above:

Step 1:

Step 2:

Step 3:

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Step 5:

Step 6:

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About Author

Zack is a 4th year Computer Science student at Georgia Tech. He specializes in cybersecurity and computational theory. In his free time, Zack likes to stay in shape, spend time with friends, and read about the latest frightening cyber threats.

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  Timothy Durham was convicted of rape when a test showed his DNA genotype matched the DNA recovered from the crime scene. Eye witnesses testified that Durham was in a different state at the time of the incident, but he wasn’t exonerated from his 3000 year sentence until follow up DNA analysis showed that the preliminary forensic analysis used in the trial was misinterpreted (Thomas et al.). DNA testing serves an invaluable part in our criminal justice system, but it comes with complications and bias that can lead to wrongful convictions and misinformed juries.  

Background:

Forensic genetics aims to identify the origin of a biological sample. It can be traced back to Karl Landsteiner who discovered human ABO blood groups. With blood groups, suspects could be eliminated based on their blood type. For example, if a suspect had type A blood and there was type B blood left at the scene by the perpetrator, the investigators could rule out the suspect. Unfortunately, the reverse didn’t work. If the suspect and perpetrator both had the same blood type, it can’t be concluded that the suspect was guilty. Blood identifiers evolved into 8 distinct categories which improved the ability to eliminate suspects, but could only conclude that (if the system matched) it might be the perpetrator or this person might be related to the perpetrator (Jobling and Gill). The field evolved in the 1980s with the discovery of Minisatellites, lines of repetitive DNA that don’t encode proteins. Minisatellite patterns in DNA create what we know today as the DNA fingerprint, and it has a much higher accuracy rate than previous methods (Augustyn et al.). Around the same time, a breakthrough for victims of sexual assault was discovered. In the case of assault, bodily fluids of both the victim and the assaulter can get mixed. A process called differential lysis was created which could selectively increase the concentration of sperm in a fluid sample thereby allowing for accurate DNA testing (Jobling and Gill). Forensics evolved to use single-locus probe testing after several years because of its ease of interpretation. It looked at specific minisatellites that had high variability between subjects. This was the first DNA-based method to be used in a criminal investigation. Two murder homicides from several years apart were connected using DNA evidence. The evidence was then used to exonerate an innocent man, and the guilty party was found when he tried to evade mandatory DNA testing for all men in the small town. His guilt was then reinforced when the DNA profiles matched, and he gave a confession (Jobling and Gill). Finding DNA at a crime scene doesn’t guarantee that you’ll find a murderer. There are issues with non-pure samples and limited samples that make DNA work difficult, and accreditation of facilities and new techniques and take time (Jobling and Gill).  

Subjectivity of DNA Interpretation:

In a 2009 publication, the National Academy of Science discusses the possibility for subjective influence in lab based forensics. The interpretation bias happens in complex cases where, for example, there could be a mixture of DNA. A study published in 2011 looked specifically at this issue. A gang rape happened in Georgia, and the mixture in DNA from the case was used to prosecute one suspect after another suspect took a plea bargain to testify against him. With DNA mixtures, there are three possible conclusions: suspect excluded, suspect cannot be excluded, and inconclusive. If the suspect is excluded, it means that there’s little to no possibility that his DNA is part of the sample. If the suspect cannot be excluded, it means that there’s a possibility that his DNA is part of the sample. In the trial, the DNA examiners who knew the context of the case concluded that the suspect being prosecuted could not be excluded. The 2011 study gave the DNA evidence from this case and the DNA profiles of the suspects to 17 forensic examiners. These examiners had no context of the case, and they all worked independently. Of the 17, only 1 came to the same conclusion (suspect cannot be excluded) as the examiners who knew the context of the case. Twelve of the 17 concluded that the suspect who was prosecuted and convicted had no relation to the DNA found in the sample. Not only does this show that subjectivity occurs due to the context of the case, but the 17 examiners came up with different findings based on their past experiences meaning that not all DNA analysis is objective. Some organizations like the Scientific Working Group on DNA Analysis Methods say that quantitative statistical evidence should be presented with every categorical conclusion, but the US doesn’t require this (Dror and Hampikian).

False Positives:

There are two categories that cause wrongful identification in a DNA report: coincidental matches and false positives. Coincidental matches are DNA matches between two unrelated people. The probability of these matches is about 1 to 500,000 for white individuals and about 1 to 350,000 for black individuals. This small probability of coincidental matches makes people feel confident in DNA evidence. False positives, on the other hand, are matches between DNA and test DNA that result from issues in the forensics lab. There have been multiple cases of forensics labs accidentally switching victim and suspected rapist DNA which lead to convictions that were later overturned when the lab realized the error. There are also cases of misreading or misunderstanding a sample. In a case from 1993, a man was convicted of rape because someone interpreted a mixed sample of victim and perpetrator DNA as an individual sample. The conviction was again overturned. To prevent false convictions, retesting DNA is allowed, but it’s not an efficient solution. Many people are pushing for statistical information on the probability of false positives, so that the jurors can make accurate judgements based on DNA evidence (Thomas et al.).  

Conclusion:

DNA analysis is a useful tool for the criminal justice system, but subjectivity can be an issue in forensic testing. To combat wrongful convictions, there are organizations like The Innocence Project who use DNA testing to free the innocent, but reformed forensic testing standards should be put in place to prevent subjectivity and to ensure juries are correctly informed during trials. https://www.innocenceproject.org/    

Citation:

Augustyn , Adam, et al. “DNA Finerprinting.” Encyclopedia Britannica , 26 Apr. 2018, www.britannica.com/science/DNA-fingerprinting#ref153186. Dror, Itiel E., and Greg Hampikian. “Subjectivity and Bias in Forensic DNA Mixture Interpretation.” Science & Justice, vol. 51, no. 4, 2011, pp. 204–208., doi:10.1016/j.scijus.2011.08.004. Jobling, Mark A., and Peter Gill. “Encoded Evidence: DNA in Forensic Analysis.” Nature Reviews: Genetics, vol. 5, no. 10, Oct. 2004, pp. 739–751., doi:10.1038/nrg1455. Thompson, William C, et al. “How the Probability of a False Positive Affects the Value of DNA Evidence.” Journal of Forensic Sciences, vol. 48, no. 1, Jan. 2003, pp. 1–8., doi:10.1520/jfs2001171.