Fastest Rsa Key Generation Time
An 'RSA operation,' whether for encrypting or decrypting, signing or verifying, is essentially a modular exponentiation, which can be performed by a series of modular multiplications.
The size of its secret keys and the running time. 2.1 Algorithm 1. Generation of the keys: Generate two random primes p, q, and let n = pkq. Compute L = LCM (p- 1, q- 1), and find e, d which satisfies ed - 1 (rood L) and GCD(e,p) = 1. Then e, n are public keys, and d,p, q are the secret keys. Oct 02, 2015 Modulus Length. When you generate RSA keys, you will be prompted to enter a modulus length. The longer the modulus, the stronger the security. However, a longer modules take longer to generate (see the table below for sample times) and takes longer to use. The size of Key Modulus range from 360 to 2048.
In practical applications, it is common to choose a small public exponent for the public key; in fact, entire groups of users can use the same public exponent, each with a different modulus. (There are some restrictions on the prime factors of the modulus when the public exponent is fixed.) This makes encryption faster than decryption and verification faster than signing. With typical modular exponentiation algorithms, public-key operations take O(k2) steps, private-key operations take O( k3) steps, and key generation takes O(k4) steps, where k is the number of bits in the modulus. ( O-notation refers to the upper bound on the asymptotic running time of an algorithm.) 'Fast multiplication' techniques, such as FFT-based methods, require asymptotically fewer steps, though in practice they are not as common due to their great software complexity and the fact that they may actually be slower for typical key sizes.
There are many commercially available software and hardware implementations of RSA, and there are frequent announcements of newer and faster chips. On a 90 MHz Pentium, RSA Data Security's cryptographic toolkit BSAFE 3.0 (see Question 173) has a throughput for private-key operations of 21.6 Kbits per second with a 512-bit modulus and 7.4 Kbits per second with a 1024-bit modulus. The fastest RSA hardware [SV93] has a throughput greater than 300 Kbits per second with a 512-bit modulus, implying that it performs over 500 RSA private-key operations per second. (There is room in that hardware to execute two RSA 512-bit RSA operations in parallel [Sha95], hence the 600 Kbits/s speed reported in [SV93]. For 970-bit keys, the throughput is 185 Kbits/s.) It is expected that RSA speeds will reach 1 Mbits/second within a year or so.
By comparison, DES is much faster than RSA. In software, DES is generally at least 100 times as fast as RSA. In hardware, DES is between 1,000 and 10,000 times as fast, depending on the implementation. Implementations of RSA will probably narrow the gap a bit in coming years, as there are growing commercial markets, but DES will get faster as well.
This is going to hurt a little: You can do everything right and still screw up majorly.
Many of you read about the Infineon crypto module flaw. The story has been reported with variations of on the theme of “RSA algorithm weakness in Infineon chips”.
Fastest Rsa Key Generation Timeline
First, let’s get this right. This was not about a weakness in the RSA® algorithm, nor was it about Infineon’s implementation of the algorithm. Infineon did that part just fine.
The problem occurred in the way Infineon generated the prime numbers used as key material. They took shortcuts to produce the key material prime numbers, because without those shortcuts the generation of the primes would simply take too long.
“That’s stupid and irresponsible!” some may scream. As if it would be that simple. There are valid reasons to speed up prime number generation on embedded devices (Smartcards, TPM chips) used directly by end-users. The chips lack CPU power as their main job is to protect the key material, not to run video games.
When generating RSA keys (and, therefore, primes) on thousands of devices it had better be fast; people don’t like to wait. The crypto-aware end-user understands that key generation can take time, but many others will simply yank the smart card out of the reader because it “hangs”.
Using shortcuts in RSA implementations is a very common practice. For example, people often choose encryption exponents like 3, 17, or 65,537 because they lend themselves to much faster computation. In some implementations, these choices have proven problematic, but on the whole, they have proven to be a sound way to implement the RSA algorithm in practice. Like any cryptographic approach, these choices have to withstand the test of time.
It’s worth noting that these attacks are not obvious. It requires some fairly ingenious observations made by some incredibly smart people. Cryptography is hard. Implementing it correctly is even harder. On the surface, it seems that Infineon likely exercised good diligence and correctly implemented an approach for fast prime generation. This approach was only recently discovered to have a subtle, but critical, mathematical flaw. This is not the typical “I invented my own crypto!” story.
Fastest Rsa Key Generation Time Of Life
Yet here Infineon sits, after selling truckloads of chips with the faulty key generator.
Two things stand out to me:
- The problem is substantial as the attack could be performed by anyone with few resources, or know-how. All you need is the public key and some CPU time. Both are pretty easy to come by.
- The smart card chip plus firmware was certified to Common Criteria EAL 5+, which is a pretty extensive certification. The TPM module certified EAL 4+, which is also pretty high.
Looking at the Common Criteria website, which lists all certified products, there are several Infineon chips and libraries included. While I am not sure this is the exact chip in question it serves well as an example for the points I make below. This Infineon chip was certified to EAL5+.
Some may be shocked that this flaw could slip thru such an extensive certification process. Understanding how certifications actually work removes the shock factor.
Certifications, such as Common Criteria EAL, do not certify that the product doing function X is secure in every possible way. It certifies that function X was implemented in a secure way. These are not the same thing.
Additionally, there can be confusion as to what is certified. Certifications are about the “Target of Evaluation” (ToE), which describes what will be certified. Everything not mentioned in the ToE is not included in the evaluation, even if it is closely related. In the example evaluation linked to above, the random number generator is included in the ToE, but there is no mention of the prime number check. It may be somewhere in the ToE, but remember, the certification is about the secure implementation of function X and not about the security of X itself.
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In other words: the security of the “fast prime” functionality – even if included in the –ToE was never part of the evaluation.
This should serve as a stark reminder that certification stamps, such as Common Criteria, FIPS and so on, do not mean the product is secure. Nor does it mean that installation of a certified product equals a secure deployment. It only means what the ToE explicitly states.
Infineon released a patch to end users via device and OS manufacturers for the TPMs. The smart cards need to be physically replaced. As with most patches, this patch won’t be installed immediately on every system and it takes time to reissue affected smart cards leaving a large attack window. Combined with the fact that some use cases for the affected smart cards involve digital signatures and that timestamping a legally binding digital signature is not yet mandatory, this could get interesting. One scenario could have an attacker deriving the private key sometime in the future to create valid signatures on documents (e.g. contracts). Due to the lack of timestamps it cannot be proven they were created after the compromised certificate had been revoked.
PKI (Public Key Infrastructure) and cryptography aren’t easy. Hindsight allows us to say where Infineon went wrong, but at the time the decisions were made things simply looked OK. Most of us would have made the same decision in that position.
Any cryptographic algorithm must withstand the test of time. As the oldest public-key cryptosystem, the RSA algorithm has withstood that test for the better part of 40 years. One should, therefore, continue to have confidence in it knowing that it has been thoroughly examined for so long, and continues to be used. Newer algorithms, as promising as they may seem, can be more risky since they may contain issues yet to be unearthed.