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{{Short description|Type of functions designed for being unsolvable by root-finding algorithms}}
A '''cryptographically secure pseudorandom number generator''' ('''CSPRNG''') or '''cryptographic pseudorandom number generator''' ('''CPRNG''') is a [[pseudorandom number generator]] (PRNG) with properties that make it suitable for use in [[cryptography]]. It is also referred to as a '''cryptographic random number generator''' ('''CRNG''').
== Background ==
{{Unreferenced section|date=June 2024}}
Most [[cryptography|cryptographic applications]] require [[random]] numbers, for example:
* [[key generation]]
* [[initialization vector]]s
* [[cryptographic nonce|nonces]]
* [[salt (cryptography)|salts]] in certain signature schemes, including [[ECDSA]] and [[PKCS 1|RSASSA-PSS]]
* [[tokenization (data security)|token generation]]
The "quality" of the randomness required for these applications varies. For example, creating a [[cryptographic nonce|nonce]] in some [[cryptographic protocol|protocols]] needs only uniqueness. On the other hand, the generation of a master [[key (cryptography)|key]] requires a higher quality, such as more [[entropy (computing)|entropy]]. And in the case of [[one-time pad]]s, the [[information theory|information-theoretic]] guarantee of perfect secrecy only holds if the key material comes from a true random source with high entropy, and thus just any kind of pseudorandom number generator is insufficient.▼
▲On the other hand, the generation of a master [[key (cryptography)|key]] requires a higher quality, such as more [[entropy (computing)|entropy]]. And in the case of [[one-time pad]]s, the [[information theory|information-theoretic]] guarantee of perfect secrecy only holds if the key material comes from a true random source with high entropy, and thus any kind of pseudorandom number generator is insufficient.
Ideally, the generation of random numbers in CSPRNGs uses entropy obtained from a high-quality source, generally the operating system's randomness [[API]]. However, unexpected correlations have been found in several such ostensibly independent processes. From an information-theoretic point of view, the amount of randomness, the entropy that can be generated, is equal to the entropy provided by the system. But sometimes, in practical situations,
==Requirements==
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# They pass statistical [[randomness tests]]:
#* Every CSPRNG should satisfy the [[next-bit test]]. That is, given the first <var>k</var> bits of a random sequence, there is no [[polynomial-time]] algorithm that can predict the (<var>k</var>+1)th bit with probability of success non-negligibly better than 50%.<ref name="katz">{{cite book |last1=Katz|first1=Jonathan|last2=Lindell|first2=Yehuda|date=2008|title=Introduction to Modern Cryptography|url=https://archive.org/details/Introduction_to_Modern_Cryptography|publisher=CRC press|page=[https://archive.org/details/Introduction_to_Modern_Cryptography/page/n88 70]|isbn= 978-1584885511}}</ref> [[Andrew Yao]] proved in 1982 that a generator passing the next-bit test will pass all other polynomial-time statistical tests for randomness.<ref name="yao82">[[Andrew Chi-Chih Yao]]. [https://www.di.ens.fr/users/phan/secuproofs/yao82.pdf Theory and applications of trapdoor functions]. In Proceedings of the 23rd IEEE Symposium on Foundations of Computer Science, 1982.</ref>
# They hold up well under serious attack, even when part of their initial or running state becomes available to an attacker:
#* Every CSPRNG should withstand "state compromise extension attacks".<ref name=Kelsey>{{cite book | last1=Kelsey | first1=John | last2=Schneier | first2=Bruce | last3=Wagner | first3=David | last4=Hall | first4=Chris | title=Fast Software Encryption | chapter=Cryptanalytic Attacks on Pseudorandom Number Generators | publisher=Springer Berlin Heidelberg | publication-place=Berlin, Heidelberg | year=1998 | isbn=978-3-540-64265-7 | issn=0302-9743 | doi=10.1007/3-540-69710-1_12 | url = https://www.schneier.com/wp-content/uploads/2017/10/paper-prngs.pdf }}</ref>{{rp|4}} In the event that part or all of its state has been revealed (or guessed correctly), it should be impossible to reconstruct the stream of random numbers prior to the revelation. Additionally, if there is an entropy input while running, it should be infeasible to use knowledge of the input's state to predict future conditions of the CSPRNG state.
