RFR 8181594: Efficient and constant-time modular arithmetic

Adam Petcher adam.petcher at oracle.com
Mon Feb 26 18:39:52 UTC 2018

Thanks for the initial feedback. Here is the latest webrev:


See inline below.

On 2/23/2018 12:46 PM, Xuelei Fan wrote:

> ArrayUtil.java:
> ===============
> I'm not very sure how widely this utilities will be used in the 
> future. Looks like only BigIntegerModuloP uses this classes.  I may 
> prefer to define private methods for byte array swap in 
> BigIntegerModuloP.

It is also used by XDHPublicKeyImpl (in the XDH code review). XDH public 
keys are represented as BigInteger, and I use the array reverse method 
to convert encoded keys to BigInteger.

> BigIntegerModuloP.java:
> =======================
> As this is a class for testing or ptototype purpose,  it might not be 
> a part of JDK products, like JRE.  Would you mind move it to a test 
> package if you want to keep it?

Good idea. I moved it into the same directory as TestIntegerModuloP.java.

> IntegerModuloP, IntegerModuloP_Base and MutableIntegerModuloP
> =============================================================
> In the security package/context, it may make sense to use 
> "IntegerModulo" for the referring to "integers modulo a prime value". 

In ECC, we also have curves over binary fields, and some methods in 
these interfaces/classes will not work correctly if the number of field 
elements is not prime (e.g. Fermat's little theorem is used to calculate 
a multiplicative inverse). So I would prefer that the names of these 
interfaces contain some string (e.g. "Prime" or "P") to make it clear 
that they only represent prime fields.

> The class name of "IntegerModuloP_Base" is not a general Java coding 
> style.  I may prefer a little bit name changes like:
>      IntegerModuloP_Base   -> IntegerModulo
>      IntegerModuloP        -> ImmutableIntegerModulo
>      MutableIntegerModuloP -> MutableIntegerModulo
>      IntegerFieldModuloP   -> IntegerModuloField (?)

I'm not fond of the current names, either. My goal with the naming was 
to indicate that "IntegerModuloP_Base" shouldn't be used in most cases, 
because you don't know whether it is mutable or not. So the typical 
interface to use is the immutable one, and the mutable one should only 
be used when mutability is necessary. Your suggested naming is more 
conventional, but I think it loses this aspect of my naming system.

It's only an internal API, so I'm not too picky about the names, and I 
have no problem changing it as you suggest. But I'd like to see if 
anyone else has any suggestions about the names, first.

> MutableIntegerModuloP.java
> ==========================
> void conditionalSwapWith(MutableIntegerModuloP b, int swap);
> As the 'swap' parameter can only be 0 or 1, could it be a boolean 
> parameter?

I couldn't come up with a way to implement this without branching when 
the swap parameter is boolean. See IntegerPolynomial.conditionalSwap to 
see how this is implemented in arithmetic with an int swap argument. If 
you (or anyone) can think of a way to do this with boolean, let me know.

I added a sentence to the comment above conditionalSwapWith that 
describes why it is an int instead of a boolean.

> Except the conditionalSwapWith() method, I did not get the points why 
> we need a mutable version.  Would you please have more description of 
> this requirement?

The comment above the class definition has this sentence:

"This interface can be used to improve performance and avoid the 
allocation of a large number of temporary objects."

Do you need more information than this in the comments? The performance 
motivation is so that a.add(b).multiply(c)... can be done without 
allocating a new buffer for each operation. For example, without mutable 
field elements, an X25519 point multiplication would allocate around 
4,300 temporary arrays totaling 350,000 bytes. If I remember correctly, 
switching the X25519 implementation to mutable field elements reduced 
the point multiplication time by about half.

> IntegerModuloP_Base.java
> ========================
> default byte[] addModPowerTwo(IntegerModuloP_Base b, int len)
> void addModPowerTwo(IntegerModuloP_Base b, byte[] result);
> For the first sign of the method names, I thought it is to calculate 
> as "(this + b) ^ 2 mod m". 

To be precise, it calculates "((this % p) + (b % p)) % 2^m" (where p is 
the prime that defines the field, and m is the desired length, in bits). 
Note that the addition here is normal integer addition (not addition in 

This operation is not used in XDH, but it is used in Poly1305 to add the 
AES encryption of a nonce to a field element. So you can get more 
information about this operation by reading the Poly1305 paper/RFC.