For instance, if the PRNG under consideration produces output by computing bits of [[pi
Most PRNGs are not suitable for use as CSPRNGs and will fail on both counts. First, while most PRNGs' outputs appear random to assorted statistical tests, they do not resist determined reverse engineering. Specialized statistical tests may be found specially tuned to such a PRNG that shows the random numbers not to be truly random. Second, for most PRNGs, when their state has been revealed, all past random numbers can be retrodicted, allowing an attacker to read all past messages, as well as future ones.
CSPRNGs are designed explicitly to resist this type of [[cryptanalysis]].
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{{main|Randomness extractor}}
Santha and Vazirani proved that several bit streams with weak randomness can be combined to produce a higher-quality, quasi-random bit stream.<ref name=santha-vazirani>
{{cite conference
| author = Miklos Santha, Umesh V. Vazirani
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==Designs==
CSPRNG designs are divided into two classes:
#
#
===Designs based on cryptographic primitives===
* A secure [[block cipher]] can be converted into a CSPRNG by running it in [[Block cipher modes of operation|counter mode]] using, for example, a special construct that the [[NIST]] in SP 800-90A calls [[
**
{{cite web |first1=George |last1=Cox |first2=Charles |last2=Dike |first3=DJ |last3=Johnston |url=http://www.hotchips.org/wp-content/uploads/hc_archives/hc23/HC23.18.2-security/HC23.18.210-Random-Numbers-Cox-Intel-e.pdf |title=Intel's Digital Random Number Generator (DRNG) |date=2011}}</ref>
** The NIST CTR_DRBG scheme erases the key ''after'' the requested randomness is output by running additional cycles. This is wasteful from a performance perspective, but does not immediately cause issues with forward secrecy. However, realizing the performance implications, the NIST recommends an "extended AES-CTR-DRBG interface" for its [[NIST Post-Quantum Cryptography Standardization|Post-Quantum Cryptography Project]] submissions. This interface allows multiple sets of randomness to be generated without intervening erasure, only erasing when the user explicitly signals the end of requests. As a result, the key could remain in memory for an extended time if the "extended interface" is misused. Newer "fast-key-erasure" RNGs erase the key with randomness as soon as randomness is requested.<ref name="fke">{{cite web |last1=Bernstein |first1=Daniel J. |title=2017.07.23: Fast-key-erasure random-number generators: An effort to clean up several messes simultaneously. #rng #forwardsecrecy #urandom #cascade #hmac #rekeying #proofs |url=https://blog.cr.yp.to/20170723-random.html}}</ref>
* A stream cipher can be converted into a CSPRNG. This has been done with RC4, [[ISAAC (cipher)|ISAAC]], and [[ChaCha20]], to name a few.
* A cryptographically secure [[cryptographic hash function|hash]] might also be a base of a good CSPRNG, using, for example, a construct that NIST calls [[
* An [[HMAC]] primitive can be used as a base of a CSPRNG, for example, as part of the construct that NIST calls [[
===Number-theoretic designs===
* The [[Blum Blum Shub]] algorithm has a security proof based on the difficulty of the [[quadratic residuosity problem]]. Since the only known way to solve that problem is to factor the modulus, it is generally regarded that the difficulty of [[integer factorization]] provides a conditional security proof for the Blum Blum Shub algorithm. However the algorithm is very inefficient and therefore impractical unless extreme security is needed.
* The [[Blum–Micali algorithm]] has a security proof based on the difficulty of the [[discrete logarithm problem]] but is also very inefficient.
* Daniel Brown of [[Certicom]]
===Practical schemes===
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* Implementations of [[/dev/random]] in Unix-like systems.
** [[Yarrow algorithm|Yarrow]], which attempts to evaluate the entropic quality of its seeding inputs, and uses SHA-1 and 3DES internally. Yarrow was used in [[macOS]] and other Apple OS' up until about December 2019, after which it switched to Fortuna.
** [[Fortuna (PRNG)|Fortuna]], the successor to Yarrow, which does not attempt to evaluate the entropic quality of its inputs; it uses SHA-256 and "any good block cipher". Fortuna is used in FreeBSD. Apple changed to Fortuna for most or all Apple
** The Linux kernel CSPRNG, which uses ChaCha20 to generate data,<ref>{{cite web |publisher=Github |url=https://github.com/torvalds/linux/blob/e192be9d9a30555aae2ca1dc3aad37cba484cd4a/drivers/char/random.c |title=Github commit of random.c |date=July 2, 2016}}</ref> and [[BLAKE2s]] to ingest entropy.<ref>{{Cite web |title=Linux 5.17 Random Number Generator Seeing Speed-Ups, Switching From SHA1 To BLAKE2s - Phoronix |url=https://www.phoronix.com/news/Linux-5.17-RNG |website=www.phoronix.com}}</ref>
* ''[[arc4random]]'', a CSPRNG in Unix-like systems that seeds from {{tt|/dev/random}}. It originally is based on [[RC4]], but all main implementations now use [[ChaCha20]].<ref>{{cite web |publisher=CVS |url=http://cvsweb.openbsd.org/cgi-bin/cvsweb/src/lib/libc/crypt/arc4random.c?rev=1.25&content-type=text/x-cvsweb-markup |title=CVS log of arc4random.c |date=October 1, 2013}}</ref><ref>{{cite web |publisher=CVS |url=http://cvsweb.netbsd.org/bsdweb.cgi/src/lib/libc/gen/arc4random.c?rev=1.26&content-type=text/x-cvsweb-markup&only_with_tag=MAIN |title=CVS log of arc4random.c |date=November 16, 2014}}</ref>
* ''[[CryptGenRandom]]'', part of [[Microsoft]]'s [[Cryptographic Application Programming Interface|CryptoAPI]], offered on Windows. Different versions of Windows use different implementations.
* [[American National Standards Institute|ANSI]] X9.17 standard (''Financial Institution Key Management (wholesale)''), which has been adopted as a [[Federal Information Processing Standard|FIPS]] standard as well. It takes as input a [[Triple DES|TDEA]] ([[Triple DES#Keying options|keying option 2]]) key bundle ''k'' and (the initial value of) a 64-bit [[random seed]] ''s''.<ref>{{cite book |url=http://www.cacr.math.uwaterloo.ca/hac |title=Handbook of Applied Cryptography |first1=Alfred |last1=Menezes |first2=Paul |last2=van Oorschot |first3=Scott |last3=Vanstone |author-link1=Alfred Menezes |author-link2=Paul van Oorschot |author-link3=Scott Vanstone |publisher=CRC Press |year=1996 |chapter=Chapter 5: Pseudorandom Bits and Sequences |chapter-url=http://www.cacr.math.uwaterloo.ca/hac/about/chap5.pdf}}</ref> Each time a random number is required, it executes the following steps:{{
|1=
|2=
|3=
|4=
{{cite book
| first1=Adam |last1=Young |first2=Moti |last2=Yung
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| url = http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0764549758.html
| isbn = 978-0-7645-4975-5
}}</ref> If the key ''k'' is leaked, the entire X9.17 stream can be predicted; this weakness is cited as a reason for creating Yarrow.<ref name="report1999">{{cite book |last1=Kelsey |first1=John |last2=Schneier |first2=Bruce |last3=Ferguson |first3=Niels |