> Besides, what's the benefits of the two methods?  Could we just use:
>       this.add(b).asByteArray()

No, because that would calculate "((this + b) mod p) mod 2^m". The value 
of (this + b) can be larger than p, so this would not produce the 
desired result.

> I guess, but not very sure, it is for constant time calculation. If 
> the function is required, could it be renamed as:
>       // the result is inside of the size range
>       IntegerModuloP addModSize(IntegerModuloP_Base b, int size)
> Or
>       // the result is wrapped if outside of the size range
>       IntegerModuloP addOnWrap(IntegerModuloP_Base b, int size)
> and the use may look like:
>       this.addModSize(b, size).asByteArray()

Any attempt to perform the addition in IntegerModuloP and then pull out 
the byte array will not work. This class can only represent field 
elements, so the sum would be in the field, which is not what we want.

> Will review the rest when I understand more about the interfaces design.
> Thanks,
> Xuelei
> On 1/30/2018 8:52 AM, Adam Petcher wrote:
>> +core-libs-dev
>> On 1/26/2018 4:06 PM, Adam Petcher wrote:
>>> JBS: https://bugs.openjdk.java.net/browse/JDK-8181594
>>> Webrev: http://cr.openjdk.java.net/~apetcher/8181594/webrev.00/
>>> This is a code review for the field arithmetic that will be used in 
>>> implementations of X25519/X448 key agreement, the Poly1305 
>>> authenticator, and EdDSA signatures. I believe that the library has 
>>> all the features necessary for X25519/X448 and Poly1305, and I 
>>> expect at most a couple of minor enhancements will be required to 
>>> support EdDSA. There is no public API for this library, so we can 
>>> change it in the future to suit the needs of new algorithms without 
>>> breaking compatibility with external code. Still, I made an attempt 
>>> to clearly structure and document the (internal) API, and I want to 
>>> make sure it is understandable and easy to use.
>>> This is not a general-purpose modular arithmetic library. It will 
>>> only work well in circumstances where the sequence of operations is 
>>> restricted, and where the prime that defines the field has some 
>>> useful structure. Moreover, each new field will require some 
>>> field-specific code that takes into account the structure of the 
>>> prime and the way the field is used in the application. The initial 
>>> implementation includes a field for Poly1305 and the fields for 
>>> X25519/X448 which should also work for EdDSA.
>>> The benefits of using this library are that it is much more 
>>> efficient than using similar operations in BigInteger. Also, many 
>>> operations are branch-free, making them suitable for use in a 
>>> side-channel resistant implementation that does not branch on secrets.
>>> To provide some context, I have attached a code snippet describing 
>>> how this library can be used. The snippet is the constant-time 
>>> Montgomery ladder from my X25519/X448 implementation, which I expect 
>>> to be out for review soon. X25519/X448 only uses standard arithmetic 
>>> operations, and the more unusual features (e.g. add modulo a power 
>>> of 2) are needed by Poly1305.
>>> The field arithmetic (for all fields) is implemented using a 32-bit 
>>> representation similar to the one described in the Ed448 paper[1] 
>>> (in the "Implementation on 32-bit platforms" section). Though my 
>>> implementation uses signed limbs, and grade-school multiplication 
>>> instead of Karatsuba. The argument for correctness is essentially 
>>> the same for all three fields: the magnitude of each 64-bit limb is 
>>> at most 2^(k-1) after reduction, except for the last limb which may 
>>> have a magnitude of up to 2^k. The values of k are between 26 to 28 
>>> (depending on the field), and we can calculate that the maximum 
>>> magnitude for any limb during an add-multiply-carry-reduce sequence 
>>> is always less than 2^63. Therefore, no overflow occurs and all 
>>> operations are correct.
>>> Process note: this enhancement is part of JEP 324 (Key Agreement 
>>> with Curve25519 and Curve448). When this code review is complete, 
>>> nothing will happen until all other work for this JEP is complete, 
>>> and the JEP is accepted as part of some release. This means that 
>>> this code will be pushed to the repo along with the X25519/X448 code 
>>> that uses it.
>>> [1] https://eprint.iacr.org/2015/625.pdf

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