All these above-mentioned schemes, save for X9.17, also mix the state of a CSPRNG with an additional source of entropy. They are therefore not "pure" pseudorandom number generators, in the sense that the output is not completely determined by their initial state. This addition aims to prevent attacks even if the initial state is compromised.{{efn|The use of entropy-mixing after CSPRNG initialization has been question by [[Daniel J. Bernstein]].<ref name=":1">{{cite web |author=Daniel J. Bernstein |author-link=Daniel J. Bernstein |date=2014-02-05 |title=cr.yp.to: 2014.02.05: Entropy Attacks! |url=http://blog.cr.yp.to/20140205-entropy.html |quote=Is there any serious argument that adding new entropy all the time is a good thing? The Linux /dev/urandom manual page claims that without new entropy the user is "theoretically vulnerable to a cryptographic attack", but (as I've mentioned in various venues) this is a ludicrous argument}}</ref>}}
==Standards==
Several CSPRNGs have been standardized. For example
* FIPS 186-4<ref>{{Cite web|url=http://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.186-4.pdf|title=FIPS 186-4}}</ref>
* [[NIST SP 800-90A]]
{{block indent|This withdrawn standard has four PRNGs. Two of them are uncontroversial and proven: CSPRNGs named Hash_DRBG<ref>{{Cite web| url=https://eprint.iacr.org/2007/345.pdf | first=Wilson | last=Kan | title=Analysis of Underlying Assumptions in NIST DRBGs | date=September 4, 2007 | access-date=November 19, 2016 }}</ref> and HMAC_DRBG.<ref>{{Cite web | url = https://www.cs.cmu.edu/~kqy/resources/thesis.pdf | title = The Notorious PRG: Formal verification of the HMAC-DRBG pseudorandom number generator | first = Katherine Qinru | last = Ye | date = April 2016 | access-date=November 19, 2016 }}</ref>
▲--><p>When the maximum number of bits output from this PRNG is equal to the 2<sup>blocksize</sup>, the resulting output delivers the mathematically expected security level that the key size would be expected to generate, but the output is shown to not be indistinguishable from a true random number generator.<ref name="CTR_DRBG"/> When the maximum number of bits output from this PRNG is less than it, the expected security level is delivered and the output appears to be indistinguishable from a true random number generator.<ref name="CTR_DRBG"/></p><!--
▲--><p>It is noted in the next revision that claimed [[security strength]] for CTR_DRBG depends on limiting the total number of generate requests and the bits provided per generate request.</p><!--
}}
* NIST SP 800-90A Rev.1
▲--><p>The fourth and final PRNG in this standard is named [[Dual EC DRBG]]. It has been shown to not be cryptographically secure and is believed to have a [[Kleptography|kleptographic]] NSA backdoor.<ref>{{Cite news| url=http://bits.blogs.nytimes.com/2013/09/10/government-announces-steps-to-restore-confidence-on-encryption-standards/| first=Nicole | last=Perlroth | newspaper=The New York Times | title=Government Announces Steps to Restore Confidence on Encryption Standards | date= September 10, 2013 | access-date = November 19, 2016 | url-access=limited}}</ref></p>
* ANSI X9.17-1985 Appendix C
* ANSI X9.31-1998 Appendix A.2.4
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There are also standards for statistical testing of new CSPRNG designs:
* ''A Statistical Test Suite for Random and Pseudorandom Number Generators'', NIST Special Publication 800-22.<ref>{{Cite journal |
==Security flaws==▼
==NSA kleptographic backdoor in the Dual_EC_DRBG PRNG==▼
▲===NSA kleptographic backdoor in the Dual_EC_DRBG PRNG===
{{main|Dual_EC_DRBG}}
''[[The Guardian]]'' and ''[[The New York Times]]''
▲==Security flaws==
===DUHK attack===
On October 23, 2017, [[Shaanan Cohney]], [[Matthew D. Green|Matthew Green]], and [[Nadia Heninger]], [[cryptographer]]s at
===Japanese PURPLE cipher machine===
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{{DEFAULTSORT:Cryptographically Secure Pseudorandom Number Generator}}
[[Category:Cryptographic algorithms]]
[[Category:Cryptographically secure pseudorandom number generators]]
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